https://www.math.wisc.edu/wiki/api.php?action=feedcontributions&user=Valko&feedformat=atom UW-Math Wiki - User contributions [en] 2019-11-17T14:39:56Z User contributions MediaWiki 1.30.1 https://www.math.wisc.edu/wiki/index.php?title=Problem_Solver%27s_Toolbox&diff=18232 Problem Solver's Toolbox 2019-10-22T18:05:09Z <p>Valko: /* General ideas */</p> <hr /> <div>The goal of this page is to collect simple problem solving strategies and tools. We hope that students interested in the Wisconsin Math Talent Search would find the described ideas useful. <br /> This page and the discussed topics can be used as a starting point for future exploration.<br /> <br /> <br /> == General ideas ==<br /> <br /> <br /> There is no universal recipe for math problems that would work every time, that's what makes math fun! There are however a number of general strategies that could be useful in most cases, here is a short list of them. (Many of these ideas were popularized by the Hungarian born mathematician George Pólya in his book [https://en.wikipedia.org/wiki/How_to_Solve_It How to Solve It].)<br /> * Make sure that you understand the problem. <br /> * If possible, draw a figure. <br /> * Can you connect the problem to a problem you have solved before? <br /> * If you have to show something for all numbers (or up to a large number) then try to check the statement for small values first.<br /> * Can you solve the problem in a special case first? Can you solve a modified version of the problem first? <br /> * Is there some symmetry in the problem that you can exploit? <br /> * Is it possible to work backward? <br /> * Does it help to consider an extreme case of the problem?<br /> * Is it possible to generalize the problem? (Sometimes the generalized is easier to solve.)<br /> <br /> == Modular arithmetic ==<br /> <br /> <br /> When we have to divide two integers, they don't always divide evenly, and there is a quotient and a remainder. For example when we divide 10 by 3 we get a remainder of 1.<br /> It turns out that these remainders behave very well under addition, subtraction, and multiplication. We say two numbers are the same &quot;modulo &lt;math&gt;m&lt;/math&gt;&quot; if they have the same remainder when divided by &lt;math&gt;m&lt;/math&gt;. If &lt;math&gt;a&lt;/math&gt; and &lt;math&gt;x&lt;/math&gt; are the same modulo &lt;math&gt;m&lt;/math&gt;, and &lt;math&gt;b&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; are the same modulo &lt;math&gt;m&lt;/math&gt;, then &lt;math&gt;a+b&lt;/math&gt; and &lt;math&gt;x+y&lt;/math&gt; are the same modulo &lt;math&gt;m&lt;/math&gt;, and similarly for subtraction and multiplication. <br /> <br /> For example, 5 is the same as 1 modulo 4, and hence &lt;math&gt;5\cdot 5 \cdot 5 \cdot 5=5^4&lt;/math&gt; is the same as &lt;math&gt;1\cdot 1\cdot 1\cdot 1=1&lt;/math&gt; modulo &lt;math&gt;4&lt;/math&gt;. Same way you can show that &lt;math&gt;5^{1000}&lt;/math&gt; has a remainder of 1 when we divide it by 4.<br /> <br /> Modular arithmetic often makes calculation much simpler. For example, see [https://www.math.wisc.edu/talent/sites/default/files/Talent16-2q.pdf 2016-17 Set #2 Problem 3].<br /> <br /> See [http://artofproblemsolving.com/wiki/index.php?title=Modular_arithmetic/Introduction Art of Problem Solving's introduction to modular arithmetic] for more information.<br /> <br /> == Mathematical induction ==<br /> <br /> Suppose that you want to prove a statement for all positive integers, for example that for each positive integer &lt;math&gt;n&lt;/math&gt; the following is true: &lt;math display=&quot;block&quot;&gt;1\cdot 2+2\cdot 3+3\cdot 4+\cdots+n\cdot (n+1)=\frac{n(n+1)(n+2)}{3}.\qquad\qquad(*) &lt;/math&gt;<br /> <br /> Mathematical induction provides a tool for doing this. You need to show the following two things:<br /> # (Base case) The statement is true for &lt;math&gt;n=1&lt;/math&gt;. <br /> # (Induction step) If the statement is true for &lt;math&gt;n&lt;/math&gt; then it must be true for &lt;math&gt;n+1&lt;/math&gt; as well.<br /> <br /> If we can show both of these parts, then it follows that the statement is true for all positive integer &lt;math&gt;n&lt;/math&gt;. Why? The first part (the base case) shows that the statement is true for &lt;math&gt;n=1&lt;/math&gt;. But then by the second part (the induction step) the statement must be true for &lt;math&gt;n=2&lt;/math&gt; as well. Using the second part again and again we see that the statement is true for &lt;math&gt;n=3, 4, 5, \cdots&lt;/math&gt; and repeating this sufficiently times we can prove that the statement is true for any fixed value of &lt;math&gt;n&lt;/math&gt;. <br /> <br /> Often the idea of induction is demonstrated as a version of Domino effect'. Imagine that you have an infinite row of dominos numbered with the positive integers, where if &lt;math&gt;n&lt;/math&gt;th domino falls then the next one will fall as well (this is the induction step). If we make the first domino fall (this is the base case) then eventually all other dominos will fall as well. <br /> <br /> * Try to use induction to show the identity &lt;math&gt;(*)&lt;/math&gt; above for all positive integer &lt;math&gt;n&lt;/math&gt;.<br /> * You can also use induction to show a statement for all integers &lt;math&gt;n\ge 5&lt;/math&gt;. Then for your base case you have to show that the statement is true for &lt;math&gt;n=5&lt;/math&gt;. (The induction step is the same.)<br /> <br /> See this page from [https://www.mathsisfun.com/algebra/mathematical-induction.html Math Is Fun] for some simple applications of induction.<br /> <br /> == Proof by contradiction ==<br /> <br /> This is a commonly used problem solving method. Suppose that you have to prove a certain statement. Now pretend that the statement is not true and try to derive (as a consequence) a false statement. The found false statement shows that your assumption about the original statement was incorrect: thus the original statement must be true. <br /> <br /> Here is a simple example: we will prove that the product of three consecutive positive integers cannot be a prime number. Assume the opposite: that means that there is a positive integer &lt;math&gt;n&lt;/math&gt; so that &lt;math&gt;n(n+1)(n+2)&lt;/math&gt; is a prime. But among three consecutive integers we will always have a multiple of 2, and also a multiple of 3. Thus the product of the three numbers must be divisible by both 2 and 3, and hence &lt;math&gt;n(n+1)(n+2)&lt;/math&gt; cannot be a prime. This contradicts our assumption that &lt;math&gt;n(n+1)(n+2)&lt;/math&gt; is a prime, which shows that our assumption had to be incorrect. <br /> <br /> Proof by contradiction can be used for example in [https://www.math.wisc.edu/talent/sites/default/files/Talent16-1q.pdf 2016-17 Set #1 Problem 4].<br /> <br /> == Pigeonhole Principle ==<br /> <br /> The Pigeonhole Principle is one of the simplest tools in mathematics, but it can be very powerful. Suppose that &lt;math&gt;n&lt;m&lt;/math&gt; are positive integers, and we have &lt;math&gt;m&lt;/math&gt; objects and &lt;math&gt;n&lt;/math&gt; boxes. The Pigeonhole Principle states that If we place each of the &lt;math&gt;m&lt;/math&gt; objects into one of the &lt;math&gt;n&lt;/math&gt; boxes then there must be at least one box with at least two objects in it. <br /> The statement can be proved by contradiction: if we can find an arrangement of objects so that each box has less than two objects in it, then each box would contain at most one object, and hence we had at most &lt;math&gt;n&lt;/math&gt; objects all together. This is a contradiction, which means that the original statement must be correct. <br /> <br /> The Pigeonhole Principle is often used in the following, more general form. Suppose that &lt;math&gt;n, m, k&lt;/math&gt; are positive integers with &lt;math&gt;n k&lt; m &lt;/math&gt;. If we place each of &lt;math&gt;m&lt;/math&gt; objects into one of &lt;math&gt;n&lt;/math&gt; boxes then there must be at least one box with at least &lt;math&gt;k+1&lt;/math&gt; objects in it. Try to prove this version by contradiction.<br /> <br /> Here is a simple application: if we roll a die 13 times then there must be a number that appears at least three times. Here each die roll correspond to an object, each of the 6 possible outcomes correspond to a possible box. Since &lt;math&gt;2\cdot 6&lt;13&lt;/math&gt;, we must have a box with at least &lt;math&gt;2+1=3&lt;/math&gt; objects. In other words: there will be number that appears at least three times. <br /> <br /> Pigeonhole Principle can be used for example in [https://www.math.wisc.edu/talent/sites/default/files/T14-1q_0_0.pdf 2014-15 Set #1 Problem 4].<br /> <br /> == Angles in the circle ==<br /> <br /> The following theorems are often useful when working with geometry problems. [[File:Thales_thm.jpg|250px|thumb|right|An illustration of Thales' Theorem. O is the center of the circle.]] <br /> <br /> '''Thales' Theorem''' <br /> <br /> Suppose that the distinct points &lt;math&gt;A, B, C&lt;/math&gt; are all on a circle, and &lt;math&gt;AB&lt;/math&gt; is a diameter of of the circle. Then the angle &lt;math&gt;ACB&lt;/math&gt; is &lt;math&gt;90^{\text{o}}&lt;/math&gt;. In other words: the triangle &lt;math&gt;\triangle ABC&lt;/math&gt; is a right triangle with hypotenuse &lt;math&gt;AB&lt;/math&gt;. <br /> <br /> The theorem can be proved with a little bit of angle-chasing'. Denote the center of the circle by &lt;math&gt;O&lt;/math&gt;. Then &lt;math&gt;AO, BO, CO&lt;/math&gt; are all radii of the circle, so they have the same length. Thus &lt;math&gt;\triangle AOC&lt;/math&gt; and &lt;math&gt;\triangle BOC&lt;/math&gt; are both isosceles triangles. Now try labeling the various angles in the picture and you should quickly arrive to a proof. (You can find the worked out proof at the [https://en.wikipedia.org/wiki/Thales%27_theorem wiki page] of the theorem, but it is more fun if you figure it out on your own!)<br /> <br /> The converse of Thales's theorem states that if &lt;math&gt;\triangle ABC&lt;/math&gt; is a right triangle with hypotenuse &lt;math&gt;AB&lt;/math&gt; then we can draw a circle with a center that is the midpoint of &lt;math&gt;AB&lt;/math&gt; that passes through &lt;math&gt;A, B, C&lt;/math&gt;.<br /> <br /> <br /> The Inscribed Angle Theorem below is a generalization of Thales' Theorem. <br /> <br /> <br /> '''The Inscribed Angle Theorem'''<br /> <br /> Suppose that the distinct points &lt;math&gt;A, B, C&lt;/math&gt; are all on a circle and let &lt;math&gt;O&lt;/math&gt; be the center of the circle. Then depending on the position of these points we have the following statements:<br /> <br /> * If &lt;math&gt;O&lt;/math&gt; is on the line &lt;math&gt;AB&lt;/math&gt; then &lt;math&gt;\angle ACB=90^{\text{o}}&lt;/math&gt;. (This is just Thales' theorem again.)<br /> * If &lt;math&gt;O&lt;/math&gt; and &lt;math&gt;C&lt;/math&gt; are both on the same side of the line &lt;math&gt;AB&lt;/math&gt; then the inscribed angle &lt;math&gt;\angle ACB&lt;/math&gt; is half of &lt;math&gt;360^{\text{o}}&lt;/math&gt; minus the central angle &lt;math&gt;\angle AOB&lt;/math&gt;: <br /> &lt;math display=&quot;block&quot;&gt; 2 \angle ACB= \angle AOB.&lt;/math&gt;<br /> * If &lt;math&gt;O&lt;/math&gt; and &lt;math&gt;C&lt;/math&gt; are on the opposite sides of the line &lt;math&gt;AB&lt;/math&gt; then the inscribed angle &lt;math&gt;\angle ACB&lt;/math&gt; is half of the central angle &lt;math&gt;\angle AOB&lt;/math&gt;: <br /> &lt;math display=&quot;block&quot;&gt; 2 \angle ACB= 360^{\text{o}}-\angle AOB.&lt;/math&gt;<br /> <br /> If we measure the central angle &lt;math&gt;\angle AOB&lt;/math&gt; the right way' then we don't need to separate the three cases. In the first case the central angle is just &lt;math&gt;180^{\text{o}}&lt;/math&gt;, and the inscribed angle is exactly the half of that. In the third case if we define the central angle to be &lt;math&gt;360^{\text{o}}-\angle AOB&lt;/math&gt; then again we get that the inscribed angle is half of the central angle. <br /> <br /> <br /> The theorem can be proved with angle-chasing, using the same idea that was described for Thales' theorem. See the [https://en.wikipedia.org/wiki/Inscribed_angle wiki page] for the proof (but first try to do it on your own!).<br /> <br /> <br /> '''Applications to cyclic quadrilaterals'''<br /> <br /> The following statements (and their converses) are useful applications of the Inscribed Angle theorem.<br /> <br /> <br /> 1. Suppose that the points &lt;math&gt;A, B, C, D&lt;/math&gt; form a cyclic quadrilateral, this means that we can draw a circle going through the four points. &lt;math&gt;AB&lt;/math&gt; divides the circle into two arcs. If the points &lt;math&gt;C&lt;/math&gt; and &lt;math&gt;D&lt;/math&gt; are in the same arc (meaning that they are on the same side of &lt;math&gt;AB&lt;/math&gt;) then <br /> &lt;math display=&quot;block&quot;&gt; \angle ACB= \angle ADB.&lt;/math&gt;<br /> The converse of this statement is also true: if &lt;math&gt;A, B, C, D&lt;/math&gt; are distinct points, the points &lt;math&gt;C, D&lt;/math&gt; are on the same side of the line &lt;math&gt;AB&lt;/math&gt; and &lt;math&gt;\angle ACB= \angle ADB<br /> &lt;/math&gt; then we can draw a circle around &lt;math&gt;A, B, C, D&lt;/math&gt;, in other words &lt;math&gt;ABCD&lt;/math&gt; is a cyclic quadrilateral.<br /> <br /> 2. Suppose that &lt;math&gt;ABCD&lt;/math&gt; is a cyclic quadrilateral. Then the sum of any two opposite angles is equal to &lt;math&gt;180^{\text{o}}&lt;/math&gt;. This means that <br /> &lt;math display=&quot;block&quot;&gt; \angle ABC+\angle CDA= 180^{\text{o}}, \quad \text{and}\quad \angle BCD+\angle DAB= 180^{\text{o}}. \qquad\qquad (**)&lt;/math&gt;<br /> <br /> The converse of the previous statement is also true: suppose that &lt;math&gt;ABCD&lt;/math&gt; is a quadrilateral with angles satisfying the equations &lt;math&gt;(**)&lt;/math&gt;. Then &lt;math&gt;ABCD&lt;/math&gt; is a cyclic quadrilateral: we can draw a circle that passes through the four points.<br /> <br /> The Inscribed Angle Theorem and the statements about cyclic quadrilaterals can be used for example in [https://www.math.wisc.edu/talent/sites/default/files/Talent15-4q.pdf 2015-16 Set #4 Problem 5].</div> Valko https://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=18226 Probability Seminar 2019-10-21T21:56:54Z <p>Valko: </p> <hr /> <div>__NOTOC__<br /> <br /> = Fall 2019 =<br /> <br /> &lt;b&gt;Thursdays in 901 Van Vleck Hall at 2:30 PM&lt;/b&gt;, unless otherwise noted. <br /> &lt;b&gt;We usually end for questions at 3:20 PM.&lt;/b&gt;<br /> <br /> If you would like to sign up for the email list to receive seminar announcements then please send an email to <br /> [mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br /> <br /> <br /> == September 12, 2019, [https://perso.univ-rennes1.fr/victor.kleptsyn/ Victor Kleptsyn], CNRS and University of Rennes 1 ==<br /> '''Furstenberg theorem: now with a parameter!'''<br /> <br /> The classical Furstenberg theorem describes the (almost sure) behaviour of a random product of independent matrices; their norms turn out to grow exponentially. In our joint work with A. Gorodetski, we study what happens if the random matrices depend on an additional parameter. <br /> It turns out that in this new situation, the conclusion changes. Namely, under some conditions, there almost surely exists a (random) &quot;exceptional&quot; set on parameters where the lower limit for the Lyapunov exponent vanishes.<br /> Our results are related to the Anderson localization in dimension one, providing a purely dynamical viewpoint on its proof. I will also speak about some generalizations and related open questions.<br /> <br /> == September 19, 2019, [http://math.columbia.edu/~xuanw Xuan Wu], Columbia University==<br /> <br /> '''A Gibbs resampling method for discrete log-gamma line ensemble.'''<br /> <br /> In this talk we will construct the discrete log-gamma line ensemble, which is assocaited with inverse gamma polymer model. This log-gamma line ensemble enjoys a random walk Gibbs resampling invariance that follows from the integrable nature of the inverse gamma polymer model via geometric RSK correspondance. By exploiting such resampling invariance, we show the tightness of this log-gamma line ensemble under weak noise scaling. Furthermore, a Gibbs property, as enjoyed by KPZ line ensemble, holds for all subsequential limits.<br /> <br /> == October 10, 2019, NO SEMINAR - [https://sites.math.northwestern.edu/mwp/ Midwest Probability Colloquium] ==<br /> <br /> == October 17, 2019, [https://www.usna.edu/Users/math/hottovy/index.php Scott Hottovy], USNA ==<br /> <br /> ''' Simplified dynamics for noisy systems with delays.'''<br /> <br /> Many biological and physical systems include some type of random noise with a temporal delay. For example, many sperm cells travel in a random motion where their velocity changes according to a chemical signal. This chemotaxis is transmitted through a delay in the system. That is, the sperm notices chemical gradients after a certain time has elapsed. In this case, the delay causes the sperm to aggregate around the egg. In this talk I will consider a general stochastic differential delay equation (SDDE) with state-dependent colored noises and derive its limit as the time delays and the correlation times of the noises go to zero. The analysis leads to a much simpler Stochastic Differential Equation to study. The work is motivated by an experiment involving an electrical circuit with noisy, delayed feedback. The main methods used in the proof are a theorem about convergence of solutions of stochastic differential equations by Kurtz and Protter and a maximal inequality for sums of a stationary sequence of random variables by Peligrad and Utev.<br /> <br /> == October 24, 2019, [https://math.temple.edu/~brider/ Brian Rider], Temple University ==<br /> <br /> '''A general beta crossover ensemble'''<br /> <br /> I'll describe an operator limit for a family of general beta ensembles which exhibit a double-scaling. In particular, a free parameter in the system provides for a crossover between the more well-known &quot;soft&quot; and &quot;hard&quot; edge point processes. This new limit operator takes as input the Riccati diffusion associated with the Stochastic Airy Operator. I like to suggest that this hints at a hierarchy of random operators analogous to the Painlevé hierarchy observed at the level of correlation functions for double-scaling ensembles most widely studied at beta = 2. Full disclosure: the result remains partially conjectural due to an unresolved uniqueness question, but I’ll provide lots of evidence to convince you we have the right answer. Joint work with Jose Ramírez (Univ. Costa Rica).<br /> <br /> == October 31, 2019, Vadim Gorin, UW Madison==<br /> <br /> '''Shift invariance for the six-vertex model and directed polymers.'''<br /> <br /> I will explain a recently discovered mysterious property in a variety of stochastic systems ranging from the six-vertex model and to the directed polymers, last passage percolation, Kardar-Parisi-Zhang equation, and Airy sheet. Vaguely speaking, the property says that the multi-point joint distributions are unchanged when some (but not necessarily all!) points of observations are shifted. The property leads to explicit computations for the previously inaccessible joint distributions in all these settings.<br /> <br /> == November 7, 2019, [https://people.kth.se/~tobergg/ Tomas Berggren], KTH Stockholm ==<br /> <br /> == November 14, 2019, [https://math.mit.edu/directory/profile.php?pid=2076 Benjamin Landon], MIT ==<br /> <br /> == November 21, 2019, Tung Nguyen, UW Madison ==<br /> <br /> == November 28, 2019, Thanksgiving (no seminar) ==<br /> <br /> <br /> == ==<br /> <br /> [[Past Seminars]]</div> Valko https://www.math.wisc.edu/wiki/index.php?title=Problem_Solver%27s_Toolbox&diff=18214 Problem Solver's Toolbox 2019-10-19T15:13:41Z <p>Valko: /* Mathematical induction */</p> <hr /> <div>The goal of this page is to collect simple problem solving strategies and tools. We hope that students interested in the Wisconsin Math Talent Search would find the described ideas useful. <br /> This page and the discussed topics can be used as a starting point for future exploration.<br /> <br /> <br /> == General ideas ==<br /> <br /> <br /> There is no universal recipe for math problems that would work every time, that's what makes math fun! There are however a number of general strategies that could be useful in most cases, here is a short list of them. (Many of these ideas were popularized by the Hungarian born mathematician George Pólya in his book [https://en.wikipedia.org/wiki/How_to_Solve_It How to Solve It].)<br /> * Make sure that you understand the problem. <br /> * If possible, draw a figure. <br /> * Can you connect the problem to a problem you have solved before? <br /> * If you have to show something for all numbers (or a large number) then try to check the statement for small values first.<br /> * Can you solve the problem in a special case first? Can you solve a modified version of the problem first? <br /> * Is there some symmetry in the problem that you can exploit? <br /> * Is it possible to work backward? <br /> * Does it help to consider an extreme case of the problem?<br /> * Is it possible to generalize the problem? (Sometimes the generalized is easier to solve.)<br /> <br /> == Modular arithmetic ==<br /> <br /> <br /> When we have to divide two integers, they don't always divide evenly, and there is a quotient and a remainder. For example when we divide 10 by 3 we get a remainder of 1.<br /> It turns out that these remainders behave very well under addition, subtraction, and multiplication. We say two numbers are the same &quot;modulo &lt;math&gt;m&lt;/math&gt;&quot; if they have the same remainder when divided by &lt;math&gt;m&lt;/math&gt;. If &lt;math&gt;a&lt;/math&gt; and &lt;math&gt;x&lt;/math&gt; are the same modulo &lt;math&gt;m&lt;/math&gt;, and &lt;math&gt;b&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; are the same modulo &lt;math&gt;m&lt;/math&gt;, then &lt;math&gt;a+b&lt;/math&gt; and &lt;math&gt;x+y&lt;/math&gt; are the same modulo &lt;math&gt;m&lt;/math&gt;, and similarly for subtraction and multiplication. <br /> <br /> For example, 5 is the same as 1 modulo 4, and hence &lt;math&gt;5\cdot 5 \cdot 5 \cdot 5=5^4&lt;/math&gt; is the same as &lt;math&gt;1\cdot 1\cdot 1\cdot 1=1&lt;/math&gt; modulo &lt;math&gt;4&lt;/math&gt;. Same way you can show that &lt;math&gt;5^{1000}&lt;/math&gt; has a remainder of 1 when we divide it by 4.<br /> <br /> Modular arithmetic often makes calculation much simpler. For example, see [https://www.math.wisc.edu/talent/sites/default/files/Talent16-2q.pdf 2016-17 Set #2 Problem 3].<br /> <br /> See [http://artofproblemsolving.com/wiki/index.php?title=Modular_arithmetic/Introduction Art of Problem Solving's introduction to modular arithmetic] for more information.<br /> <br /> == Mathematical induction ==<br /> <br /> Suppose that you want to prove a statement for all positive integers, for example that for each positive integer &lt;math&gt;n&lt;/math&gt; the following is true: &lt;math display=&quot;block&quot;&gt;1\cdot 2+2\cdot 3+3\cdot 4+\cdots+n\cdot (n+1)=\frac{n(n+1)(n+2)}{3}.\qquad\qquad(*) &lt;/math&gt;<br /> <br /> Mathematical induction provides a tool for doing this. You need to show the following two things:<br /> # (Base case) The statement is true for &lt;math&gt;n=1&lt;/math&gt;. <br /> # (Induction step) If the statement is true for &lt;math&gt;n&lt;/math&gt; then it must be true for &lt;math&gt;n+1&lt;/math&gt; as well.<br /> <br /> If we can show both of these parts, then it follows that the statement is true for all positive integer &lt;math&gt;n&lt;/math&gt;. Why? The first part (the base case) shows that the statement is true for &lt;math&gt;n=1&lt;/math&gt;. But then by the second part (the induction step) the statement must be true for &lt;math&gt;n=2&lt;/math&gt; as well. Using the second part again and again we see that the statement is true for &lt;math&gt;n=3, 4, 5, \cdots&lt;/math&gt; and repeating this sufficiently times we can prove that the statement is true for any fixed value of &lt;math&gt;n&lt;/math&gt;. <br /> <br /> Often the idea of induction is demonstrated as a version of Domino effect'. Imagine that you have an infinite row of dominos numbered with the positive integers, where if &lt;math&gt;n&lt;/math&gt;th domino falls then the next one will fall as well (this is the induction step). If we make the first domino fall (this is the base case) then eventually all other dominos will fall as well. <br /> <br /> * Try to use induction to show the identity &lt;math&gt;(*)&lt;/math&gt; above for all positive integer &lt;math&gt;n&lt;/math&gt;.<br /> * You can also use induction to show a statement for all integers &lt;math&gt;n\ge 5&lt;/math&gt;. Then for your base case you have to show that the statement is true for &lt;math&gt;n=5&lt;/math&gt;. (The induction step is the same.)<br /> <br /> See this page from [https://www.mathsisfun.com/algebra/mathematical-induction.html Math Is Fun] for some simple applications of induction.<br /> <br /> == Proof by contradiction ==<br /> <br /> This is a commonly used problem solving method. Suppose that you have to prove a certain statement. Now pretend that the statement is not true and try to derive (as a consequence) a false statement. The found false statement shows that your assumption about the original statement was incorrect: thus the original statement must be true. <br /> <br /> Here is a simple example: we will prove that the product of three consecutive positive integers cannot be a prime number. Assume the opposite: that means that there is a positive integer &lt;math&gt;n&lt;/math&gt; so that &lt;math&gt;n(n+1)(n+2)&lt;/math&gt; is a prime. But among three consecutive integers we will always have a multiple of 2, and also a multiple of 3. Thus the product of the three numbers must be divisible by both 2 and 3, and hence &lt;math&gt;n(n+1)(n+2)&lt;/math&gt; cannot be a prime. This contradicts our assumption that &lt;math&gt;n(n+1)(n+2)&lt;/math&gt; is a prime, which shows that our assumption had to be incorrect. <br /> <br /> Proof by contradiction can be used for example in [https://www.math.wisc.edu/talent/sites/default/files/Talent16-1q.pdf 2016-17 Set #1 Problem 4].<br /> <br /> == Pigeonhole Principle ==<br /> <br /> The Pigeonhole Principle is one of the simplest tools in mathematics, but it can be very powerful. Suppose that &lt;math&gt;n&lt;m&lt;/math&gt; are positive integers, and we have &lt;math&gt;m&lt;/math&gt; objects and &lt;math&gt;n&lt;/math&gt; boxes. The Pigeonhole Principle states that If we place each of the &lt;math&gt;m&lt;/math&gt; objects into one of the &lt;math&gt;n&lt;/math&gt; boxes then there must be at least one box with at least two objects in it. <br /> The statement can be proved by contradiction: if we can find an arrangement of objects so that each box has less than two objects in it, then each box would contain at most one object, and hence we had at most &lt;math&gt;n&lt;/math&gt; objects all together. This is a contradiction, which means that the original statement must be correct. <br /> <br /> The Pigeonhole Principle is often used in the following, more general form. Suppose that &lt;math&gt;n, m, k&lt;/math&gt; are positive integers with &lt;math&gt;n k&lt; m &lt;/math&gt;. If we place each of &lt;math&gt;m&lt;/math&gt; objects into one of &lt;math&gt;n&lt;/math&gt; boxes then there must be at least one box with at least &lt;math&gt;k+1&lt;/math&gt; objects in it. Try to prove this version by contradiction.<br /> <br /> Here is a simple application: if we roll a die 13 times then there must be a number that appears at least three times. Here each die roll correspond to an object, each of the 6 possible outcomes correspond to a possible box. Since &lt;math&gt;2\cdot 6&lt;13&lt;/math&gt;, we must have a box with at least &lt;math&gt;2+1=3&lt;/math&gt; objects. In other words: there will be number that appears at least three times. <br /> <br /> Pigeonhole Principle can be used for example in [https://www.math.wisc.edu/talent/sites/default/files/T14-1q_0_0.pdf 2014-15 Set #1 Problem 4].<br /> <br /> == Angles in the circle ==<br /> <br /> The following theorems are often useful when working with geometry problems. [[File:Thales_thm.jpg|250px|thumb|right|An illustration of Thales' Theorem. O is the center of the circle.]] <br /> <br /> '''Thales' Theorem''' <br /> <br /> Suppose that the distinct points &lt;math&gt;A, B, C&lt;/math&gt; are all on a circle, and &lt;math&gt;AB&lt;/math&gt; is a diameter of of the circle. Then the angle &lt;math&gt;ACB&lt;/math&gt; is &lt;math&gt;90^{\text{o}}&lt;/math&gt;. In other words: the triangle &lt;math&gt;\triangle ABC&lt;/math&gt; is a right triangle with hypotenuse &lt;math&gt;AB&lt;/math&gt;. <br /> <br /> The theorem can be proved with a little bit of angle-chasing'. Denote the center of the circle by &lt;math&gt;O&lt;/math&gt;. Then &lt;math&gt;AO, BO, CO&lt;/math&gt; are all radii of the circle, so they have the same length. Thus &lt;math&gt;\triangle AOC&lt;/math&gt; and &lt;math&gt;\triangle BOC&lt;/math&gt; are both isosceles triangles. Now try labeling the various angles in the picture and you should quickly arrive to a proof. (You can find the worked out proof at the [https://en.wikipedia.org/wiki/Thales%27_theorem wiki page] of the theorem, but it is more fun if you figure it out on your own!)<br /> <br /> The converse of Thales's theorem states that if &lt;math&gt;\triangle ABC&lt;/math&gt; is a right triangle with hypotenuse &lt;math&gt;AB&lt;/math&gt; then we can draw a circle with a center that is the midpoint of &lt;math&gt;AB&lt;/math&gt; that passes through &lt;math&gt;A, B, C&lt;/math&gt;.<br /> <br /> <br /> The Inscribed Angle Theorem below is a generalization of Thales' Theorem. <br /> <br /> <br /> '''The Inscribed Angle Theorem'''<br /> <br /> Suppose that the distinct points &lt;math&gt;A, B, C&lt;/math&gt; are all on a circle and let &lt;math&gt;O&lt;/math&gt; be the center of the circle. Then depending on the position of these points we have the following statements:<br /> <br /> * If &lt;math&gt;O&lt;/math&gt; is on the line &lt;math&gt;AB&lt;/math&gt; then &lt;math&gt;\angle ACB=90^{\text{o}}&lt;/math&gt;. (This is just Thales' theorem again.)<br /> * If &lt;math&gt;O&lt;/math&gt; and &lt;math&gt;C&lt;/math&gt; are both on the same side of the line &lt;math&gt;AB&lt;/math&gt; then the inscribed angle &lt;math&gt;\angle ACB&lt;/math&gt; is half of &lt;math&gt;360^{\text{o}}&lt;/math&gt; minus the central angle &lt;math&gt;\angle AOB&lt;/math&gt;: <br /> &lt;math display=&quot;block&quot;&gt; 2 \angle ACB= \angle AOB.&lt;/math&gt;<br /> * If &lt;math&gt;O&lt;/math&gt; and &lt;math&gt;C&lt;/math&gt; are on the opposite sides of the line &lt;math&gt;AB&lt;/math&gt; then the inscribed angle &lt;math&gt;\angle ACB&lt;/math&gt; is half of the central angle &lt;math&gt;\angle AOB&lt;/math&gt;: <br /> &lt;math display=&quot;block&quot;&gt; 2 \angle ACB= 360^{\text{o}}-\angle AOB.&lt;/math&gt;<br /> <br /> If we measure the central angle &lt;math&gt;\angle AOB&lt;/math&gt; the right way' then we don't need to separate the three cases. In the first case the central angle is just &lt;math&gt;180^{\text{o}}&lt;/math&gt;, and the inscribed angle is exactly the half of that. In the third case if we define the central angle to be &lt;math&gt;360^{\text{o}}-\angle AOB&lt;/math&gt; then again we get that the inscribed angle is half of the central angle. <br /> <br /> <br /> The theorem can be proved with angle-chasing, using the same idea that was described for Thales' theorem. See the [https://en.wikipedia.org/wiki/Inscribed_angle wiki page] for the proof (but first try to do it on your own!).<br /> <br /> <br /> '''Applications to cyclic quadrilaterals'''<br /> <br /> The following statements (and their converses) are useful applications of the Inscribed Angle theorem.<br /> <br /> <br /> 1. Suppose that the points &lt;math&gt;A, B, C, D&lt;/math&gt; form a cyclic quadrilateral, this means that we can draw a circle going through the four points. &lt;math&gt;AB&lt;/math&gt; divides the circle into two arcs. If the points &lt;math&gt;C&lt;/math&gt; and &lt;math&gt;D&lt;/math&gt; are in the same arc (meaning that they are on the same side of &lt;math&gt;AB&lt;/math&gt;) then <br /> &lt;math display=&quot;block&quot;&gt; \angle ACB= \angle ADB.&lt;/math&gt;<br /> The converse of this statement is also true: if &lt;math&gt;A, B, C, D&lt;/math&gt; are distinct points, the points &lt;math&gt;C, D&lt;/math&gt; are on the same side of the line &lt;math&gt;AB&lt;/math&gt; and &lt;math&gt;\angle ACB= \angle ADB<br /> &lt;/math&gt; then we can draw a circle around &lt;math&gt;A, B, C, D&lt;/math&gt;, in other words &lt;math&gt;ABCD&lt;/math&gt; is a cyclic quadrilateral.<br /> <br /> 2. Suppose that &lt;math&gt;ABCD&lt;/math&gt; is a cyclic quadrilateral. Then the sum of any two opposite angles is equal to &lt;math&gt;180^{\text{o}}&lt;/math&gt;. This means that <br /> &lt;math display=&quot;block&quot;&gt; \angle ABC+\angle CDA= 180^{\text{o}}, \quad \text{and}\quad \angle BCD+\angle DAB= 180^{\text{o}}. \qquad\qquad (**)&lt;/math&gt;<br /> <br /> The converse of the previous statement is also true: suppose that &lt;math&gt;ABCD&lt;/math&gt; is a quadrilateral with angles satisfying the equations &lt;math&gt;(**)&lt;/math&gt;. Then &lt;math&gt;ABCD&lt;/math&gt; is a cyclic quadrilateral: we can draw a circle that passes through the four points.<br /> <br /> The Inscribed Angle Theorem and the statements about cyclic quadrilaterals can be used for example in [https://www.math.wisc.edu/talent/sites/default/files/Talent15-4q.pdf 2015-16 Set #4 Problem 5].</div> Valko https://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=18142 Probability Seminar 2019-10-11T00:04:17Z <p>Valko: /* October 24, 2019, Brian Rider, Temple University */</p> <hr /> <div>__NOTOC__<br /> <br /> = Fall 2019 =<br /> <br /> &lt;b&gt;Thursdays in 901 Van Vleck Hall at 2:30 PM&lt;/b&gt;, unless otherwise noted. <br /> &lt;b&gt;We usually end for questions at 3:20 PM.&lt;/b&gt;<br /> <br /> If you would like to sign up for the email list to receive seminar announcements then please send an email to <br /> [mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br /> <br /> <br /> == September 12, 2019, [https://perso.univ-rennes1.fr/victor.kleptsyn/ Victor Kleptsyn], CNRS and University of Rennes 1 ==<br /> '''Furstenberg theorem: now with a parameter!'''<br /> <br /> The classical Furstenberg theorem describes the (almost sure) behaviour of a random product of independent matrices; their norms turn out to grow exponentially. In our joint work with A. Gorodetski, we study what happens if the random matrices depend on an additional parameter. <br /> It turns out that in this new situation, the conclusion changes. Namely, under some conditions, there almost surely exists a (random) &quot;exceptional&quot; set on parameters where the lower limit for the Lyapunov exponent vanishes.<br /> Our results are related to the Anderson localization in dimension one, providing a purely dynamical viewpoint on its proof. I will also speak about some generalizations and related open questions.<br /> <br /> == September 19, 2019, [http://math.columbia.edu/~xuanw Xuan Wu], Columbia University==<br /> <br /> '''A Gibbs resampling method for discrete log-gamma line ensemble.'''<br /> <br /> In this talk we will construct the discrete log-gamma line ensemble, which is assocaited with inverse gamma polymer model. This log-gamma line ensemble enjoys a random walk Gibbs resampling invariance that follows from the integrable nature of the inverse gamma polymer model via geometric RSK correspondance. By exploiting such resampling invariance, we show the tightness of this log-gamma line ensemble under weak noise scaling. Furthermore, a Gibbs property, as enjoyed by KPZ line ensemble, holds for all subsequential limits.<br /> <br /> == October 10, 2019, NO SEMINAR - [https://sites.math.northwestern.edu/mwp/ Midwest Probability Colloquium] ==<br /> <br /> == October 17, 2019, [https://www.usna.edu/Users/math/hottovy/index.php Scott Hottovy], USNA ==<br /> <br /> ''' Simplified dynamics for noisy systems with delays.'''<br /> <br /> Many biological and physical systems include some type of random noise with a temporal delay. For example, many sperm cells travel in a random motion where their velocity changes according to a chemical signal. This chemotaxis is transmitted through a delay in the system. That is, the sperm notices chemical gradients after a certain time has elapsed. In this case, the delay causes the sperm to aggregate around the egg. In this talk I will consider a general stochastic differential delay equation (SDDE) with state-dependent colored noises and derive its limit as the time delays and the correlation times of the noises go to zero. The analysis leads to a much simpler Stochastic Differential Equation to study. The work is motivated by an experiment involving an electrical circuit with noisy, delayed feedback. The main methods used in the proof are a theorem about convergence of solutions of stochastic differential equations by Kurtz and Protter and a maximal inequality for sums of a stationary sequence of random variables by Peligrad and Utev.<br /> <br /> == October 24, 2019, [https://math.temple.edu/~brider/ Brian Rider], Temple University ==<br /> <br /> '''A general beta crossover ensemble'''<br /> <br /> I'll describe an operator limit for a family of general beta ensembles which exhibit a double-scaling. In particular, a free parameter in the system provides for a crossover between the more well-known &quot;soft&quot; and &quot;hard&quot; edge point processes. This new limit operator takes as input the Riccati diffusion associated with the Stochastic Airy Operator. I like to suggest that this hints at a hierarchy of random operators analogous to the Painlevé hierarchy observed at the level of correlation functions for double-scaling ensembles most widely studied at beta = 2. Full disclosure: the result remains partially conjectural due to an unresolved uniqueness question, but I’ll provide lots of evidence to convince you we have the right answer. Joint work with Jose Ramírez (Univ. Costa Rica).<br /> <br /> == October 31, 2019, [http://math.mit.edu/~elmos/ Elchanan Mossel], MIT ==<br /> <br /> == November 7, 2019, [https://people.kth.se/~tobergg/ Tomas Berggren], KTH Stockholm ==<br /> <br /> == November 14, 2019, [https://math.mit.edu/directory/profile.php?pid=2076 Benjamin Landon], MIT ==<br /> <br /> == November 21, 2019, Tung Nguyen, UW Madison ==<br /> <br /> == November 28, 2019, Thanksgiving (no seminar) ==<br /> <br /> == December 5, 2019, Vadim Gorin, UW Madison ==<br /> <br /> <br /> <br /> &lt;!--<br /> <br /> == &lt;span style=&quot;color:red&quot;&gt; Wednesday, February 6 at 4:00pm in Van Vleck 911&lt;/span&gt; , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br /> <br /> Title: '''When particle systems meet PDEs'''<br /> <br /> Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br /> <br /> <br /> == &lt;span style=&quot;color:red&quot;&gt;'''Tuesday''' &lt;/span&gt;, May 7, Van Vleck 901, 2:25pm, Duncan Dauvergne (Toronto) ==<br /> <br /> <br /> &lt;div style=&quot;width:250px;height:50px;border:5px solid black&quot;&gt;<br /> &lt;b&gt;&lt;span style=&quot;color:red&quot;&gt;&amp;emsp; Please note the unusual day. <br /> &amp;emsp; &lt;/span&gt;&lt;/b&gt;<br /> &lt;/div&gt;<br /> Title: '''The directed landscape'''<br /> <br /> Abstract: I will describe the construction of the full scaling limit of (Brownian) last passage percolation: the directed landscape. The directed landscape can be thought of as a random scale-invariant directed' metric on the plane, and last passage paths converge to directed geodesics in this metric. The directed landscape is expected to be a universal scaling limit for general last passage and random growth models (i.e. TASEP, the KPZ equation, the longest increasing subsequence in a random permutation). Joint work with Janosch Ormann and Balint Virag.<br /> --&gt;<br /> <br /> == ==<br /> <br /> [[Past Seminars]]</div> Valko https://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=18141 Probability Seminar 2019-10-11T00:04:01Z <p>Valko: /* October 24, 2019, Brian Rider, Temple University */</p> <hr /> <div>__NOTOC__<br /> <br /> = Fall 2019 =<br /> <br /> &lt;b&gt;Thursdays in 901 Van Vleck Hall at 2:30 PM&lt;/b&gt;, unless otherwise noted. <br /> &lt;b&gt;We usually end for questions at 3:20 PM.&lt;/b&gt;<br /> <br /> If you would like to sign up for the email list to receive seminar announcements then please send an email to <br /> [mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br /> <br /> <br /> == September 12, 2019, [https://perso.univ-rennes1.fr/victor.kleptsyn/ Victor Kleptsyn], CNRS and University of Rennes 1 ==<br /> '''Furstenberg theorem: now with a parameter!'''<br /> <br /> The classical Furstenberg theorem describes the (almost sure) behaviour of a random product of independent matrices; their norms turn out to grow exponentially. In our joint work with A. Gorodetski, we study what happens if the random matrices depend on an additional parameter. <br /> It turns out that in this new situation, the conclusion changes. Namely, under some conditions, there almost surely exists a (random) &quot;exceptional&quot; set on parameters where the lower limit for the Lyapunov exponent vanishes.<br /> Our results are related to the Anderson localization in dimension one, providing a purely dynamical viewpoint on its proof. I will also speak about some generalizations and related open questions.<br /> <br /> == September 19, 2019, [http://math.columbia.edu/~xuanw Xuan Wu], Columbia University==<br /> <br /> '''A Gibbs resampling method for discrete log-gamma line ensemble.'''<br /> <br /> In this talk we will construct the discrete log-gamma line ensemble, which is assocaited with inverse gamma polymer model. This log-gamma line ensemble enjoys a random walk Gibbs resampling invariance that follows from the integrable nature of the inverse gamma polymer model via geometric RSK correspondance. By exploiting such resampling invariance, we show the tightness of this log-gamma line ensemble under weak noise scaling. Furthermore, a Gibbs property, as enjoyed by KPZ line ensemble, holds for all subsequential limits.<br /> <br /> == October 10, 2019, NO SEMINAR - [https://sites.math.northwestern.edu/mwp/ Midwest Probability Colloquium] ==<br /> <br /> == October 17, 2019, [https://www.usna.edu/Users/math/hottovy/index.php Scott Hottovy], USNA ==<br /> <br /> ''' Simplified dynamics for noisy systems with delays.'''<br /> <br /> Many biological and physical systems include some type of random noise with a temporal delay. For example, many sperm cells travel in a random motion where their velocity changes according to a chemical signal. This chemotaxis is transmitted through a delay in the system. That is, the sperm notices chemical gradients after a certain time has elapsed. In this case, the delay causes the sperm to aggregate around the egg. In this talk I will consider a general stochastic differential delay equation (SDDE) with state-dependent colored noises and derive its limit as the time delays and the correlation times of the noises go to zero. The analysis leads to a much simpler Stochastic Differential Equation to study. The work is motivated by an experiment involving an electrical circuit with noisy, delayed feedback. The main methods used in the proof are a theorem about convergence of solutions of stochastic differential equations by Kurtz and Protter and a maximal inequality for sums of a stationary sequence of random variables by Peligrad and Utev.<br /> <br /> == October 24, 2019, [https://math.temple.edu/~brider/ Brian Rider], Temple University ==<br /> <br /> Title: A general beta crossover ensemble<br /> <br /> Abstract: I'll describe an operator limit for a family of general beta ensembles which exhibit a double-scaling. In particular, a free parameter in the system provides for a crossover between the more well-known &quot;soft&quot; and &quot;hard&quot; edge point processes. This new limit operator takes as input the Riccati diffusion associated with the Stochastic Airy Operator. I like to suggest that this hints at a hierarchy of random operators analogous to the Painlevé hierarchy observed at the level of correlation functions for double-scaling ensembles most widely studied at beta = 2. Full disclosure: the result remains partially conjectural due to an unresolved uniqueness question, but I’ll provide lots of evidence to convince you we have the right answer. Joint work with Jose Ramírez (Univ. Costa Rica).<br /> <br /> == October 31, 2019, [http://math.mit.edu/~elmos/ Elchanan Mossel], MIT ==<br /> <br /> == November 7, 2019, [https://people.kth.se/~tobergg/ Tomas Berggren], KTH Stockholm ==<br /> <br /> == November 14, 2019, [https://math.mit.edu/directory/profile.php?pid=2076 Benjamin Landon], MIT ==<br /> <br /> == November 21, 2019, Tung Nguyen, UW Madison ==<br /> <br /> == November 28, 2019, Thanksgiving (no seminar) ==<br /> <br /> == December 5, 2019, Vadim Gorin, UW Madison ==<br /> <br /> <br /> <br /> &lt;!--<br /> <br /> == &lt;span style=&quot;color:red&quot;&gt; Wednesday, February 6 at 4:00pm in Van Vleck 911&lt;/span&gt; , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br /> <br /> Title: '''When particle systems meet PDEs'''<br /> <br /> Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br /> <br /> <br /> == &lt;span style=&quot;color:red&quot;&gt;'''Tuesday''' &lt;/span&gt;, May 7, Van Vleck 901, 2:25pm, Duncan Dauvergne (Toronto) ==<br /> <br /> <br /> &lt;div style=&quot;width:250px;height:50px;border:5px solid black&quot;&gt;<br /> &lt;b&gt;&lt;span style=&quot;color:red&quot;&gt;&amp;emsp; Please note the unusual day. <br /> &amp;emsp; &lt;/span&gt;&lt;/b&gt;<br /> &lt;/div&gt;<br /> Title: '''The directed landscape'''<br /> <br /> Abstract: I will describe the construction of the full scaling limit of (Brownian) last passage percolation: the directed landscape. The directed landscape can be thought of as a random scale-invariant directed' metric on the plane, and last passage paths converge to directed geodesics in this metric. The directed landscape is expected to be a universal scaling limit for general last passage and random growth models (i.e. TASEP, the KPZ equation, the longest increasing subsequence in a random permutation). Joint work with Janosch Ormann and Balint Virag.<br /> --&gt;<br /> <br /> == ==<br /> <br /> [[Past Seminars]]</div> Valko https://www.math.wisc.edu/wiki/index.php?title=Probability&diff=17903 Probability 2019-09-16T19:56:41Z <p>Valko: </p> <hr /> <div>__NOTOC__<br /> <br /> = '''Probability at UW-Madison''' =<br /> <br /> &lt;br&gt;<br /> <br /> == Tenured and tenure-track faculty ==<br /> <br /> [http://www.math.wisc.edu/~anderson/ David Anderson] (Duke, 2005) applied probability, numerical methods, mathematical biology.<br /> <br /> Vadim Gorin (Moscow, 2011) integrable probability, random matrices, asymptotic representation theory<br /> <br /> [http://www.math.wisc.edu/~roch/ Sebastien Roch] (UC Berkeley, 2007) applied probability, mathematical biology, theoretical computer science.<br /> <br /> [http://www.math.wisc.edu/~seppalai/ Timo Seppäläinen] (Minnesota, 1991) motion in a random medium, random growth models, interacting particle systems, large deviation theory.<br /> <br /> [http://www.math.wisc.edu/~hshen3/ Hao Shen] (Princeton, 2013) stochastic partial differential equations, mathematical physics, integrable probability<br /> <br /> [http://www.math.wisc.edu/~valko/ Benedek Valko] (Budapest, 2004) interacting particle systems, random matrices.<br /> <br /> <br /> <br /> == Emeriti ==<br /> <br /> [http://psoup.math.wisc.edu/kitchen.html David Griffeath] (Cornell, 1976)<br /> <br /> [http://www.math.wisc.edu/~kuelbs Jim Kuelbs] (Minnesota, 1965)<br /> <br /> [http://www.math.wisc.edu/~kurtz Tom Kurtz] (Stanford, 1967)<br /> <br /> Peter Ney (Columbia, 1961)<br /> <br /> Josh Chover (Michigan, 1952)<br /> <br /> <br /> == Postdocs ==<br /> <br /> Scott Smith (Maryland, 2016)<br /> <br /> <br /> <br /> == Graduate students ==<br /> <br /> <br /> [http://www.math.wisc.edu/~kehlert/ Kurt Ehlert] <br /> <br /> [http://www.math.wisc.edu/~kang Dae Han Kang]<br /> <br /> [https://sites.google.com/a/wisc.edu/brandon-legried/ Brandon Legried]<br /> <br /> Yun Li<br /> <br /> [http://sites.google.com/a/wisc.edu/tung-nguyen/ Tung Nguyen]<br /> <br /> [http://www.math.wisc.edu/~cyuan25/ Chaojie Yuan]<br /> <br /> <br /> <br /> == [[Probability Seminar]] ==<br /> <br /> Thursdays at 2:25pm, VV901<br /> <br /> ==[[Graduate student reading seminar]]==<br /> <br /> Email list: join-grad_prob_seminar@lists.wisc.edu<br /> <br /> Tuesdays, 2:30pm, 901 Van Vleck<br /> <br /> == [[Probability group timetable]]==<br /> <br /> == [[Undergraduate courses in probability]]==<br /> <br /> == Graduate Courses in Probability ==<br /> <br /> <br /> <br /> '''2019 Fall'''<br /> <br /> Math/Stat 733 Theory of Probability I<br /> <br /> <br /> <br /> <br /> '''2020 Spring'''<br /> <br /> Math/Stat 734 Theory of Probability II <br /> <br /> Math 833 Topics in Probability</div> Valko https://www.math.wisc.edu/wiki/index.php?title=Graduate_student_reading_seminar&diff=17813 Graduate student reading seminar 2019-09-11T18:11:13Z <p>Valko: </p> <hr /> <div>(... in probability)<br /> <br /> <br /> Email list: join-grad_prob_seminar@lists.wisc.edu<br /> <br /> ==2018 Fall==<br /> <br /> Tuesday 2:30pm, 901 Van Vleck<br /> <br /> 9/24, 10/1: Xiao<br /> <br /> 10/8, 10/15: Jakwang<br /> <br /> 10/22, 10/29: Evan<br /> <br /> 11/5, 11/12: Chaojie<br /> <br /> 12/3, 12/10: Tung<br /> <br /> ==2019 Spring==<br /> <br /> Tuesday 2:30pm, 901 Van Vleck<br /> <br /> 2/5: Timo<br /> <br /> 2/12, 2/19: Evan<br /> <br /> 2/26, 3/5: Chaojie<br /> <br /> 3/12, 3/26: Kurt<br /> <br /> 4/2, 4/9: Yu<br /> <br /> 4/16, 4/23: Max<br /> <br /> 4/30, 5/7: Xiao<br /> <br /> ==2018 Fall==<br /> <br /> Tuesday 2:30pm, 901 Van Vleck<br /> <br /> <br /> The topic this semester is large deviation theory. Send me (BV) an email, if you want access to the shared Box folder with some reading material. <br /> <br /> <br /> 9/25, 10/2: Dae Han<br /> <br /> 10/9, 10/16: Kurt<br /> <br /> 10/23, 10/30: Stephen Davis<br /> <br /> 11/6, 11/13: Brandon Legried <br /> <br /> 11/20, 11/27: Shuqi Yu<br /> <br /> 12/4, 12/11: Yun Li<br /> <br /> ==2018 Spring==<br /> <br /> Tuesday 2:30pm, B135 Van Vleck<br /> <br /> <br /> Preliminary schedule:<br /> <br /> 2/20, 2/27: Yun<br /> <br /> 3/6, 3/13: Greg<br /> <br /> 3/20, 4/3: Yu<br /> <br /> 4/10, 4/17: Shuqi<br /> <br /> 4/24, 5/1: Tony<br /> <br /> ==2017 Fall==<br /> <br /> Tuesday 2:30pm, 214 Ingraham Hall<br /> <br /> <br /> Preliminary schedule: <br /> <br /> 9/26, 10/3: Hans<br /> <br /> 10/10, 10/17: Guo<br /> <br /> 10/24, 10/31: Chaoji<br /> <br /> 11/7, 11/14: Yun <br /> <br /> 11/21, 11/28: Kurt<br /> <br /> 12/5, 12/12: Christian<br /> <br /> <br /> <br /> <br /> ==2017 Spring==<br /> <br /> Tuesday 2:25pm, B211<br /> <br /> 1/31, 2/7: Fan<br /> <br /> I will talk about the Hanson-Wright inequality, which is a large deviation estimate for random variable of the form X^* A X, where X is a random vector with independent subgaussian entries and A is an arbitrary deterministic matrix. In the first talk, I will present a beautiful proof given by Mark Rudelson and Roman Vershynin. In the second talk, I will talk about some applications of this inequality.<br /> <br /> Reference: M. Rudelson and R. Vershynin, Hanson-Wright inequality and sub-gaussian concentration, Electron. Commun. Probab. Volume 18 (2013).<br /> <br /> 3/7, 3/14 : Jinsu<br /> <br /> Title : Donsker's Theorem and its application.<br /> Donsker's Theorem roughly says normalized random walk with linear interpolation on time interval [0,1] weakly converges to the Brownian motion B[0,1] in C([0,1]). It is sometimes called Donsker's invariance principle or the functional central limit theorem. I will show main ideas for the proof of this theorem tomorrow and show a couple of applications in my 2nd talk.<br /> <br /> Reference : https://www.math.utah.edu/~davar/ps-pdf-files/donsker.pdf<br /> <br /> ==2016 Fall==<br /> <br /> 9/27 Daniele<br /> <br /> Stochastic reaction networks.<br /> <br /> Stochastic reaction networks are continuous time Markov chain models used primarily in biochemistry. I will define them, prove some results that connect them to related deterministic models and introduce some open questions. <br /> <br /> 10/4 Jessica<br /> <br /> 10/11, 10/18: Dae Han<br /> <br /> 10/25, 11/1: Jinsu<br /> <br /> Coupling of Markov processes.<br /> <br /> When we have two distributions on same probability space, we can think of a pair whose marginal probability is each of two distributions.<br /> This pairing can be used to estimate the total variation distance between two distributions. This idea is called coupling method.<br /> I am going to introduce basic concepts,ideas and applications of coupling for Markov processes.<br /> <br /> Links of References<br /> <br /> http://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf<br /> <br /> http://websites.math.leidenuniv.nl/probability/lecturenotes/CouplingLectures.pdf<br /> <br /> 11/8, 11/15: Hans<br /> <br /> 11/22, 11/29: Keith<br /> <br /> Surprisingly Determinental: DPPs and some asymptotics of ASEP <br /> <br /> I'll be reading and presenting some recent papers of Alexei Borodin and a few collaborators which have uncovered certain equivalences between determinental point processes and non-determinental processes.<br /> <br /> <br /> ==2016 Spring==<br /> <br /> Tuesday, 2:25pm, B321 Van Vleck<br /> <br /> <br /> 3/29, 4/5: Fan Yang<br /> <br /> I will talk about the ergodic decomposition theorem (EDT). More specifically, given a compact metric space X and a continuous transformation T on it, the theorem shows that any T-invariant measure on X can be decomposed into a convex combination of ergodic measures. In the first talk I introduced the EDT and some related facts. In the second talk, I will talk about the conditional measures, and prove that the ergodic measures in EDT are indeed the conditional measures.<br /> <br /> <br /> 2/16 : Jinsu<br /> <br /> Lyapunov function for Markov Processes.<br /> <br /> For ODE, we can show stability of the trajectory using Lyapunov functions.<br /> <br /> There is an analogy for Markov Processes. I'd like to talk about the existence of stationary distribution with Lyapunov function.<br /> <br /> In some cases, it is also possible to show the rate of convergence to the stationary distribution.<br /> <br /> ==2015 Fall==<br /> <br /> This semester we will focus on tools and methods.<br /> <br /> [https://www.math.wisc.edu/wiki/images/a/ac/Reading_seminar_2015.pdf Seminar notes] ([https://www.dropbox.com/s/f4km7pevwfb1vbm/Reading%20seminar%202015.tex?dl=1 tex file], [https://www.dropbox.com/s/lg7kcgyf3nsukbx/Reading_seminar_2015.bib?dl=1 bib file])<br /> <br /> 9/15, 9/22: Elnur<br /> <br /> I will talk about large deviation theory and its applications. For the first talk, my plan is to introduce Gartner-Ellis theorem and show a few applications of it to finite state discrete time Markov chains.<br /> <br /> 9/29, 10/6, 10/13 :Dae Han<br /> <br /> 10/20, 10/27, 11/3: Jessica<br /> <br /> I will first present an overview of concentration of measure and concentration inequalities with a focus on the connection with related topics in analysis and geometry. Then, I will present Log-Sobolev inequalities and their connection to concentration of measure. <br /> <br /> 11/10, 11/17: Hao Kai<br /> <br /> 11/24, 12/1, 12/8, 12/15: Chris<br /> <br /> : <br /> <br /> <br /> <br /> <br /> <br /> 2016 Spring:<br /> <br /> 2/2, 2/9: Louis<br /> <br /> <br /> 2/16, 2/23: Jinsu<br /> <br /> 3/1, 3/8: Hans<br /> <br /> ==2015 Spring==<br /> <br /> <br /> 2/3, 2/10: Scott<br /> <br /> An Introduction to Entropy for Random Variables<br /> <br /> In these lectures I will introduce entropy for random variables and present some simple, finite state-space, examples to gain some intuition. We will prove the <br /> MacMillan Theorem using entropy and the law of large numbers. Then I will introduce relative entropy and prove the Markov Chain Convergence Theorem. Finally I will <br /> define entropy for a discrete time process. The lecture notes can be found at http://www.math.wisc.edu/~shottovy/EntropyLecture.pdf.<br /> <br /> 2/17, 2/24: Dae Han<br /> <br /> 3/3, 3/10: Hans<br /> <br /> 3/17, 3/24: In Gun<br /> <br /> 4/7, 4/14: Jinsu<br /> <br /> 4/21, 4/28: Chris N.<br /> <br /> <br /> <br /> <br /> <br /> <br /> ==2014 Fall==<br /> <br /> 9/23: Dave<br /> <br /> I will go over Mike Giles’ 2008 paper “Multi-level Monte Carlo path simulation.” This paper introduced a new Monte Carlo method to approximate expectations of SDEs (driven by Brownian motions) that is significantly more efficient than what was the state of the art. This work opened up a whole new field in the numerical analysis of stochastic processes as the basic idea is quite flexible and has found a variety of applications including SDEs driven by Brownian motions, Levy-driven SDEs, SPDEs, and models from biology<br /> <br /> 9/30: Benedek<br /> <br /> A very quick introduction to Stein's method. <br /> <br /> I will give a brief introduction to Stein's method, mostly based on the the first couple of sections of the following survey article:<br /> <br /> Ross, N. (2011). Fundamentals of Stein’s method. Probability Surveys, 8, 210-293. <br /> <br /> The following webpage has a huge collection of resources if you want to go deeper: https://sites.google.com/site/yvikswan/about-stein-s-method<br /> <br /> <br /> Note that the Midwest Probability Colloquium (http://www.math.northwestern.edu/mwp/) will have a tutorial program on Stein's method this year. <br /> <br /> 10/7, 10/14: Chris J.<br /> [http://www.math.wisc.edu/~janjigia/research/MartingaleProblemNotes.pdf An introduction to the (local) martingale problem.]<br /> <br /> <br /> 10/21, 10/28: Dae Han<br /> <br /> 11/4, 11/11: Elnur<br /> <br /> 11/18, 11/25: Chris N. Free Probability with an emphasis on C* and Von Neumann Algebras<br /> <br /> 12/2, 12/9: Yun Zhai<br /> <br /> ==2014 Spring==<br /> <br /> <br /> 1/28: Greg<br /> <br /> 2/04, 2/11: Scott <br /> <br /> [http://www.math.wisc.edu/~shottovy/BLT.pdf Reflected Brownian motion, Occupation time, and applications.] <br /> <br /> 2/18: Phil-- Examples of structure results in probability theory.<br /> <br /> 2/25, 3/4: Beth-- Derivative estimation for discrete time Markov chains<br /> <br /> 3/11, 3/25: Chris J [http://www.math.wisc.edu/~janjigia/research/stationarytalk.pdf Some classical results on stationary distributions of Markov processes]<br /> <br /> 4/1, 4/8: Chris N <br /> <br /> 4/15, 4/22: Yu Sun<br /> <br /> 4/29. 5/6: Diane<br /> <br /> ==2013 Fall==<br /> <br /> 9/24, 10/1: Chris<br /> [http://www.math.wisc.edu/~janjigia/research/metastabilitytalk.pdf A light introduction to metastability]<br /> <br /> 10/8, Dae Han<br /> Majoring multiplicative cascades for directed polymers in random media<br /> <br /> 10/15, 10/22: no reading seminar<br /> <br /> 10/29, 11/5: Elnur<br /> Limit fluctuations of last passage times <br /> <br /> 11/12: Yun<br /> Helffer-Sjostrand representation and Brascamp-Lieb inequality for stochastic interface models<br /> <br /> 11/19, 11/26: Yu Sun<br /> <br /> 12/3, 12/10: Jason<br /> <br /> ==2013 Spring==<br /> <br /> 2/13: Elnur <br /> <br /> Young diagrams, RSK correspondence, corner growth models, distribution of last passage times. <br /> <br /> 2/20: Elnur<br /> <br /> 2/27: Chris<br /> <br /> A brief introduction to enlargement of filtration and the Dufresne identity<br /> [http://www.math.wisc.edu/~janjigia/research/Presentation%20Notes.pdf Notes]<br /> <br /> 3/6: Chris<br /> <br /> 3/13: Dae Han<br /> <br /> An introduction to random polymers<br /> <br /> 3/20: Dae Han<br /> <br /> Directed polymers in a random environment: path localization and strong disorder<br /> <br /> 4/3: Diane<br /> <br /> Scale and Speed for honest 1 dimensional diffusions<br /> <br /> References: &lt;br&gt;<br /> Rogers &amp; Williams - Diffusions, Markov Processes and Martingales &lt;br&gt;<br /> Ito &amp; McKean - Diffusion Processes and their Sample Paths &lt;br&gt;<br /> Breiman - Probability &lt;br&gt;<br /> http://www.statslab.cam.ac.uk/~beresty/Articles/diffusions.pdf<br /> <br /> 4/10: Diane<br /> <br /> 4/17: Yun<br /> <br /> Introduction to stochastic interface models<br /> <br /> 4/24: Yun<br /> <br /> Dynamics and Gaussian equilibrium sytems<br /> <br /> 5/1: This reading seminar will be shifted because of a probability seminar.<br /> <br /> <br /> 5/8: Greg, Maso<br /> <br /> The Bethe ansatz vs. The Replica Trick. This lecture is an overview of the two <br /> approaches. See [http://arxiv.org/abs/1212.2267] for a nice overview.<br /> <br /> 5/15: Greg, Maso<br /> <br /> Rigorous use of the replica trick.</div> Valko https://www.math.wisc.edu/wiki/index.php?title=Talk:Probability_Seminar&diff=17779 Talk:Probability Seminar 2019-09-06T19:19:27Z <p>Valko: Blanked the page</p> <hr /> <div></div> Valko https://www.math.wisc.edu/wiki/index.php?title=Problem_Solver%27s_Toolbox&diff=17683 Problem Solver's Toolbox 2019-08-27T19:04:18Z <p>Valko: </p> <hr /> <div>The goal of this page is to collect simple problem solving strategies and tools. We hope that students interested in the Wisconsin Math Talent Search would find the described ideas useful. <br /> This page and the discussed topics can be used as a starting point for future exploration.<br /> <br /> <br /> == General ideas ==<br /> <br /> <br /> There is no universal recipe for math problems that would work every time, that's what makes math fun! There are however a number of general strategies that could be useful in most cases, here is a short list of them. (Many of these ideas were popularized by the Hungarian born mathematician George Pólya in his book [https://en.wikipedia.org/wiki/How_to_Solve_It How to Solve It].)<br /> * Make sure that you understand the problem. <br /> * If possible, draw a figure. <br /> * Can you connect the problem to a problem you have solved before? <br /> * If you have to show something for all numbers (or a large number) then try to check the statement for small values first.<br /> * Can you solve the problem in a special case first? Can you solve a modified version of the problem first? <br /> * Is there some symmetry in the problem that you can exploit? <br /> * Is it possible to work backward? <br /> * Does it help to consider an extreme case of the problem?<br /> * Is it possible to generalize the problem? (Sometimes the generalized is easier to solve.)<br /> <br /> == Modular arithmetic ==<br /> <br /> <br /> When we have to divide two integers, they don't always divide evenly, and there is a quotient and a remainder. For example when we divide 10 by 3 we get a remainder of 1.<br /> It turns out that these remainders behave very well under addition, subtraction, and multiplication. We say two numbers are the same &quot;modulo &lt;math&gt;m&lt;/math&gt;&quot; if they have the same remainder when divided by &lt;math&gt;m&lt;/math&gt;. If &lt;math&gt;a&lt;/math&gt; and &lt;math&gt;x&lt;/math&gt; are the same modulo &lt;math&gt;m&lt;/math&gt;, and &lt;math&gt;b&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; are the same modulo &lt;math&gt;m&lt;/math&gt;, then &lt;math&gt;a+b&lt;/math&gt; and &lt;math&gt;x+y&lt;/math&gt; are the same modulo &lt;math&gt;m&lt;/math&gt;, and similarly for subtraction and multiplication. <br /> <br /> For example, 5 is the same as 1 modulo 4, and hence &lt;math&gt;5\cdot 5 \cdot 5 \cdot 5=5^4&lt;/math&gt; is the same as &lt;math&gt;1\cdot 1\cdot 1\cdot 1=1&lt;/math&gt; modulo &lt;math&gt;4&lt;/math&gt;. Same way you can show that &lt;math&gt;5^{1000}&lt;/math&gt; has a remainder of 1 when we divide it by 4.<br /> <br /> Modular arithmetic often makes calculation much simpler. For example, see [https://www.math.wisc.edu/talent/sites/default/files/Talent16-2q.pdf 2016-17 Set #2 Problem 3].<br /> <br /> See [http://artofproblemsolving.com/wiki/index.php?title=Modular_arithmetic/Introduction Art of Problem Solving's introduction to modular arithmetic] for more information.<br /> <br /> == Mathematical induction ==<br /> <br /> Suppose that you want to prove a statement for all positive integers, for example that for each positive integer &lt;math&gt;n&lt;/math&gt; the following is true: &lt;math display=&quot;block&quot;&gt;1\cdot 2+2\cdot 3+3\cdot 4+\cdots+n\cdot (n-1)=\frac{n(n+1)(n+2)}{3}.\qquad\qquad(*) &lt;/math&gt;<br /> Mathematical induction provides a tool for doing this. You need to show the following two things:<br /> # (Base case) The statement is true for &lt;math&gt;n=1&lt;/math&gt;. <br /> # (Induction step) If the statement is true for &lt;math&gt;n&lt;/math&gt; then it must be true for &lt;math&gt;n+1&lt;/math&gt; as well.<br /> <br /> If we can show both of these parts, then it follows that the statement is true for all positive integer &lt;math&gt;n&lt;/math&gt;. Why? The first part (the base case) shows that the statement is true for &lt;math&gt;n=1&lt;/math&gt;. But then by the second part (the induction step) the statement must be true for &lt;math&gt;n=2&lt;/math&gt; as well. Using the second part again and again we see that the statement is true for &lt;math&gt;n=3, 4, 5, \cdots&lt;/math&gt; and repeating this sufficiently times we can prove that the statement is true for any fixed value of &lt;math&gt;n&lt;/math&gt;. <br /> <br /> Often the idea of induction is demonstrated as a version of Domino effect'. Imagine that you have an infinite row of dominos numbered with the positive integers, where if &lt;math&gt;n&lt;/math&gt;th domino falls then the next one will fall as well (this is the induction step). If we make the first domino fall (this is the base case) then eventually all other dominos will fall as well. <br /> <br /> * Try to use induction to show the identity &lt;math&gt;(*)&lt;/math&gt; above for all positive integer &lt;math&gt;n&lt;/math&gt;.<br /> * You can also use induction to show a statement for all integers &lt;math&gt;n\ge 5&lt;/math&gt;. Then for your base case you have to show that the statement is true for &lt;math&gt;n=5&lt;/math&gt;. (The induction step is the same.)<br /> <br /> See this page from [https://www.mathsisfun.com/algebra/mathematical-induction.html Math Is Fun] for some simple applications of induction.<br /> <br /> == Proof by contradiction ==<br /> <br /> This is a commonly used problem solving method. Suppose that you have to prove a certain statement. Now pretend that the statement is not true and try to derive (as a consequence) a false statement. The found false statement shows that your assumption about the original statement was incorrect: thus the original statement must be true. <br /> <br /> Here is a simple example: we will prove that the product of three consecutive positive integers cannot be a prime number. Assume the opposite: that means that there is a positive integer &lt;math&gt;n&lt;/math&gt; so that &lt;math&gt;n(n+1)(n+2)&lt;/math&gt; is a prime. But among three consecutive integers we will always have a multiple of 2, and also a multiple of 3. Thus the product of the three numbers must be divisible by both 2 and 3, and hence &lt;math&gt;n(n+1)(n+2)&lt;/math&gt; cannot be a prime. This contradicts our assumption that &lt;math&gt;n(n+1)(n+2)&lt;/math&gt; is a prime, which shows that our assumption had to be incorrect. <br /> <br /> Proof by contradiction can be used for example in [https://www.math.wisc.edu/talent/sites/default/files/Talent16-1q.pdf 2016-17 Set #1 Problem 4].<br /> <br /> == Pigeonhole Principle ==<br /> <br /> The Pigeonhole Principle is one of the simplest tools in mathematics, but it can be very powerful. Suppose that &lt;math&gt;n&lt;m&lt;/math&gt; are positive integers, and we have &lt;math&gt;m&lt;/math&gt; objects and &lt;math&gt;n&lt;/math&gt; boxes. The Pigeonhole Principle states that If we place each of the &lt;math&gt;m&lt;/math&gt; objects into one of the &lt;math&gt;n&lt;/math&gt; boxes then there must be at least one box with at least two objects in it. <br /> The statement can be proved by contradiction: if we can find an arrangement of objects so that each box has less than two objects in it, then each box would contain at most one object, and hence we had at most &lt;math&gt;n&lt;/math&gt; objects all together. This is a contradiction, which means that the original statement must be correct. <br /> <br /> The Pigeonhole Principle is often used in the following, more general form. Suppose that &lt;math&gt;n, m, k&lt;/math&gt; are positive integers with &lt;math&gt;n k&lt; m &lt;/math&gt;. If we place each of &lt;math&gt;m&lt;/math&gt; objects into one of &lt;math&gt;n&lt;/math&gt; boxes then there must be at least one box with at least &lt;math&gt;k+1&lt;/math&gt; objects in it. Try to prove this version by contradiction.<br /> <br /> Here is a simple application: if we roll a die 13 times then there must be a number that appears at least three times. Here each die roll correspond to an object, each of the 6 possible outcomes correspond to a possible box. Since &lt;math&gt;2\cdot 6&lt;13&lt;/math&gt;, we must have a box with at least &lt;math&gt;2+1=3&lt;/math&gt; objects. In other words: there will be number that appears at least three times. <br /> <br /> Pigeonhole Principle can be used for example in [https://www.math.wisc.edu/talent/sites/default/files/T14-1q_0_0.pdf 2014-15 Set #1 Problem 4].<br /> <br /> == Angles in the circle ==<br /> <br /> The following theorems are often useful when working with geometry problems. [[File:Thales_thm.jpg|250px|thumb|right|An illustration of Thales' Theorem. O is the center of the circle.]] <br /> <br /> '''Thales' Theorem''' <br /> <br /> Suppose that the distinct points &lt;math&gt;A, B, C&lt;/math&gt; are all on a circle, and &lt;math&gt;AB&lt;/math&gt; is a diameter of of the circle. Then the angle &lt;math&gt;ACB&lt;/math&gt; is &lt;math&gt;90^{\text{o}}&lt;/math&gt;. In other words: the triangle &lt;math&gt;\triangle ABC&lt;/math&gt; is a right triangle with hypotenuse &lt;math&gt;AB&lt;/math&gt;. <br /> <br /> The theorem can be proved with a little bit of angle-chasing'. Denote the center of the circle by &lt;math&gt;O&lt;/math&gt;. Then &lt;math&gt;AO, BO, CO&lt;/math&gt; are all radii of the circle, so they have the same length. Thus &lt;math&gt;\triangle AOC&lt;/math&gt; and &lt;math&gt;\triangle BOC&lt;/math&gt; are both isosceles triangles. Now try labeling the various angles in the picture and you should quickly arrive to a proof. (You can find the worked out proof at the [https://en.wikipedia.org/wiki/Thales%27_theorem wiki page] of the theorem, but it is more fun if you figure it out on your own!)<br /> <br /> The converse of Thales's theorem states that if &lt;math&gt;\triangle ABC&lt;/math&gt; is a right triangle with hypotenuse &lt;math&gt;AB&lt;/math&gt; then we can draw a circle with a center that is the midpoint of &lt;math&gt;AB&lt;/math&gt; that passes through &lt;math&gt;A, B, C&lt;/math&gt;.<br /> <br /> <br /> The Inscribed Angle Theorem below is a generalization of Thales' Theorem. <br /> <br /> <br /> '''The Inscribed Angle Theorem'''<br /> <br /> Suppose that the distinct points &lt;math&gt;A, B, C&lt;/math&gt; are all on a circle and let &lt;math&gt;O&lt;/math&gt; be the center of the circle. Then depending on the position of these points we have the following statements:<br /> <br /> * If &lt;math&gt;O&lt;/math&gt; is on the line &lt;math&gt;AB&lt;/math&gt; then &lt;math&gt;\angle ACB=90^{\text{o}}&lt;/math&gt;. (This is just Thales' theorem again.)<br /> * If &lt;math&gt;O&lt;/math&gt; and &lt;math&gt;C&lt;/math&gt; are both on the same side of the line &lt;math&gt;AB&lt;/math&gt; then the inscribed angle &lt;math&gt;\angle ACB&lt;/math&gt; is half of &lt;math&gt;360^{\text{o}}&lt;/math&gt; minus the central angle &lt;math&gt;\angle AOB&lt;/math&gt;: <br /> &lt;math display=&quot;block&quot;&gt; 2 \angle ACB= \angle AOB.&lt;/math&gt;<br /> * If &lt;math&gt;O&lt;/math&gt; and &lt;math&gt;C&lt;/math&gt; are on the opposite sides of the line &lt;math&gt;AB&lt;/math&gt; then the inscribed angle &lt;math&gt;\angle ACB&lt;/math&gt; is half of the central angle &lt;math&gt;\angle AOB&lt;/math&gt;: <br /> &lt;math display=&quot;block&quot;&gt; 2 \angle ACB= 360^{\text{o}}-\angle AOB.&lt;/math&gt;<br /> <br /> If we measure the central angle &lt;math&gt;\angle AOB&lt;/math&gt; the right way' then we don't need to separate the three cases. In the first case the central angle is just &lt;math&gt;180^{\text{o}}&lt;/math&gt;, and the inscribed angle is exactly the half of that. In the third case if we define the central angle to be &lt;math&gt;360^{\text{o}}-\angle AOB&lt;/math&gt; then again we get that the inscribed angle is half of the central angle. <br /> <br /> <br /> The theorem can be proved with angle-chasing, using the same idea that was described for Thales' theorem. See the [https://en.wikipedia.org/wiki/Inscribed_angle wiki page] for the proof (but first try to do it on your own!).<br /> <br /> <br /> '''Applications to cyclic quadrilaterals'''<br /> <br /> The following statements (and their converses) are useful applications of the Inscribed Angle theorem.<br /> <br /> <br /> 1. Suppose that the points &lt;math&gt;A, B, C, D&lt;/math&gt; form a cyclic quadrilateral, this means that we can draw a circle going through the four points. &lt;math&gt;AB&lt;/math&gt; divides the circle into two arcs. If the points &lt;math&gt;C&lt;/math&gt; and &lt;math&gt;D&lt;/math&gt; are in the same arc (meaning that they are on the same side of &lt;math&gt;AB&lt;/math&gt;) then <br /> &lt;math display=&quot;block&quot;&gt; \angle ACB= \angle ADB.&lt;/math&gt;<br /> The converse of this statement is also true: if &lt;math&gt;A, B, C, D&lt;/math&gt; are distinct points, the points &lt;math&gt;C, D&lt;/math&gt; are on the same side of the line &lt;math&gt;AB&lt;/math&gt; and &lt;math&gt;\angle ACB= \angle ADB<br /> &lt;/math&gt; then we can draw a circle around &lt;math&gt;A, B, C, D&lt;/math&gt;, in other words &lt;math&gt;ABCD&lt;/math&gt; is a cyclic quadrilateral.<br /> <br /> 2. Suppose that &lt;math&gt;ABCD&lt;/math&gt; is a cyclic quadrilateral. Then the sum of any two opposite angles is equal to &lt;math&gt;180^{\text{o}}&lt;/math&gt;. This means that <br /> &lt;math display=&quot;block&quot;&gt; \angle ABC+\angle CDA= 180^{\text{o}}, \quad \text{and}\quad \angle BCD+\angle DAB= 180^{\text{o}}. \qquad\qquad (**)&lt;/math&gt;<br /> <br /> The converse of the previous statement is also true: suppose that &lt;math&gt;ABCD&lt;/math&gt; is a quadrilateral with angles satisfying the equations &lt;math&gt;(**)&lt;/math&gt;. Then &lt;math&gt;ABCD&lt;/math&gt; is a cyclic quadrilateral: we can draw a circle that passes through the four points.<br /> <br /> The Inscribed Angle Theorem and the statements about cyclic quadrilaterals can be used for example in [https://www.math.wisc.edu/talent/sites/default/files/Talent15-4q.pdf 2015-16 Set #4 Problem 5].</div> Valko https://www.math.wisc.edu/wiki/index.php?title=Undergraduate_courses_in_probability&diff=17642 Undergraduate courses in probability 2019-08-15T20:20:06Z <p>Valko: </p> <hr /> <div>'''431 - Introduction to the theory of probability'''<br /> <br /> Math 431 is an introduction to probability theory, the part of mathematics that studies random phenomena. We model simple random experiments mathematically and learn techniques for studying these models. Topics covered include methods of counting (combinatorics), axioms of probability, random variables, the most important discrete and continuous probability distributions, expectations, moment generating functions, conditional probability and conditional expectations, multivariate distributions, Markov's and Chebyshev's inequalities, laws of large numbers, and the central limit theorem.<br /> <br /> Probability theory is ubiquitous in natural science, social science and engineering, so this course can be valuable in conjunction with many different majors. 431 is not a course in statistics. Statistics is a discipline mainly concerned with analyzing and representing data. Probability theory forms the mathematical foundation of statistics, but the two disciplines are separate.<br /> <br /> The course is offered every semester, including the summer. <br /> <br /> ''Prerequisite'': Math 234. <br /> <br /> &lt;span style=&quot;color:#0000FF&quot;&gt; '''Who should take this class?'''&lt;/span&gt; A well rounded undergraduate experience in math should include some probability theory. Math 431 is our introductory probability class with no high level prerequisites. <br /> <br /> <br /> <br /> '''531 - Probability theory'''<br /> <br /> The course is a rigorous introduction to probability theory on an advanced undergraduate level. Only a minimal amount of measure theory is used (in particular, Lebesgue integrals will not be needed). The course gives an introduction to the basics (Kolmogorov axioms, conditional probability and independence, random variables, expectation) and discusses some of the classical results of probability theory with proofs (DeMoivre-Laplace limit theorems, the study of simple random walk on Z, applications of generating functions).<br /> <br /> The course is offered every spring.<br /> <br /> ''Prerequisite'': a proof based analysis course (Math 376, Math 421 or Math 521). <br /> <br /> &lt;span style=&quot;color:#0000FF&quot;&gt; '''Who should take this class?'''&lt;/span&gt; Students who would like to get a rigorous introduction to probability. It could also provide a stepping stone for our 600 level stochastic processes courses. (The course can be taken even after taking Math 431.)<br /> <br /> <br /> <br /> '''605 - Stochastic methods in biology'''<br /> <br /> Math 605 provides an introduction to stochastic processes. It introduces both discrete and continuous time Markov chains, and some aspects of renewal theory. The course focuses on biological applications of these mathematical models including: the Wright-Fischer model, birth and death processes, branching processes, and many models from intracellular biochemistry. This course is similar to Math 632 in content. However, unlike in Math 632, simulation plays a vital role in the study of the requisite processes in Math 605, with Matlab the software package of choice. <br /> <br /> The course is offered every two years in the fall semester. <br /> <br /> ''Prerequisite'': Math 431, a basic knowledge of linear algebra and linear differential equations (e.g. Math 319, Math 340, Math 341)<br /> <br /> &lt;span style=&quot;color:#0000FF&quot;&gt; '''Who should take this class?'''&lt;/span&gt; Anybody who is interested in stochastic processes and would like to learn more about applications in the biosciences, and especially intracellular biochemical processes.<br /> <br /> <br /> '''632 - Introduction to stochastic processes'''<br /> <br /> Math 632 gives an introduction to Markov chains and Markov processes with discrete state spaces and their applications. Particular models studied include birth-death chains, queuing models, random walks and branching processes. Selected topics from renewal theory, martingales, and Brownian motion are also included, but vary from semester to semester to meet the needs of different audiences. <br /> <br /> ''Prerequisite'': Intro to probability (Math 309, 431 or 531)+ a linear algebra or an intro to proofs class (320, 340, 341, 375, 421)<br /> <br /> &lt;span style=&quot;color:#0000FF&quot;&gt; '''Who should take this class?'''&lt;/span&gt; Math 632 is the natural next step after an introductory probability course. It could be useful for an Option 1 math major interested in higher level probability and it is also a great fit for many of our [https://www.math.wisc.edu/undergraduate/option-2-sample-packages Option 2 packages]. <br /> <br /> <br /> <br /> '''635 - Introduction to Brownian motion and stochastic calculus'''<br /> <br /> Math 635 is an introduction to Brownian motion and stochastic calculus without a measure theory prerequisite. Topics touched upon include sample path properties of Brownian motion, Itô stochastic integrals, Itô's formula, stochastic differential equations and their solutions. As an application we will discuss the Black-Scholes formula of mathematical finance.<br /> <br /> The course is offered every two years in the spring semester. <br /> <br /> ''Prerequisite'': Math 521 and Math 632<br /> <br /> &lt;span style=&quot;color:#0000FF&quot;&gt; '''Who should take this class?'''&lt;/span&gt; Anybody with an interest in higher level probability. It is especially useful for those who are planning to study financial math on a graduate level. <br /> <br /> <br /> &lt;!--[[File:Probability_courses_1.jpg|600px]]--&gt;</div> Valko https://www.math.wisc.edu/wiki/index.php?title=Probability&diff=17641 Probability 2019-08-15T20:17:07Z <p>Valko: </p> <hr /> <div>__NOTOC__<br /> <br /> = '''Probability at UW-Madison''' =<br /> <br /> &lt;br&gt;<br /> <br /> == Tenured and tenure-track faculty ==<br /> <br /> [http://www.math.wisc.edu/~anderson/ David Anderson] (Duke, 2005) applied probability, numerical methods, mathematical biology.<br /> <br /> Vadim Gorin<br /> <br /> [http://www.math.wisc.edu/~roch/ Sebastien Roch] (UC Berkeley, 2007) applied probability, mathematical biology, theoretical computer science.<br /> <br /> [http://www.math.wisc.edu/~seppalai/ Timo Seppäläinen] (Minnesota, 1991) motion in a random medium, random growth models, interacting particle systems, large deviation theory.<br /> <br /> [http://www.math.wisc.edu/~hshen3/ Hao Shen] (Princeton, 2013) stochastic partial differential equations, mathematical physics, integrable probability<br /> <br /> [http://www.math.wisc.edu/~valko/ Benedek Valko] (Budapest, 2004) interacting particle systems, random matrices.<br /> <br /> <br /> <br /> <br /> == Emeriti ==<br /> <br /> [http://psoup.math.wisc.edu/kitchen.html David Griffeath] (Cornell, 1976)<br /> <br /> [http://www.math.wisc.edu/~kuelbs Jim Kuelbs] (Minnesota, 1965)<br /> <br /> [http://www.math.wisc.edu/~kurtz Tom Kurtz] (Stanford, 1967)<br /> <br /> Peter Ney (Columbia, 1961)<br /> <br /> Josh Chover (Michigan, 1952)<br /> <br /> == Graduate students ==<br /> <br /> <br /> [http://www.math.wisc.edu/~kehlert/ Kurt Ehlert] <br /> <br /> [http://www.math.wisc.edu/~kang Dae Han Kang]<br /> <br /> [https://sites.google.com/a/wisc.edu/brandon-legried/ Brandon Legried]<br /> <br /> Yun Li<br /> <br /> [http://sites.google.com/a/wisc.edu/tung-nguyen/ Tung Nguyen]<br /> <br /> [http://www.math.wisc.edu/~cyuan25/ Chaojie Yuan]<br /> <br /> <br /> <br /> == [[Probability Seminar]] ==<br /> <br /> Thursdays at 2:25pm, VV901<br /> <br /> ==[[Graduate student reading seminar]]==<br /> <br /> Email list: join-grad_prob_seminar@lists.wisc.edu<br /> <br /> Tuesdays, 2:30pm, 901 Van Vleck<br /> <br /> == [[Probability group timetable]]==<br /> <br /> == [[Undergraduate courses in probability]]==<br /> <br /> == Graduate Courses in Probability ==<br /> <br /> <br /> <br /> '''2019 Fall'''<br /> <br /> Math/Stat 733 Theory of Probability I<br /> <br /> <br /> <br /> <br /> '''2020 Spring'''<br /> <br /> Math/Stat 734 Theory of Probability II <br /> <br /> Math 833 Topics in Probability</div> Valko https://www.math.wisc.edu/wiki/index.php?title=Probability_group_timetable&diff=17585 Probability group timetable 2019-07-25T18:00:15Z <p>Valko: </p> <hr /> <div>2019 Fall<br /> <br /> <br /> {| border=&quot;2&quot;<br /> | ||Monday||Tuesday||Wednesday||Thursday||Friday<br /> |-<br /> | 9-10|| || || || || <br /> |- <br /> | 10-11|| || || || || <br /> |-<br /> | 11-12|| || || || ||<br /> |-<br /> | 12-1|| || || || || <br /> |-<br /> | 1-2|| || || || ||<br /> |-<br /> | 2-3|| || graduate probability seminar (2:25) || || probability seminar (2:25) || <br /> |-<br /> | 3-4|| || || || || <br /> |-<br /> | 4-5|| || || || || colloquium<br /> |-<br /> | 5-6|| || || || ||<br /> |}<br /> <br /> <br /> &lt;!-- <br /> {| border=&quot;2&quot;<br /> | ||Monday||Tuesday||Wednesday||Thursday||Friday<br /> |-<br /> | 9-10|| Timo 431, Kurt 222|| Benedek 431, Sebastien 632, Louis 735 (9:30), Kurt CS719 || Timo 431, Kurt 222 || Benedek 431, Sebastien 632, Louis 735 (9:30), Kurt CS719 || Timo 431<br /> |-<br /> | 10-11|| Kurt 222, Hans 234 || Phil out all day, Kurt 735 || Kurt 222, Hans 234 || Kurt 735 || Phil out all day, Hans 234 <br /> |-<br /> | 11-12|| Jinsu 375, Kurt 222, Hans 846, Christian 846 || Jinsu 375, Kurt 703 || Jinsu 375, Kurt 222, Hans 846, Christian 846 || Jinsu 375, Kurt 703 || Hans 846, Christian 846<br /> |-<br /> | 12-1|| Dave 431, Jinsu 375 || Kurt 703 (12:15) || Dave 431, Jinsu 375 || Kurt 703 (12:15) || Dave 431 <br /> |-<br /> | 1-2|| || Sebastien 632, Benedek 733, Jinsu 801, Hans 234 || || Sebastien 632, Benedek 733, Jinsu 801, Hans 234 ||<br /> |-<br /> | 2-3|| Daniele 431 (2:25) || graduate probability seminar (2:25) || Daniele 431 (2:25) || probability seminar (2:25) || Daniele 431 (2:25)<br /> |-<br /> | 3-4|| || Kurt 222, Hans 234 || || Kurt 222, Hans 234 || <br /> |-<br /> | 4-5|| || || || || colloquium<br /> |-<br /> | 5-6|| || || || ||<br /> |}<br /> --&gt;<br /> <br /> &lt;!--<br /> {| border=&quot;2&quot;<br /> | ||Monday||Tuesday||Wednesday||Thursday||Friday<br /> |-<br /> | 9-10|| Phil out all day || Benedek 531 (9:30)|| || Benedek 531 (9:30) || Phil out all day<br /> |-<br /> | 10-11||Jinsu 722, Louis 431 || || Jinsu 722, Louis 431|| ||Jinsu 722, Louis 431<br /> |-<br /> | 11-12|| || Hans 820 || || Hans 820 ||<br /> |-<br /> | 12-1|| Jinsu 222, Louis 632 || ||Jinsu 222, Louis 632 || || Jinsu 222, Louis 632<br /> |-<br /> | 1-2|| Jinsu 222, Hans 851 || Benedek OH, Hans 843 || Jinsu 222, Hans 851|| Hans 843 ||Jinsu 222, Hans 851<br /> |-<br /> | 2-3|| || graduate probability seminar (2:25) || Louis (Seb) || probability seminar (2:25) ||<br /> |-<br /> | 3-4|| ||Benedek (OH (3:30) || Benedek OH || || <br /> |-<br /> | 4-5|| || || Louis (OH 4:30)|| Louis (OH 4:30)|| colloquium<br /> |-<br /> | 5-6|| || || || ||<br /> |}<br /> --&gt;</div> Valko https://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=17413 Probability Seminar 2019-05-01T14:46:06Z <p>Valko: /* Tuesday , May 7, Van Vleck 901, 2:25pm, Duncan Dauvergne (Toronto) */</p> <hr /> <div>__NOTOC__<br /> <br /> = Spring 2019 =<br /> <br /> &lt;b&gt;Thursdays in 901 Van Vleck Hall at 2:25 PM&lt;/b&gt;, unless otherwise noted. <br /> &lt;b&gt;We usually end for questions at 3:15 PM.&lt;/b&gt;<br /> <br /> If you would like to sign up for the email list to receive seminar announcements then please send an email to <br /> [mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br /> <br /> <br /> <br /> == January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br /> <br /> Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br /> <br /> Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda&gt;0$.<br /> Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br /> <br /> == &lt;span style=&quot;color:red&quot;&gt; Wednesday, February 6 at 4:00pm in Van Vleck 911&lt;/span&gt; , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br /> <br /> Title: '''When particle systems meet PDEs'''<br /> <br /> Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br /> <br /> == February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br /> <br /> Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br /> <br /> Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br /> <br /> == February 14, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], UW-Madison==<br /> <br /> Title: '''Geometry of the corner growth model'''<br /> <br /> Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).<br /> <br /> == February 21, [https://people.kth.se/~holcomb/ Diane Holcomb], KTH ==<br /> <br /> <br /> Title: '''On the centered maximum of the Sine beta process'''<br /> <br /> <br /> Abstract: There has been a great deal or recent work on the asymptotics of the maximum of characteristic polynomials or random matrices. Other recent work studies the analogous result for log-correlated Gaussian fields. Here we will discuss a maximum result for the centered counting function of the Sine beta process. The Sine beta process arises as the local limit in the bulk of a beta-ensemble, and was originally described as the limit of a generalization of the Gaussian Unitary Ensemble by Valko and Virag with an equivalent process identified as a limit of the circular beta ensembles by Killip and Stoiciu. A brief introduction to the Sine process as well as some ideas from the proof of the maximum will be covered. This talk is on joint work with Elliot Paquette.<br /> <br /> == Probability related talk in PDE Geometric Analysis seminar: &lt;br&gt; Monday, February 22 3:30pm to 4:30pm, Van Vleck 901, Xiaoqin Guo, UW-Madison ==<br /> <br /> Title: Quantitative homogenization in a balanced random environment<br /> <br /> Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison).<br /> <br /> == &lt;span style=&quot;color:red&quot;&gt; Wednesday, February 27 at 1:10pm&lt;/span&gt; [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br /> <br /> <br /> &lt;div style=&quot;width:520px;height:50px;border:5px solid black&quot;&gt;<br /> &lt;b&gt;&lt;span style=&quot;color:red&quot;&gt;&amp;emsp; Please note the unusual day and time. <br /> &amp;emsp; &lt;/span&gt;&lt;/b&gt;<br /> &lt;/div&gt;<br /> <br /> Title: '''Functional Limit Laws for Recurrent Excited Random Walks'''<br /> <br /> Abstract:<br /> <br /> Excited random walks (also called cookie random walks) are model for self-interacting random motion where the transition probabilities are dependent on the local time at the current location. While self-interacting random walks are typically very difficult to study, many results for (one-dimensional) excited random walks are remarkably explicit. In particular, one can easily (by hand) calculate a parameter of the model that will determine many features of the random walk: recurrence/transience, non-zero limiting speed, limiting distributions and more. In this talk I will prove functional limit laws for one-dimensional excited random walks that are recurrent. For certain values of the parameters in the model the random walks under diffusive scaling converge to a Brownian motion perturbed at its extremum. This was known previously for the case of excited random walks with boundedly many cookies per site, but we are able to generalize this to excited random walks with periodic cookie stacks. In this more general case, it is much less clear why perturbed Brownian motion should be the correct scaling limit. This is joint work with Elena Kosygina.<br /> <br /> &lt;!-- == March 7, TBA == --&gt;<br /> <br /> &lt;!-- == March 14, TBA == --&gt;<br /> <br /> == March 21, Spring Break, No seminar ==<br /> <br /> == March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br /> <br /> Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br /> <br /> Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the ''character ratio'': <br /> <br /> $$<br /> \text{trace}(\rho(g))/\text{dim}(\rho),<br />$$<br /> <br /> for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant ''rank''. This talk will discuss the notion of rank for $GL_n$ over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br /> <br /> == April 4, [https://www.math.wisc.edu/~pmwood/ Philip Matchett Wood], [http://www.math.wisc.edu/ UW-Madison] ==<br /> <br /> Title: '''Outliers in the spectrum for products of independent random matrices'''<br /> <br /> Abstract: For fixed positive integers m, we consider the product of m independent n by n random matrices with iid entries as in the limit as n tends to infinity. Under suitable assumptions on the entries of each matrix, it is known that the limiting empirical distribution of the eigenvalues is described by the m-th power of the circular law. Moreover, this same limiting distribution continues to hold if each iid random matrix is additively perturbed by a bounded rank deterministic error. However, the bounded rank perturbations may create one or more outlier eigenvalues. We describe the asymptotic location of the outlier eigenvalues, which extends a result of Terence Tao for the case of a single iid matrix. Our methods also allow us to consider several other types of perturbations, including multiplicative perturbations. Joint work with Natalie Coston and Sean O'Rourke.<br /> <br /> == April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Procaccia], [http://www.math.tamu.edu/index.html Texas A&amp;M] ==<br /> <br /> '''Title: Stabilization of Diffusion Limited Aggregation in a Wedge.''' <br /> <br /> Abstract: We prove a discrete Beurling estimate for the harmonic measure in a wedge in $\mathbf{Z}^2$, and use it to show that Diffusion Limited Aggregation (DLA) in a wedge of angle smaller than $\pi/4$ stabilizes. This allows to consider the infinite DLA and questions about the number of arms, growth and dimension. I will present some conjectures and open problems.<br /> <br /> == April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br /> <br /> <br /> Title: '''Large Deviations Theory for Chemical Reaction Networks'''<br /> <br /> Abstract:<br /> The microscopic dynamics of well-stirred networks of chemical reactions are modeled as jump Markov processes. At large volume, one may expect in this framework to have a straightforward application of large deviation theory. This is not at all true, for the jump rates of this class of models are typically neither globally Lipschitz, nor bounded away from zero, with both blowup and absorption as quite possible scenarios. In joint work with Amir Dembo and Jean-Pierre Eckmann, we utilize Lyapunov stability theory to bypass this challenges and to characterize a large class of network topologies that satisfy the full Wentzell-Freidlin theory of asymptotic rates of exit from domains of attraction. Under the assumption of positive recurrence these results also allow for the estimation of transitions times between metastable states of this class of processes.<br /> <br /> == April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br /> <br /> Title: '''Beyond Mean-Field Limits: Local Dynamics on Sparse Graphs'''<br /> <br /> Abstract: Many applications can be modeled as a large system of homogeneous interacting particle systems on a graph in which the infinitesimal evolution of each particle depends on its own state and the empirical distribution of the states of neighboring particles. When the graph is a clique, it is well known that the dynamics of a typical particle converges in the limit, as the number of vertices goes to infinity, to a nonlinear Markov process, often referred to as the McKean-Vlasov or mean-field limit. In this talk, we focus on the complementary case of scaling limits of dynamics on certain sequences of sparse graphs, including regular trees and sparse Erdos-Renyi graphs, and obtain a novel characterization of the dynamics of the neighborhood of a typical particle. This is based on various joint works with Ankan Ganguly, Dan Lacker and Ruoyu Wu.<br /> <br /> == Friday, April 26, Colloquium, Van Vleck 911 from 4pm to 5pm, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br /> <br /> Title: '''Tales of Random Projections'''<br /> <br /> Abstract: The interplay between geometry and probability in high-dimensional spaces is a subject of active research. Classical theorems in probability theory such as the central limit theorem and Cramer’s theorem can be viewed as providing information about certain scalar projections of high-dimensional product measures. In this talk we will describe the behavior of random projections of more general (possibly non-product) high-dimensional measures, which are of interest in diverse fields, ranging from asymptotic convex geometry to high-dimensional statistics. Although the study of (typical) projections of high-dimensional measures dates back to Borel, only recently has a theory begun to emerge, which in particular identifies the role of certain geometric assumptions that lead to better behaved projections. A particular question of interest is to identify what properties of the high-dimensional measure are captured by its lower-dimensional projections. While fluctuations of these projections have been studied over the past decade, we describe more recent work on the tail behavior of multidimensional projections, and associated conditional limit theorems.<br /> <br /> == &lt;span style=&quot;color:red&quot;&gt;'''Tuesday''' &lt;/span&gt;, May 7, Van Vleck 901, 2:25pm, Duncan Dauvergne (Toronto) ==<br /> <br /> <br /> &lt;div style=&quot;width:220px;height:50px;border:5px solid black&quot;&gt;<br /> &lt;b&gt;&lt;span style=&quot;color:red&quot;&gt;&amp;emsp; Please note the unusual day. <br /> &amp;emsp; &lt;/span&gt;&lt;/b&gt;<br /> &lt;/div&gt;<br /> Title: '''The directed landscape'''<br /> <br /> Abstract: I will describe the construction of the full scaling limit of (Brownian) last passage percolation: the directed landscape. The directed landscape can be thought of as a random scale-invariant directed' metric on the plane, and last passage paths converge to directed geodesics in this metric. The directed landscape is expected to be a universal scaling limit for general last passage and random growth models (i.e. TASEP, the KPZ equation, the longest increasing subsequence in a random permutation). Joint work with Janosch Ormann and Balint Virag.<br /> <br /> <br /> &lt;!--<br /> ==&lt;span style=&quot;color:red&quot;&gt; Friday, August 10, 10am, B239 Van Vleck &lt;/span&gt; András Mészáros, Central European University, Budapest ==<br /> <br /> <br /> Title: '''The distribution of sandpile groups of random regular graphs'''<br /> <br /> Abstract:<br /> We study the distribution of the sandpile group of random &lt;math&gt;d&lt;/math&gt;-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the &lt;math&gt;p&lt;/math&gt;-Sylow subgroup of the sandpile group is a given &lt;math&gt;p&lt;/math&gt;-group &lt;math&gt;P&lt;/math&gt;, is proportional to &lt;math&gt;|\operatorname{Aut}(P)|^{-1}&lt;/math&gt;. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br /> <br /> Our results extends a recent theorem of Huang saying that the adjacency matrices of random &lt;math&gt;d&lt;/math&gt;-regular directed graphs are invertible with high probability to the undirected case.<br /> <br /> <br /> ==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br /> <br /> Title: '''Stochastic quantization of Yang-Mills'''<br /> <br /> Abstract:<br /> &quot;Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br /> In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br /> <br /> --&gt;<br /> <br /> == ==<br /> <br /> [[Past Seminars]]</div> Valko https://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=17412 Probability Seminar 2019-05-01T14:45:55Z <p>Valko: /* Tuesday , May 7, Van Vleck 901, 2:25pm, Duncan Dauvergne (Toronto) */</p> <hr /> <div>__NOTOC__<br /> <br /> = Spring 2019 =<br /> <br /> &lt;b&gt;Thursdays in 901 Van Vleck Hall at 2:25 PM&lt;/b&gt;, unless otherwise noted. <br /> &lt;b&gt;We usually end for questions at 3:15 PM.&lt;/b&gt;<br /> <br /> If you would like to sign up for the email list to receive seminar announcements then please send an email to <br /> [mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br /> <br /> <br /> <br /> == January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br /> <br /> Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br /> <br /> Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda&gt;0$.<br /> Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br /> <br /> == &lt;span style=&quot;color:red&quot;&gt; Wednesday, February 6 at 4:00pm in Van Vleck 911&lt;/span&gt; , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br /> <br /> Title: '''When particle systems meet PDEs'''<br /> <br /> Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br /> <br /> == February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br /> <br /> Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br /> <br /> Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br /> <br /> == February 14, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], UW-Madison==<br /> <br /> Title: '''Geometry of the corner growth model'''<br /> <br /> Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).<br /> <br /> == February 21, [https://people.kth.se/~holcomb/ Diane Holcomb], KTH ==<br /> <br /> <br /> Title: '''On the centered maximum of the Sine beta process'''<br /> <br /> <br /> Abstract: There has been a great deal or recent work on the asymptotics of the maximum of characteristic polynomials or random matrices. Other recent work studies the analogous result for log-correlated Gaussian fields. Here we will discuss a maximum result for the centered counting function of the Sine beta process. The Sine beta process arises as the local limit in the bulk of a beta-ensemble, and was originally described as the limit of a generalization of the Gaussian Unitary Ensemble by Valko and Virag with an equivalent process identified as a limit of the circular beta ensembles by Killip and Stoiciu. A brief introduction to the Sine process as well as some ideas from the proof of the maximum will be covered. This talk is on joint work with Elliot Paquette.<br /> <br /> == Probability related talk in PDE Geometric Analysis seminar: &lt;br&gt; Monday, February 22 3:30pm to 4:30pm, Van Vleck 901, Xiaoqin Guo, UW-Madison ==<br /> <br /> Title: Quantitative homogenization in a balanced random environment<br /> <br /> Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison).<br /> <br /> == &lt;span style=&quot;color:red&quot;&gt; Wednesday, February 27 at 1:10pm&lt;/span&gt; [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br /> <br /> <br /> &lt;div style=&quot;width:520px;height:50px;border:5px solid black&quot;&gt;<br /> &lt;b&gt;&lt;span style=&quot;color:red&quot;&gt;&amp;emsp; Please note the unusual day and time. <br /> &amp;emsp; &lt;/span&gt;&lt;/b&gt;<br /> &lt;/div&gt;<br /> <br /> Title: '''Functional Limit Laws for Recurrent Excited Random Walks'''<br /> <br /> Abstract:<br /> <br /> Excited random walks (also called cookie random walks) are model for self-interacting random motion where the transition probabilities are dependent on the local time at the current location. While self-interacting random walks are typically very difficult to study, many results for (one-dimensional) excited random walks are remarkably explicit. In particular, one can easily (by hand) calculate a parameter of the model that will determine many features of the random walk: recurrence/transience, non-zero limiting speed, limiting distributions and more. In this talk I will prove functional limit laws for one-dimensional excited random walks that are recurrent. For certain values of the parameters in the model the random walks under diffusive scaling converge to a Brownian motion perturbed at its extremum. This was known previously for the case of excited random walks with boundedly many cookies per site, but we are able to generalize this to excited random walks with periodic cookie stacks. In this more general case, it is much less clear why perturbed Brownian motion should be the correct scaling limit. This is joint work with Elena Kosygina.<br /> <br /> &lt;!-- == March 7, TBA == --&gt;<br /> <br /> &lt;!-- == March 14, TBA == --&gt;<br /> <br /> == March 21, Spring Break, No seminar ==<br /> <br /> == March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br /> <br /> Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br /> <br /> Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the ''character ratio'': <br /> <br /> $$<br /> \text{trace}(\rho(g))/\text{dim}(\rho),<br />$$<br /> <br /> for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant ''rank''. This talk will discuss the notion of rank for $GL_n$ over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br /> <br /> == April 4, [https://www.math.wisc.edu/~pmwood/ Philip Matchett Wood], [http://www.math.wisc.edu/ UW-Madison] ==<br /> <br /> Title: '''Outliers in the spectrum for products of independent random matrices'''<br /> <br /> Abstract: For fixed positive integers m, we consider the product of m independent n by n random matrices with iid entries as in the limit as n tends to infinity. Under suitable assumptions on the entries of each matrix, it is known that the limiting empirical distribution of the eigenvalues is described by the m-th power of the circular law. Moreover, this same limiting distribution continues to hold if each iid random matrix is additively perturbed by a bounded rank deterministic error. However, the bounded rank perturbations may create one or more outlier eigenvalues. We describe the asymptotic location of the outlier eigenvalues, which extends a result of Terence Tao for the case of a single iid matrix. Our methods also allow us to consider several other types of perturbations, including multiplicative perturbations. Joint work with Natalie Coston and Sean O'Rourke.<br /> <br /> == April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Procaccia], [http://www.math.tamu.edu/index.html Texas A&amp;M] ==<br /> <br /> '''Title: Stabilization of Diffusion Limited Aggregation in a Wedge.''' <br /> <br /> Abstract: We prove a discrete Beurling estimate for the harmonic measure in a wedge in $\mathbf{Z}^2$, and use it to show that Diffusion Limited Aggregation (DLA) in a wedge of angle smaller than $\pi/4$ stabilizes. This allows to consider the infinite DLA and questions about the number of arms, growth and dimension. I will present some conjectures and open problems.<br /> <br /> == April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br /> <br /> <br /> Title: '''Large Deviations Theory for Chemical Reaction Networks'''<br /> <br /> Abstract:<br /> The microscopic dynamics of well-stirred networks of chemical reactions are modeled as jump Markov processes. At large volume, one may expect in this framework to have a straightforward application of large deviation theory. This is not at all true, for the jump rates of this class of models are typically neither globally Lipschitz, nor bounded away from zero, with both blowup and absorption as quite possible scenarios. In joint work with Amir Dembo and Jean-Pierre Eckmann, we utilize Lyapunov stability theory to bypass this challenges and to characterize a large class of network topologies that satisfy the full Wentzell-Freidlin theory of asymptotic rates of exit from domains of attraction. Under the assumption of positive recurrence these results also allow for the estimation of transitions times between metastable states of this class of processes.<br /> <br /> == April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br /> <br /> Title: '''Beyond Mean-Field Limits: Local Dynamics on Sparse Graphs'''<br /> <br /> Abstract: Many applications can be modeled as a large system of homogeneous interacting particle systems on a graph in which the infinitesimal evolution of each particle depends on its own state and the empirical distribution of the states of neighboring particles. When the graph is a clique, it is well known that the dynamics of a typical particle converges in the limit, as the number of vertices goes to infinity, to a nonlinear Markov process, often referred to as the McKean-Vlasov or mean-field limit. In this talk, we focus on the complementary case of scaling limits of dynamics on certain sequences of sparse graphs, including regular trees and sparse Erdos-Renyi graphs, and obtain a novel characterization of the dynamics of the neighborhood of a typical particle. This is based on various joint works with Ankan Ganguly, Dan Lacker and Ruoyu Wu.<br /> <br /> == Friday, April 26, Colloquium, Van Vleck 911 from 4pm to 5pm, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br /> <br /> Title: '''Tales of Random Projections'''<br /> <br /> Abstract: The interplay between geometry and probability in high-dimensional spaces is a subject of active research. Classical theorems in probability theory such as the central limit theorem and Cramer’s theorem can be viewed as providing information about certain scalar projections of high-dimensional product measures. In this talk we will describe the behavior of random projections of more general (possibly non-product) high-dimensional measures, which are of interest in diverse fields, ranging from asymptotic convex geometry to high-dimensional statistics. Although the study of (typical) projections of high-dimensional measures dates back to Borel, only recently has a theory begun to emerge, which in particular identifies the role of certain geometric assumptions that lead to better behaved projections. A particular question of interest is to identify what properties of the high-dimensional measure are captured by its lower-dimensional projections. While fluctuations of these projections have been studied over the past decade, we describe more recent work on the tail behavior of multidimensional projections, and associated conditional limit theorems.<br /> <br /> == &lt;span style=&quot;color:red&quot;&gt;'''Tuesday''' &lt;/span&gt;, May 7, Van Vleck 901, 2:25pm, Duncan Dauvergne (Toronto) ==<br /> <br /> <br /> &lt;div style=&quot;width:220px;height:50px;border:5px solid black&quot;&gt;<br /> &lt;b&gt;&lt;span style=&quot;color:red&quot;&gt;&amp;emsp; Please note the unusual day. <br /> &amp;emsp; &lt;/span&gt;&lt;/b&gt;<br /> &lt;/div&gt;<br /> Title: The directed landscape<br /> <br /> Abstract: I will describe the construction of the full scaling limit of (Brownian) last passage percolation: the directed landscape. The directed landscape can be thought of as a random scale-invariant directed' metric on the plane, and last passage paths converge to directed geodesics in this metric. The directed landscape is expected to be a universal scaling limit for general last passage and random growth models (i.e. TASEP, the KPZ equation, the longest increasing subsequence in a random permutation). Joint work with Janosch Ormann and Balint Virag.<br /> <br /> <br /> &lt;!--<br /> ==&lt;span style=&quot;color:red&quot;&gt; Friday, August 10, 10am, B239 Van Vleck &lt;/span&gt; András Mészáros, Central European University, Budapest ==<br /> <br /> <br /> Title: '''The distribution of sandpile groups of random regular graphs'''<br /> <br /> Abstract:<br /> We study the distribution of the sandpile group of random &lt;math&gt;d&lt;/math&gt;-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the &lt;math&gt;p&lt;/math&gt;-Sylow subgroup of the sandpile group is a given &lt;math&gt;p&lt;/math&gt;-group &lt;math&gt;P&lt;/math&gt;, is proportional to &lt;math&gt;|\operatorname{Aut}(P)|^{-1}&lt;/math&gt;. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br /> <br /> Our results extends a recent theorem of Huang saying that the adjacency matrices of random &lt;math&gt;d&lt;/math&gt;-regular directed graphs are invertible with high probability to the undirected case.<br /> <br /> <br /> ==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br /> <br /> Title: '''Stochastic quantization of Yang-Mills'''<br /> <br /> Abstract:<br /> &quot;Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br /> In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br /> <br /> --&gt;<br /> <br /> == ==<br /> <br /> [[Past Seminars]]</div> Valko https://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=17344 Probability Seminar 2019-04-18T18:07:12Z <p>Valko: /* May 2, TBA */</p> <hr /> <div>__NOTOC__<br /> <br /> = Spring 2019 =<br /> <br /> &lt;b&gt;Thursdays in 901 Van Vleck Hall at 2:25 PM&lt;/b&gt;, unless otherwise noted. <br /> &lt;b&gt;We usually end for questions at 3:15 PM.&lt;/b&gt;<br /> <br /> If you would like to sign up for the email list to receive seminar announcements then please send an email to <br /> [mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br /> <br /> <br /> <br /> == January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br /> <br /> Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br /> <br /> Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda&gt;0$.<br /> Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br /> <br /> == &lt;span style=&quot;color:red&quot;&gt; Wednesday, February 6 at 4:00pm in Van Vleck 911&lt;/span&gt; , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br /> <br /> Title: '''When particle systems meet PDEs'''<br /> <br /> Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br /> <br /> == February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br /> <br /> Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br /> <br /> Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br /> <br /> == February 14, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], UW-Madison==<br /> <br /> Title: '''Geometry of the corner growth model'''<br /> <br /> Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).<br /> <br /> == February 21, [https://people.kth.se/~holcomb/ Diane Holcomb], KTH ==<br /> <br /> <br /> Title: '''On the centered maximum of the Sine beta process'''<br /> <br /> <br /> Abstract: There has been a great deal or recent work on the asymptotics of the maximum of characteristic polynomials or random matrices. Other recent work studies the analogous result for log-correlated Gaussian fields. Here we will discuss a maximum result for the centered counting function of the Sine beta process. The Sine beta process arises as the local limit in the bulk of a beta-ensemble, and was originally described as the limit of a generalization of the Gaussian Unitary Ensemble by Valko and Virag with an equivalent process identified as a limit of the circular beta ensembles by Killip and Stoiciu. A brief introduction to the Sine process as well as some ideas from the proof of the maximum will be covered. This talk is on joint work with Elliot Paquette.<br /> <br /> == Probability related talk in PDE Geometric Analysis seminar: &lt;br&gt; Monday, February 22 3:30pm to 4:30pm, Van Vleck 901, Xiaoqin Guo, UW-Madison ==<br /> <br /> Title: Quantitative homogenization in a balanced random environment<br /> <br /> Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison).<br /> <br /> == &lt;span style=&quot;color:red&quot;&gt; Wednesday, February 27 at 1:10pm&lt;/span&gt; [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br /> <br /> <br /> &lt;div style=&quot;width:520px;height:50px;border:5px solid black&quot;&gt;<br /> &lt;b&gt;&lt;span style=&quot;color:red&quot;&gt;&amp;emsp; Please note the unusual day and time. <br /> &amp;emsp; &lt;/span&gt;&lt;/b&gt;<br /> &lt;/div&gt;<br /> <br /> Title: '''Functional Limit Laws for Recurrent Excited Random Walks'''<br /> <br /> Abstract:<br /> <br /> Excited random walks (also called cookie random walks) are model for self-interacting random motion where the transition probabilities are dependent on the local time at the current location. While self-interacting random walks are typically very difficult to study, many results for (one-dimensional) excited random walks are remarkably explicit. In particular, one can easily (by hand) calculate a parameter of the model that will determine many features of the random walk: recurrence/transience, non-zero limiting speed, limiting distributions and more. In this talk I will prove functional limit laws for one-dimensional excited random walks that are recurrent. For certain values of the parameters in the model the random walks under diffusive scaling converge to a Brownian motion perturbed at its extremum. This was known previously for the case of excited random walks with boundedly many cookies per site, but we are able to generalize this to excited random walks with periodic cookie stacks. In this more general case, it is much less clear why perturbed Brownian motion should be the correct scaling limit. This is joint work with Elena Kosygina.<br /> <br /> &lt;!-- == March 7, TBA == --&gt;<br /> <br /> &lt;!-- == March 14, TBA == --&gt;<br /> <br /> == March 21, Spring Break, No seminar ==<br /> <br /> == March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br /> <br /> Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br /> <br /> Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the ''character ratio'': <br /> <br /> $$<br /> \text{trace}(\rho(g))/\text{dim}(\rho),<br />$$<br /> <br /> for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant ''rank''. This talk will discuss the notion of rank for $GL_n$ over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br /> <br /> == April 4, [https://www.math.wisc.edu/~pmwood/ Philip Matchett Wood], [http://www.math.wisc.edu/ UW-Madison] ==<br /> <br /> Title: '''Outliers in the spectrum for products of independent random matrices'''<br /> <br /> Abstract: For fixed positive integers m, we consider the product of m independent n by n random matrices with iid entries as in the limit as n tends to infinity. Under suitable assumptions on the entries of each matrix, it is known that the limiting empirical distribution of the eigenvalues is described by the m-th power of the circular law. Moreover, this same limiting distribution continues to hold if each iid random matrix is additively perturbed by a bounded rank deterministic error. However, the bounded rank perturbations may create one or more outlier eigenvalues. We describe the asymptotic location of the outlier eigenvalues, which extends a result of Terence Tao for the case of a single iid matrix. Our methods also allow us to consider several other types of perturbations, including multiplicative perturbations. Joint work with Natalie Coston and Sean O'Rourke.<br /> <br /> == April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Procaccia], [http://www.math.tamu.edu/index.html Texas A&amp;M] ==<br /> <br /> '''Title: Stabilization of Diffusion Limited Aggregation in a Wedge.''' <br /> <br /> Abstract: We prove a discrete Beurling estimate for the harmonic measure in a wedge in $\mathbf{Z}^2$, and use it to show that Diffusion Limited Aggregation (DLA) in a wedge of angle smaller than $\pi/4$ stabilizes. This allows to consider the infinite DLA and questions about the number of arms, growth and dimension. I will present some conjectures and open problems.<br /> <br /> == April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br /> <br /> <br /> Title: '''Large Deviations Theory for Chemical Reaction Networks'''<br /> <br /> Abstract:<br /> The microscopic dynamics of well-stirred networks of chemical reactions are modeled as jump Markov processes. At large volume, one may expect in this framework to have a straightforward application of large deviation theory. This is not at all true, for the jump rates of this class of models are typically neither globally Lipschitz, nor bounded away from zero, with both blowup and absorption as quite possible scenarios. In joint work with Amir Dembo and Jean-Pierre Eckmann, we utilize Lyapunov stability theory to bypass this challenges and to characterize a large class of network topologies that satisfy the full Wentzell-Freidlin theory of asymptotic rates of exit from domains of attraction. Under the assumption of positive recurrence these results also allow for the estimation of transitions times between metastable states of this class of processes.<br /> <br /> == April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br /> <br /> == April 26, Colloquium, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br /> <br /> Title: '''Tales of Random Projections'''<br /> <br /> Abstract: The interplay between geometry and probability in high-dimensional spaces is a subject of active research. Classical theorems in probability theory such as the central limit theorem and Cramer’s theorem can be viewed as providing information about certain scalar projections of high-dimensional product measures. In this talk we will describe the behavior of random projections of more general (possibly non-product) high-dimensional measures, which are of interest in diverse fields, ranging from asymptotic convex geometry to high-dimensional statistics. Although the study of (typical) projections of high-dimensional measures dates back to Borel, only recently has a theory begun to emerge, which in particular identifies the role of certain geometric assumptions that lead to better behaved projections. A particular question of interest is to identify what properties of the high-dimensional measure are captured by its lower-dimensional projections. While fluctuations of these projections have been studied over the past decade, we describe more recent work on the tail behavior of multidimensional projections, and associated conditional limit theorems.<br /> <br /> == May 7, '''Tuesday''', Duncan Dauvergne (Toronto) ==<br /> <br /> <br /> &lt;!--<br /> ==&lt;span style=&quot;color:red&quot;&gt; Friday, August 10, 10am, B239 Van Vleck &lt;/span&gt; András Mészáros, Central European University, Budapest ==<br /> <br /> <br /> Title: '''The distribution of sandpile groups of random regular graphs'''<br /> <br /> Abstract:<br /> We study the distribution of the sandpile group of random &lt;math&gt;d&lt;/math&gt;-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the &lt;math&gt;p&lt;/math&gt;-Sylow subgroup of the sandpile group is a given &lt;math&gt;p&lt;/math&gt;-group &lt;math&gt;P&lt;/math&gt;, is proportional to &lt;math&gt;|\operatorname{Aut}(P)|^{-1}&lt;/math&gt;. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br /> <br /> Our results extends a recent theorem of Huang saying that the adjacency matrices of random &lt;math&gt;d&lt;/math&gt;-regular directed graphs are invertible with high probability to the undirected case.<br /> <br /> <br /> ==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br /> <br /> Title: '''Stochastic quantization of Yang-Mills'''<br /> <br /> Abstract:<br /> &quot;Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br /> In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br /> <br /> --&gt;<br /> <br /> == ==<br /> <br /> [[Past Seminars]]</div> Valko https://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=17343 Probability Seminar 2019-04-18T18:06:12Z <p>Valko: /* April 26, Colloquium, Kavita Ramanan, Brown */</p> <hr /> <div>__NOTOC__<br /> <br /> = Spring 2019 =<br /> <br /> &lt;b&gt;Thursdays in 901 Van Vleck Hall at 2:25 PM&lt;/b&gt;, unless otherwise noted. <br /> &lt;b&gt;We usually end for questions at 3:15 PM.&lt;/b&gt;<br /> <br /> If you would like to sign up for the email list to receive seminar announcements then please send an email to <br /> [mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br /> <br /> <br /> <br /> == January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br /> <br /> Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br /> <br /> Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda&gt;0$.<br /> Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br /> <br /> == &lt;span style=&quot;color:red&quot;&gt; Wednesday, February 6 at 4:00pm in Van Vleck 911&lt;/span&gt; , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br /> <br /> Title: '''When particle systems meet PDEs'''<br /> <br /> Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br /> <br /> == February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br /> <br /> Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br /> <br /> Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br /> <br /> == February 14, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], UW-Madison==<br /> <br /> Title: '''Geometry of the corner growth model'''<br /> <br /> Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).<br /> <br /> == February 21, [https://people.kth.se/~holcomb/ Diane Holcomb], KTH ==<br /> <br /> <br /> Title: '''On the centered maximum of the Sine beta process'''<br /> <br /> <br /> Abstract: There has been a great deal or recent work on the asymptotics of the maximum of characteristic polynomials or random matrices. Other recent work studies the analogous result for log-correlated Gaussian fields. Here we will discuss a maximum result for the centered counting function of the Sine beta process. The Sine beta process arises as the local limit in the bulk of a beta-ensemble, and was originally described as the limit of a generalization of the Gaussian Unitary Ensemble by Valko and Virag with an equivalent process identified as a limit of the circular beta ensembles by Killip and Stoiciu. A brief introduction to the Sine process as well as some ideas from the proof of the maximum will be covered. This talk is on joint work with Elliot Paquette.<br /> <br /> == Probability related talk in PDE Geometric Analysis seminar: &lt;br&gt; Monday, February 22 3:30pm to 4:30pm, Van Vleck 901, Xiaoqin Guo, UW-Madison ==<br /> <br /> Title: Quantitative homogenization in a balanced random environment<br /> <br /> Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison).<br /> <br /> == &lt;span style=&quot;color:red&quot;&gt; Wednesday, February 27 at 1:10pm&lt;/span&gt; [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br /> <br /> <br /> &lt;div style=&quot;width:520px;height:50px;border:5px solid black&quot;&gt;<br /> &lt;b&gt;&lt;span style=&quot;color:red&quot;&gt;&amp;emsp; Please note the unusual day and time. <br /> &amp;emsp; &lt;/span&gt;&lt;/b&gt;<br /> &lt;/div&gt;<br /> <br /> Title: '''Functional Limit Laws for Recurrent Excited Random Walks'''<br /> <br /> Abstract:<br /> <br /> Excited random walks (also called cookie random walks) are model for self-interacting random motion where the transition probabilities are dependent on the local time at the current location. While self-interacting random walks are typically very difficult to study, many results for (one-dimensional) excited random walks are remarkably explicit. In particular, one can easily (by hand) calculate a parameter of the model that will determine many features of the random walk: recurrence/transience, non-zero limiting speed, limiting distributions and more. In this talk I will prove functional limit laws for one-dimensional excited random walks that are recurrent. For certain values of the parameters in the model the random walks under diffusive scaling converge to a Brownian motion perturbed at its extremum. This was known previously for the case of excited random walks with boundedly many cookies per site, but we are able to generalize this to excited random walks with periodic cookie stacks. In this more general case, it is much less clear why perturbed Brownian motion should be the correct scaling limit. This is joint work with Elena Kosygina.<br /> <br /> &lt;!-- == March 7, TBA == --&gt;<br /> <br /> &lt;!-- == March 14, TBA == --&gt;<br /> <br /> == March 21, Spring Break, No seminar ==<br /> <br /> == March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br /> <br /> Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br /> <br /> Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the ''character ratio'': <br /> <br /> $$<br /> \text{trace}(\rho(g))/\text{dim}(\rho),<br />$$<br /> <br /> for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant ''rank''. This talk will discuss the notion of rank for $GL_n$ over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br /> <br /> == April 4, [https://www.math.wisc.edu/~pmwood/ Philip Matchett Wood], [http://www.math.wisc.edu/ UW-Madison] ==<br /> <br /> Title: '''Outliers in the spectrum for products of independent random matrices'''<br /> <br /> Abstract: For fixed positive integers m, we consider the product of m independent n by n random matrices with iid entries as in the limit as n tends to infinity. Under suitable assumptions on the entries of each matrix, it is known that the limiting empirical distribution of the eigenvalues is described by the m-th power of the circular law. Moreover, this same limiting distribution continues to hold if each iid random matrix is additively perturbed by a bounded rank deterministic error. However, the bounded rank perturbations may create one or more outlier eigenvalues. We describe the asymptotic location of the outlier eigenvalues, which extends a result of Terence Tao for the case of a single iid matrix. Our methods also allow us to consider several other types of perturbations, including multiplicative perturbations. Joint work with Natalie Coston and Sean O'Rourke.<br /> <br /> == April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Procaccia], [http://www.math.tamu.edu/index.html Texas A&amp;M] ==<br /> <br /> '''Title: Stabilization of Diffusion Limited Aggregation in a Wedge.''' <br /> <br /> Abstract: We prove a discrete Beurling estimate for the harmonic measure in a wedge in $\mathbf{Z}^2$, and use it to show that Diffusion Limited Aggregation (DLA) in a wedge of angle smaller than $\pi/4$ stabilizes. This allows to consider the infinite DLA and questions about the number of arms, growth and dimension. I will present some conjectures and open problems.<br /> <br /> == April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br /> <br /> <br /> Title: '''Large Deviations Theory for Chemical Reaction Networks'''<br /> <br /> Abstract:<br /> The microscopic dynamics of well-stirred networks of chemical reactions are modeled as jump Markov processes. At large volume, one may expect in this framework to have a straightforward application of large deviation theory. This is not at all true, for the jump rates of this class of models are typically neither globally Lipschitz, nor bounded away from zero, with both blowup and absorption as quite possible scenarios. In joint work with Amir Dembo and Jean-Pierre Eckmann, we utilize Lyapunov stability theory to bypass this challenges and to characterize a large class of network topologies that satisfy the full Wentzell-Freidlin theory of asymptotic rates of exit from domains of attraction. Under the assumption of positive recurrence these results also allow for the estimation of transitions times between metastable states of this class of processes.<br /> <br /> == April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br /> <br /> == April 26, Colloquium, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br /> <br /> Title: '''Tales of Random Projections'''<br /> <br /> Abstract: The interplay between geometry and probability in high-dimensional spaces is a subject of active research. Classical theorems in probability theory such as the central limit theorem and Cramer’s theorem can be viewed as providing information about certain scalar projections of high-dimensional product measures. In this talk we will describe the behavior of random projections of more general (possibly non-product) high-dimensional measures, which are of interest in diverse fields, ranging from asymptotic convex geometry to high-dimensional statistics. Although the study of (typical) projections of high-dimensional measures dates back to Borel, only recently has a theory begun to emerge, which in particular identifies the role of certain geometric assumptions that lead to better behaved projections. A particular question of interest is to identify what properties of the high-dimensional measure are captured by its lower-dimensional projections. While fluctuations of these projections have been studied over the past decade, we describe more recent work on the tail behavior of multidimensional projections, and associated conditional limit theorems.<br /> <br /> == May 2, TBA ==<br /> <br /> <br /> &lt;!--<br /> ==&lt;span style=&quot;color:red&quot;&gt; Friday, August 10, 10am, B239 Van Vleck &lt;/span&gt; András Mészáros, Central European University, Budapest ==<br /> <br /> <br /> Title: '''The distribution of sandpile groups of random regular graphs'''<br /> <br /> Abstract:<br /> We study the distribution of the sandpile group of random &lt;math&gt;d&lt;/math&gt;-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the &lt;math&gt;p&lt;/math&gt;-Sylow subgroup of the sandpile group is a given &lt;math&gt;p&lt;/math&gt;-group &lt;math&gt;P&lt;/math&gt;, is proportional to &lt;math&gt;|\operatorname{Aut}(P)|^{-1}&lt;/math&gt;. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br /> <br /> Our results extends a recent theorem of Huang saying that the adjacency matrices of random &lt;math&gt;d&lt;/math&gt;-regular directed graphs are invertible with high probability to the undirected case.<br /> <br /> <br /> ==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br /> <br /> Title: '''Stochastic quantization of Yang-Mills'''<br /> <br /> Abstract:<br /> &quot;Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br /> In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br /> <br /> --&gt;<br /> <br /> == ==<br /> <br /> [[Past Seminars]]</div> Valko https://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=17342 Probability Seminar 2019-04-18T18:05:59Z <p>Valko: /* April 26, Colloquium, Kavita Ramanan, Brown */</p> <hr /> <div>__NOTOC__<br /> <br /> = Spring 2019 =<br /> <br /> &lt;b&gt;Thursdays in 901 Van Vleck Hall at 2:25 PM&lt;/b&gt;, unless otherwise noted. <br /> &lt;b&gt;We usually end for questions at 3:15 PM.&lt;/b&gt;<br /> <br /> If you would like to sign up for the email list to receive seminar announcements then please send an email to <br /> [mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br /> <br /> <br /> <br /> == January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br /> <br /> Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br /> <br /> Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda&gt;0$.<br /> Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br /> <br /> == &lt;span style=&quot;color:red&quot;&gt; Wednesday, February 6 at 4:00pm in Van Vleck 911&lt;/span&gt; , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br /> <br /> Title: '''When particle systems meet PDEs'''<br /> <br /> Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br /> <br /> == February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br /> <br /> Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br /> <br /> Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br /> <br /> == February 14, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], UW-Madison==<br /> <br /> Title: '''Geometry of the corner growth model'''<br /> <br /> Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).<br /> <br /> == February 21, [https://people.kth.se/~holcomb/ Diane Holcomb], KTH ==<br /> <br /> <br /> Title: '''On the centered maximum of the Sine beta process'''<br /> <br /> <br /> Abstract: There has been a great deal or recent work on the asymptotics of the maximum of characteristic polynomials or random matrices. Other recent work studies the analogous result for log-correlated Gaussian fields. Here we will discuss a maximum result for the centered counting function of the Sine beta process. The Sine beta process arises as the local limit in the bulk of a beta-ensemble, and was originally described as the limit of a generalization of the Gaussian Unitary Ensemble by Valko and Virag with an equivalent process identified as a limit of the circular beta ensembles by Killip and Stoiciu. A brief introduction to the Sine process as well as some ideas from the proof of the maximum will be covered. This talk is on joint work with Elliot Paquette.<br /> <br /> == Probability related talk in PDE Geometric Analysis seminar: &lt;br&gt; Monday, February 22 3:30pm to 4:30pm, Van Vleck 901, Xiaoqin Guo, UW-Madison ==<br /> <br /> Title: Quantitative homogenization in a balanced random environment<br /> <br /> Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison).<br /> <br /> == &lt;span style=&quot;color:red&quot;&gt; Wednesday, February 27 at 1:10pm&lt;/span&gt; [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br /> <br /> <br /> &lt;div style=&quot;width:520px;height:50px;border:5px solid black&quot;&gt;<br /> &lt;b&gt;&lt;span style=&quot;color:red&quot;&gt;&amp;emsp; Please note the unusual day and time. <br /> &amp;emsp; &lt;/span&gt;&lt;/b&gt;<br /> &lt;/div&gt;<br /> <br /> Title: '''Functional Limit Laws for Recurrent Excited Random Walks'''<br /> <br /> Abstract:<br /> <br /> Excited random walks (also called cookie random walks) are model for self-interacting random motion where the transition probabilities are dependent on the local time at the current location. While self-interacting random walks are typically very difficult to study, many results for (one-dimensional) excited random walks are remarkably explicit. In particular, one can easily (by hand) calculate a parameter of the model that will determine many features of the random walk: recurrence/transience, non-zero limiting speed, limiting distributions and more. In this talk I will prove functional limit laws for one-dimensional excited random walks that are recurrent. For certain values of the parameters in the model the random walks under diffusive scaling converge to a Brownian motion perturbed at its extremum. This was known previously for the case of excited random walks with boundedly many cookies per site, but we are able to generalize this to excited random walks with periodic cookie stacks. In this more general case, it is much less clear why perturbed Brownian motion should be the correct scaling limit. This is joint work with Elena Kosygina.<br /> <br /> &lt;!-- == March 7, TBA == --&gt;<br /> <br /> &lt;!-- == March 14, TBA == --&gt;<br /> <br /> == March 21, Spring Break, No seminar ==<br /> <br /> == March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br /> <br /> Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br /> <br /> Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the ''character ratio'': <br /> <br /> $$<br /> \text{trace}(\rho(g))/\text{dim}(\rho),<br />$$<br /> <br /> for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant ''rank''. This talk will discuss the notion of rank for $GL_n$ over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br /> <br /> == April 4, [https://www.math.wisc.edu/~pmwood/ Philip Matchett Wood], [http://www.math.wisc.edu/ UW-Madison] ==<br /> <br /> Title: '''Outliers in the spectrum for products of independent random matrices'''<br /> <br /> Abstract: For fixed positive integers m, we consider the product of m independent n by n random matrices with iid entries as in the limit as n tends to infinity. Under suitable assumptions on the entries of each matrix, it is known that the limiting empirical distribution of the eigenvalues is described by the m-th power of the circular law. Moreover, this same limiting distribution continues to hold if each iid random matrix is additively perturbed by a bounded rank deterministic error. However, the bounded rank perturbations may create one or more outlier eigenvalues. We describe the asymptotic location of the outlier eigenvalues, which extends a result of Terence Tao for the case of a single iid matrix. Our methods also allow us to consider several other types of perturbations, including multiplicative perturbations. Joint work with Natalie Coston and Sean O'Rourke.<br /> <br /> == April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Procaccia], [http://www.math.tamu.edu/index.html Texas A&amp;M] ==<br /> <br /> '''Title: Stabilization of Diffusion Limited Aggregation in a Wedge.''' <br /> <br /> Abstract: We prove a discrete Beurling estimate for the harmonic measure in a wedge in $\mathbf{Z}^2$, and use it to show that Diffusion Limited Aggregation (DLA) in a wedge of angle smaller than $\pi/4$ stabilizes. This allows to consider the infinite DLA and questions about the number of arms, growth and dimension. I will present some conjectures and open problems.<br /> <br /> == April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br /> <br /> <br /> Title: '''Large Deviations Theory for Chemical Reaction Networks'''<br /> <br /> Abstract:<br /> The microscopic dynamics of well-stirred networks of chemical reactions are modeled as jump Markov processes. At large volume, one may expect in this framework to have a straightforward application of large deviation theory. This is not at all true, for the jump rates of this class of models are typically neither globally Lipschitz, nor bounded away from zero, with both blowup and absorption as quite possible scenarios. In joint work with Amir Dembo and Jean-Pierre Eckmann, we utilize Lyapunov stability theory to bypass this challenges and to characterize a large class of network topologies that satisfy the full Wentzell-Freidlin theory of asymptotic rates of exit from domains of attraction. Under the assumption of positive recurrence these results also allow for the estimation of transitions times between metastable states of this class of processes.<br /> <br /> == April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br /> <br /> == April 26, Colloquium, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br /> <br /> Title: Tales of Random Projections<br /> <br /> Abstract: The interplay between geometry and probability in high-dimensional spaces is a subject of active research. Classical theorems in probability theory such as the central limit theorem and Cramer’s theorem can be viewed as providing information about certain scalar projections of high-dimensional product measures. In this talk we will describe the behavior of random projections of more general (possibly non-product) high-dimensional measures, which are of interest in diverse fields, ranging from asymptotic convex geometry to high-dimensional statistics. Although the study of (typical) projections of high-dimensional measures dates back to Borel, only recently has a theory begun to emerge, which in particular identifies the role of certain geometric assumptions that lead to better behaved projections. A particular question of interest is to identify what properties of the high-dimensional measure are captured by its lower-dimensional projections. While fluctuations of these projections have been studied over the past decade, we describe more recent work on the tail behavior of multidimensional projections, and associated conditional limit theorems.<br /> <br /> == May 2, TBA ==<br /> <br /> <br /> &lt;!--<br /> ==&lt;span style=&quot;color:red&quot;&gt; Friday, August 10, 10am, B239 Van Vleck &lt;/span&gt; András Mészáros, Central European University, Budapest ==<br /> <br /> <br /> Title: '''The distribution of sandpile groups of random regular graphs'''<br /> <br /> Abstract:<br /> We study the distribution of the sandpile group of random &lt;math&gt;d&lt;/math&gt;-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the &lt;math&gt;p&lt;/math&gt;-Sylow subgroup of the sandpile group is a given &lt;math&gt;p&lt;/math&gt;-group &lt;math&gt;P&lt;/math&gt;, is proportional to &lt;math&gt;|\operatorname{Aut}(P)|^{-1}&lt;/math&gt;. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br /> <br /> Our results extends a recent theorem of Huang saying that the adjacency matrices of random &lt;math&gt;d&lt;/math&gt;-regular directed graphs are invertible with high probability to the undirected case.<br /> <br /> <br /> ==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br /> <br /> Title: '''Stochastic quantization of Yang-Mills'''<br /> <br /> Abstract:<br /> &quot;Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br /> In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br /> <br /> --&gt;<br /> <br /> == ==<br /> <br /> [[Past Seminars]]</div> Valko https://www.math.wisc.edu/wiki/index.php?title=Graduate_student_reading_seminar&diff=16989 Graduate student reading seminar 2019-02-18T23:57:30Z <p>Valko: /* 2019 Spring */</p> <hr /> <div>(... in probability)<br /> <br /> <br /> Email list: join-grad_prob_seminar@lists.wisc.edu<br /> <br /> ==2019 Spring==<br /> <br /> Tuesday 2:30pm, 901 Van Vleck<br /> <br /> 2/5: Timo<br /> <br /> 2/12, 2/19: Evan<br /> <br /> 2/26, 3/5: Chaojie<br /> <br /> 3/12, 3/26: Kurt<br /> <br /> 4/2, 4/9: Yu<br /> <br /> 4/16, 4/23: Max<br /> <br /> 4/30, 5/7: Xiao<br /> <br /> ==2018 Fall==<br /> <br /> Tuesday 2:30pm, 901 Van Vleck<br /> <br /> <br /> The topic this semester is large deviation theory. Send me (BV) an email, if you want access to the shared Box folder with some reading material. <br /> <br /> <br /> 9/25, 10/2: Dae Han<br /> <br /> 10/9, 10/16: Kurt<br /> <br /> 10/23, 10/30: Stephen Davis<br /> <br /> 11/6, 11/13: Brandon Legried <br /> <br /> 11/20, 11/27: Shuqi Yu<br /> <br /> 12/4, 12/11: Yun Li<br /> <br /> ==2018 Spring==<br /> <br /> Tuesday 2:30pm, B135 Van Vleck<br /> <br /> <br /> Preliminary schedule:<br /> <br /> 2/20, 2/27: Yun<br /> <br /> 3/6, 3/13: Greg<br /> <br /> 3/20, 4/3: Yu<br /> <br /> 4/10, 4/17: Shuqi<br /> <br /> 4/24, 5/1: Tony<br /> <br /> ==2017 Fall==<br /> <br /> Tuesday 2:30pm, 214 Ingraham Hall<br /> <br /> <br /> Preliminary schedule: <br /> <br /> 9/26, 10/3: Hans<br /> <br /> 10/10, 10/17: Guo<br /> <br /> 10/24, 10/31: Chaoji<br /> <br /> 11/7, 11/14: Yun <br /> <br /> 11/21, 11/28: Kurt<br /> <br /> 12/5, 12/12: Christian<br /> <br /> <br /> <br /> <br /> ==2017 Spring==<br /> <br /> Tuesday 2:25pm, B211<br /> <br /> 1/31, 2/7: Fan<br /> <br /> I will talk about the Hanson-Wright inequality, which is a large deviation estimate for random variable of the form X^* A X, where X is a random vector with independent subgaussian entries and A is an arbitrary deterministic matrix. In the first talk, I will present a beautiful proof given by Mark Rudelson and Roman Vershynin. In the second talk, I will talk about some applications of this inequality.<br /> <br /> Reference: M. Rudelson and R. Vershynin, Hanson-Wright inequality and sub-gaussian concentration, Electron. Commun. Probab. Volume 18 (2013).<br /> <br /> 3/7, 3/14 : Jinsu<br /> <br /> Title : Donsker's Theorem and its application.<br /> Donsker's Theorem roughly says normalized random walk with linear interpolation on time interval [0,1] weakly converges to the Brownian motion B[0,1] in C([0,1]). It is sometimes called Donsker's invariance principle or the functional central limit theorem. I will show main ideas for the proof of this theorem tomorrow and show a couple of applications in my 2nd talk.<br /> <br /> Reference : https://www.math.utah.edu/~davar/ps-pdf-files/donsker.pdf<br /> <br /> ==2016 Fall==<br /> <br /> 9/27 Daniele<br /> <br /> Stochastic reaction networks.<br /> <br /> Stochastic reaction networks are continuous time Markov chain models used primarily in biochemistry. I will define them, prove some results that connect them to related deterministic models and introduce some open questions. <br /> <br /> 10/4 Jessica<br /> <br /> 10/11, 10/18: Dae Han<br /> <br /> 10/25, 11/1: Jinsu<br /> <br /> Coupling of Markov processes.<br /> <br /> When we have two distributions on same probability space, we can think of a pair whose marginal probability is each of two distributions.<br /> This pairing can be used to estimate the total variation distance between two distributions. This idea is called coupling method.<br /> I am going to introduce basic concepts,ideas and applications of coupling for Markov processes.<br /> <br /> Links of References<br /> <br /> http://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf<br /> <br /> http://websites.math.leidenuniv.nl/probability/lecturenotes/CouplingLectures.pdf<br /> <br /> 11/8, 11/15: Hans<br /> <br /> 11/22, 11/29: Keith<br /> <br /> Surprisingly Determinental: DPPs and some asymptotics of ASEP <br /> <br /> I'll be reading and presenting some recent papers of Alexei Borodin and a few collaborators which have uncovered certain equivalences between determinental point processes and non-determinental processes.<br /> <br /> <br /> ==2016 Spring==<br /> <br /> Tuesday, 2:25pm, B321 Van Vleck<br /> <br /> <br /> 3/29, 4/5: Fan Yang<br /> <br /> I will talk about the ergodic decomposition theorem (EDT). More specifically, given a compact metric space X and a continuous transformation T on it, the theorem shows that any T-invariant measure on X can be decomposed into a convex combination of ergodic measures. In the first talk I introduced the EDT and some related facts. In the second talk, I will talk about the conditional measures, and prove that the ergodic measures in EDT are indeed the conditional measures.<br /> <br /> <br /> 2/16 : Jinsu<br /> <br /> Lyapunov function for Markov Processes.<br /> <br /> For ODE, we can show stability of the trajectory using Lyapunov functions.<br /> <br /> There is an analogy for Markov Processes. I'd like to talk about the existence of stationary distribution with Lyapunov function.<br /> <br /> In some cases, it is also possible to show the rate of convergence to the stationary distribution.<br /> <br /> ==2015 Fall==<br /> <br /> This semester we will focus on tools and methods.<br /> <br /> [https://www.math.wisc.edu/wiki/images/a/ac/Reading_seminar_2015.pdf Seminar notes] ([https://www.dropbox.com/s/f4km7pevwfb1vbm/Reading%20seminar%202015.tex?dl=1 tex file], [https://www.dropbox.com/s/lg7kcgyf3nsukbx/Reading_seminar_2015.bib?dl=1 bib file])<br /> <br /> 9/15, 9/22: Elnur<br /> <br /> I will talk about large deviation theory and its applications. For the first talk, my plan is to introduce Gartner-Ellis theorem and show a few applications of it to finite state discrete time Markov chains.<br /> <br /> 9/29, 10/6, 10/13 :Dae Han<br /> <br /> 10/20, 10/27, 11/3: Jessica<br /> <br /> I will first present an overview of concentration of measure and concentration inequalities with a focus on the connection with related topics in analysis and geometry. Then, I will present Log-Sobolev inequalities and their connection to concentration of measure. <br /> <br /> 11/10, 11/17: Hao Kai<br /> <br /> 11/24, 12/1, 12/8, 12/15: Chris<br /> <br /> : <br /> <br /> <br /> <br /> <br /> <br /> 2016 Spring:<br /> <br /> 2/2, 2/9: Louis<br /> <br /> <br /> 2/16, 2/23: Jinsu<br /> <br /> 3/1, 3/8: Hans<br /> <br /> ==2015 Spring==<br /> <br /> <br /> 2/3, 2/10: Scott<br /> <br /> An Introduction to Entropy for Random Variables<br /> <br /> In these lectures I will introduce entropy for random variables and present some simple, finite state-space, examples to gain some intuition. We will prove the <br /> MacMillan Theorem using entropy and the law of large numbers. Then I will introduce relative entropy and prove the Markov Chain Convergence Theorem. Finally I will <br /> define entropy for a discrete time process. The lecture notes can be found at http://www.math.wisc.edu/~shottovy/EntropyLecture.pdf.<br /> <br /> 2/17, 2/24: Dae Han<br /> <br /> 3/3, 3/10: Hans<br /> <br /> 3/17, 3/24: In Gun<br /> <br /> 4/7, 4/14: Jinsu<br /> <br /> 4/21, 4/28: Chris N.<br /> <br /> <br /> <br /> <br /> <br /> <br /> ==2014 Fall==<br /> <br /> 9/23: Dave<br /> <br /> I will go over Mike Giles’ 2008 paper “Multi-level Monte Carlo path simulation.” This paper introduced a new Monte Carlo method to approximate expectations of SDEs (driven by Brownian motions) that is significantly more efficient than what was the state of the art. This work opened up a whole new field in the numerical analysis of stochastic processes as the basic idea is quite flexible and has found a variety of applications including SDEs driven by Brownian motions, Levy-driven SDEs, SPDEs, and models from biology<br /> <br /> 9/30: Benedek<br /> <br /> A very quick introduction to Stein's method. <br /> <br /> I will give a brief introduction to Stein's method, mostly based on the the first couple of sections of the following survey article:<br /> <br /> Ross, N. (2011). Fundamentals of Stein’s method. Probability Surveys, 8, 210-293. <br /> <br /> The following webpage has a huge collection of resources if you want to go deeper: https://sites.google.com/site/yvikswan/about-stein-s-method<br /> <br /> <br /> Note that the Midwest Probability Colloquium (http://www.math.northwestern.edu/mwp/) will have a tutorial program on Stein's method this year. <br /> <br /> 10/7, 10/14: Chris J.<br /> [http://www.math.wisc.edu/~janjigia/research/MartingaleProblemNotes.pdf An introduction to the (local) martingale problem.]<br /> <br /> <br /> 10/21, 10/28: Dae Han<br /> <br /> 11/4, 11/11: Elnur<br /> <br /> 11/18, 11/25: Chris N. Free Probability with an emphasis on C* and Von Neumann Algebras<br /> <br /> 12/2, 12/9: Yun Zhai<br /> <br /> ==2014 Spring==<br /> <br /> <br /> 1/28: Greg<br /> <br /> 2/04, 2/11: Scott <br /> <br /> [http://www.math.wisc.edu/~shottovy/BLT.pdf Reflected Brownian motion, Occupation time, and applications.] <br /> <br /> 2/18: Phil-- Examples of structure results in probability theory.<br /> <br /> 2/25, 3/4: Beth-- Derivative estimation for discrete time Markov chains<br /> <br /> 3/11, 3/25: Chris J [http://www.math.wisc.edu/~janjigia/research/stationarytalk.pdf Some classical results on stationary distributions of Markov processes]<br /> <br /> 4/1, 4/8: Chris N <br /> <br /> 4/15, 4/22: Yu Sun<br /> <br /> 4/29. 5/6: Diane<br /> <br /> ==2013 Fall==<br /> <br /> 9/24, 10/1: Chris<br /> [http://www.math.wisc.edu/~janjigia/research/metastabilitytalk.pdf A light introduction to metastability]<br /> <br /> 10/8, Dae Han<br /> Majoring multiplicative cascades for directed polymers in random media<br /> <br /> 10/15, 10/22: no reading seminar<br /> <br /> 10/29, 11/5: Elnur<br /> Limit fluctuations of last passage times <br /> <br /> 11/12: Yun<br /> Helffer-Sjostrand representation and Brascamp-Lieb inequality for stochastic interface models<br /> <br /> 11/19, 11/26: Yu Sun<br /> <br /> 12/3, 12/10: Jason<br /> <br /> ==2013 Spring==<br /> <br /> 2/13: Elnur <br /> <br /> Young diagrams, RSK correspondence, corner growth models, distribution of last passage times. <br /> <br /> 2/20: Elnur<br /> <br /> 2/27: Chris<br /> <br /> A brief introduction to enlargement of filtration and the Dufresne identity<br /> [http://www.math.wisc.edu/~janjigia/research/Presentation%20Notes.pdf Notes]<br /> <br /> 3/6: Chris<br /> <br /> 3/13: Dae Han<br /> <br /> An introduction to random polymers<br /> <br /> 3/20: Dae Han<br /> <br /> Directed polymers in a random environment: path localization and strong disorder<br /> <br /> 4/3: Diane<br /> <br /> Scale and Speed for honest 1 dimensional diffusions<br /> <br /> References: &lt;br&gt;<br /> Rogers &amp; Williams - Diffusions, Markov Processes and Martingales &lt;br&gt;<br /> Ito &amp; McKean - Diffusion Processes and their Sample Paths &lt;br&gt;<br /> Breiman - Probability &lt;br&gt;<br /> http://www.statslab.cam.ac.uk/~beresty/Articles/diffusions.pdf<br /> <br /> 4/10: Diane<br /> <br /> 4/17: Yun<br /> <br /> Introduction to stochastic interface models<br /> <br /> 4/24: Yun<br /> <br /> Dynamics and Gaussian equilibrium sytems<br /> <br /> 5/1: This reading seminar will be shifted because of a probability seminar.<br /> <br /> <br /> 5/8: Greg, Maso<br /> <br /> The Bethe ansatz vs. The Replica Trick. This lecture is an overview of the two <br /> approaches. See [http://arxiv.org/abs/1212.2267] for a nice overview.<br /> <br /> 5/15: Greg, Maso<br /> <br /> Rigorous use of the replica trick.</div> Valko https://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=16899 Probability Seminar 2019-02-11T00:03:12Z <p>Valko: /* March 28, Shamgar Gurevitch UW-Madison */</p> <hr /> <div>__NOTOC__<br /> <br /> = Spring 2019 =<br /> <br /> &lt;b&gt;Thursdays in 901 Van Vleck Hall at 2:25 PM&lt;/b&gt;, unless otherwise noted. <br /> &lt;b&gt;We usually end for questions at 3:15 PM.&lt;/b&gt;<br /> <br /> If you would like to sign up for the email list to receive seminar announcements then please send an email to <br /> [mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br /> <br /> <br /> <br /> == January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br /> <br /> Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br /> <br /> Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda&gt;0$.<br /> Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br /> <br /> == &lt;span style=&quot;color:red&quot;&gt; Wednesday, February 6 at 4:00pm in Van Vleck 911&lt;/span&gt; , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br /> <br /> Title: '''When particle systems meet PDEs'''<br /> <br /> Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br /> <br /> == February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br /> <br /> Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br /> <br /> Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br /> <br /> == February 14, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], UW-Madison==<br /> <br /> Title: '''Geometry of the corner growth model'''<br /> <br /> Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).<br /> <br /> == February 21, [https://people.kth.se/~holcomb/ Diane Holcomb], KTH ==<br /> <br /> <br /> <br /> <br /> == &lt;span style=&quot;color:red&quot;&gt; Wednesday, February 27 at 1:10pm&lt;/span&gt; [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br /> <br /> <br /> &lt;div style=&quot;width:520px;height:50px;border:5px solid black&quot;&gt;<br /> &lt;b&gt;&lt;span style=&quot;color:red&quot;&gt;&amp;emsp; Please note the unusual day and time. <br /> &amp;emsp; &lt;/span&gt;&lt;/b&gt;<br /> &lt;/div&gt;<br /> <br /> == March 7, TBA ==<br /> <br /> == March 14, TBA ==<br /> == March 21, Spring Break, No seminar ==<br /> <br /> == March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br /> <br /> Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br /> <br /> Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the ''character ratio'': <br /> <br /> $$<br /> \text{trace}(\rho(g))/\text{dim}(\rho),<br />$$<br /> <br /> for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant ''rank''. This talk will discuss the notion of rank for $GL_n$ over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br /> <br /> == April 4, TBA ==<br /> == April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Procaccia], [http://www.math.tamu.edu/index.html Texas A&amp;M] ==<br /> <br /> == April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br /> <br /> == April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br /> <br /> == April 26, Colloquium, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br /> <br /> == April 26, TBA ==<br /> == May 2, TBA ==<br /> <br /> <br /> &lt;!--<br /> ==&lt;span style=&quot;color:red&quot;&gt; Friday, August 10, 10am, B239 Van Vleck &lt;/span&gt; András Mészáros, Central European University, Budapest ==<br /> <br /> <br /> Title: '''The distribution of sandpile groups of random regular graphs'''<br /> <br /> Abstract:<br /> We study the distribution of the sandpile group of random &lt;math&gt;d&lt;/math&gt;-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the &lt;math&gt;p&lt;/math&gt;-Sylow subgroup of the sandpile group is a given &lt;math&gt;p&lt;/math&gt;-group &lt;math&gt;P&lt;/math&gt;, is proportional to &lt;math&gt;|\operatorname{Aut}(P)|^{-1}&lt;/math&gt;. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br /> <br /> Our results extends a recent theorem of Huang saying that the adjacency matrices of random &lt;math&gt;d&lt;/math&gt;-regular directed graphs are invertible with high probability to the undirected case.<br /> <br /> <br /> ==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br /> <br /> Title: '''Stochastic quantization of Yang-Mills'''<br /> <br /> Abstract:<br /> &quot;Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br /> In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br /> <br /> --&gt;<br /> <br /> == ==<br /> <br /> [[Past Seminars]]</div> Valko https://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=16898 Probability Seminar 2019-02-11T00:02:31Z <p>Valko: /* March 28, Shamgar Gurevitch UW-Madison */</p> <hr /> <div>__NOTOC__<br /> <br /> = Spring 2019 =<br /> <br /> &lt;b&gt;Thursdays in 901 Van Vleck Hall at 2:25 PM&lt;/b&gt;, unless otherwise noted. <br /> &lt;b&gt;We usually end for questions at 3:15 PM.&lt;/b&gt;<br /> <br /> If you would like to sign up for the email list to receive seminar announcements then please send an email to <br /> [mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br /> <br /> <br /> <br /> == January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br /> <br /> Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br /> <br /> Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda&gt;0$.<br /> Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br /> <br /> == &lt;span style=&quot;color:red&quot;&gt; Wednesday, February 6 at 4:00pm in Van Vleck 911&lt;/span&gt; , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br /> <br /> Title: '''When particle systems meet PDEs'''<br /> <br /> Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br /> <br /> == February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br /> <br /> Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br /> <br /> Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br /> <br /> == February 14, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], UW-Madison==<br /> <br /> Title: '''Geometry of the corner growth model'''<br /> <br /> Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).<br /> <br /> == February 21, [https://people.kth.se/~holcomb/ Diane Holcomb], KTH ==<br /> <br /> <br /> <br /> <br /> == &lt;span style=&quot;color:red&quot;&gt; Wednesday, February 27 at 1:10pm&lt;/span&gt; [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br /> <br /> <br /> &lt;div style=&quot;width:520px;height:50px;border:5px solid black&quot;&gt;<br /> &lt;b&gt;&lt;span style=&quot;color:red&quot;&gt;&amp;emsp; Please note the unusual day and time. <br /> &amp;emsp; &lt;/span&gt;&lt;/b&gt;<br /> &lt;/div&gt;<br /> <br /> == March 7, TBA ==<br /> <br /> == March 14, TBA ==<br /> == March 21, Spring Break, No seminar ==<br /> <br /> == March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br /> <br /> Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br /> <br /> Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the ''character ratio'': <br /> <br /> $$<br /> trace(\rho(g))/dim(\rho),<br />$$<br /> <br /> for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant ''rank''. This talk will discuss the notion of rank for GLn over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br /> <br /> == April 4, TBA ==<br /> == April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Procaccia], [http://www.math.tamu.edu/index.html Texas A&amp;M] ==<br /> <br /> == April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br /> <br /> == April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br /> <br /> == April 26, Colloquium, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br /> <br /> == April 26, TBA ==<br /> == May 2, TBA ==<br /> <br /> <br /> &lt;!--<br /> ==&lt;span style=&quot;color:red&quot;&gt; Friday, August 10, 10am, B239 Van Vleck &lt;/span&gt; András Mészáros, Central European University, Budapest ==<br /> <br /> <br /> Title: '''The distribution of sandpile groups of random regular graphs'''<br /> <br /> Abstract:<br /> We study the distribution of the sandpile group of random &lt;math&gt;d&lt;/math&gt;-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the &lt;math&gt;p&lt;/math&gt;-Sylow subgroup of the sandpile group is a given &lt;math&gt;p&lt;/math&gt;-group &lt;math&gt;P&lt;/math&gt;, is proportional to &lt;math&gt;|\operatorname{Aut}(P)|^{-1}&lt;/math&gt;. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br /> <br /> Our results extends a recent theorem of Huang saying that the adjacency matrices of random &lt;math&gt;d&lt;/math&gt;-regular directed graphs are invertible with high probability to the undirected case.<br /> <br /> <br /> ==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br /> <br /> Title: '''Stochastic quantization of Yang-Mills'''<br /> <br /> Abstract:<br /> &quot;Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br /> In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br /> <br /> --&gt;<br /> <br /> == ==<br /> <br /> [[Past Seminars]]</div> Valko https://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=16897 Probability Seminar 2019-02-11T00:01:40Z <p>Valko: </p> <hr /> <div>__NOTOC__<br /> <br /> = Spring 2019 =<br /> <br /> &lt;b&gt;Thursdays in 901 Van Vleck Hall at 2:25 PM&lt;/b&gt;, unless otherwise noted. <br /> &lt;b&gt;We usually end for questions at 3:15 PM.&lt;/b&gt;<br /> <br /> If you would like to sign up for the email list to receive seminar announcements then please send an email to <br /> [mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br /> <br /> <br /> <br /> == January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br /> <br /> Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br /> <br /> Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda&gt;0$.<br /> Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br /> <br /> == &lt;span style=&quot;color:red&quot;&gt; Wednesday, February 6 at 4:00pm in Van Vleck 911&lt;/span&gt; , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br /> <br /> Title: '''When particle systems meet PDEs'''<br /> <br /> Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br /> <br /> == February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br /> <br /> Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br /> <br /> Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br /> <br /> == February 14, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], UW-Madison==<br /> <br /> Title: '''Geometry of the corner growth model'''<br /> <br /> Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).<br /> <br /> == February 21, [https://people.kth.se/~holcomb/ Diane Holcomb], KTH ==<br /> <br /> <br /> <br /> <br /> == &lt;span style=&quot;color:red&quot;&gt; Wednesday, February 27 at 1:10pm&lt;/span&gt; [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br /> <br /> <br /> &lt;div style=&quot;width:520px;height:50px;border:5px solid black&quot;&gt;<br /> &lt;b&gt;&lt;span style=&quot;color:red&quot;&gt;&amp;emsp; Please note the unusual day and time. <br /> &amp;emsp; &lt;/span&gt;&lt;/b&gt;<br /> &lt;/div&gt;<br /> <br /> == March 7, TBA ==<br /> <br /> == March 14, TBA ==<br /> == March 21, Spring Break, No seminar ==<br /> <br /> == March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br /> <br /> Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br /> <br /> Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the {\it character ratio}: <br /> <br /> $$<br /> trace(\rho(g))/dim(\rho),<br />$$<br /> <br /> for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant {\it rank}. This talk will discuss the notion of rank for GLn over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br /> <br /> == April 4, TBA ==<br /> == April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Procaccia], [http://www.math.tamu.edu/index.html Texas A&amp;M] ==<br /> <br /> == April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br /> <br /> == April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br /> <br /> == April 26, Colloquium, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br /> <br /> == April 26, TBA ==<br /> == May 2, TBA ==<br /> <br /> <br /> &lt;!--<br /> ==&lt;span style=&quot;color:red&quot;&gt; Friday, August 10, 10am, B239 Van Vleck &lt;/span&gt; András Mészáros, Central European University, Budapest ==<br /> <br /> <br /> Title: '''The distribution of sandpile groups of random regular graphs'''<br /> <br /> Abstract:<br /> We study the distribution of the sandpile group of random &lt;math&gt;d&lt;/math&gt;-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the &lt;math&gt;p&lt;/math&gt;-Sylow subgroup of the sandpile group is a given &lt;math&gt;p&lt;/math&gt;-group &lt;math&gt;P&lt;/math&gt;, is proportional to &lt;math&gt;|\operatorname{Aut}(P)|^{-1}&lt;/math&gt;. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br /> <br /> Our results extends a recent theorem of Huang saying that the adjacency matrices of random &lt;math&gt;d&lt;/math&gt;-regular directed graphs are invertible with high probability to the undirected case.<br /> <br /> <br /> ==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br /> <br /> Title: '''Stochastic quantization of Yang-Mills'''<br /> <br /> Abstract:<br /> &quot;Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br /> In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br /> <br /> --&gt;<br /> <br /> == ==<br /> <br /> [[Past Seminars]]</div> Valko https://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=16894 Probability Seminar 2019-02-10T23:31:49Z <p>Valko: /* February 14, Timo Seppäläinen */</p> <hr /> <div>__NOTOC__<br /> <br /> = Spring 2019 =<br /> <br /> &lt;b&gt;Thursdays in 901 Van Vleck Hall at 2:25 PM&lt;/b&gt;, unless otherwise noted. <br /> &lt;b&gt;We usually end for questions at 3:15 PM.&lt;/b&gt;<br /> <br /> If you would like to sign up for the email list to receive seminar announcements then please send an email to <br /> [mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br /> <br /> <br /> <br /> == January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br /> <br /> Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br /> <br /> Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda&gt;0$.<br /> Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br /> <br /> == &lt;span style=&quot;color:red&quot;&gt; Wednesday, February 6 at 4:00pm in Van Vleck 911&lt;/span&gt; , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br /> <br /> Title: '''When particle systems meet PDEs'''<br /> <br /> Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br /> <br /> == February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br /> <br /> Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br /> <br /> Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br /> <br /> == February 14, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], UW-Madison==<br /> <br /> Title: '''Geometry of the corner growth model'''<br /> <br /> Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).<br /> <br /> == February 21, TBA ==<br /> == &lt;span style=&quot;color:red&quot;&gt; Wednesday, February 27 at 1:10pm&lt;/span&gt; [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br /> <br /> <br /> &lt;div style=&quot;width:520px;height:50px;border:5px solid black&quot;&gt;<br /> &lt;b&gt;&lt;span style=&quot;color:red&quot;&gt;&amp;emsp; Please note the unusual day and time. <br /> &amp;emsp; &lt;/span&gt;&lt;/b&gt;<br /> &lt;/div&gt;<br /> <br /> == March 7, TBA ==<br /> <br /> == March 14, TBA ==<br /> == March 21, Spring Break, No seminar ==<br /> <br /> == March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br /> <br /> Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br /> <br /> Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the {\it character ratio}: <br /> <br /> $$<br /> trace(\rho(g))/dim(\rho),<br />$$<br /> <br /> for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant {\it rank}. This talk will discuss the notion of rank for GLn over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br /> <br /> == April 4, TBA ==<br /> == April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Procaccia], [http://www.math.tamu.edu/index.html Texas A&amp;M] ==<br /> <br /> == April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br /> <br /> == April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br /> <br /> == April 26, Colloquium, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br /> <br /> == April 26, TBA ==<br /> == May 2, TBA ==<br /> <br /> <br /> &lt;!--<br /> ==&lt;span style=&quot;color:red&quot;&gt; Friday, August 10, 10am, B239 Van Vleck &lt;/span&gt; András Mészáros, Central European University, Budapest ==<br /> <br /> <br /> Title: '''The distribution of sandpile groups of random regular graphs'''<br /> <br /> Abstract:<br /> We study the distribution of the sandpile group of random &lt;math&gt;d&lt;/math&gt;-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the &lt;math&gt;p&lt;/math&gt;-Sylow subgroup of the sandpile group is a given &lt;math&gt;p&lt;/math&gt;-group &lt;math&gt;P&lt;/math&gt;, is proportional to &lt;math&gt;|\operatorname{Aut}(P)|^{-1}&lt;/math&gt;. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br /> <br /> Our results extends a recent theorem of Huang saying that the adjacency matrices of random &lt;math&gt;d&lt;/math&gt;-regular directed graphs are invertible with high probability to the undirected case.<br /> <br /> <br /> ==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br /> <br /> Title: '''Stochastic quantization of Yang-Mills'''<br /> <br /> Abstract:<br /> &quot;Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br /> In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br /> <br /> --&gt;<br /> <br /> == ==<br /> <br /> [[Past Seminars]]</div> Valko https://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=16893 Probability Seminar 2019-02-10T23:31:31Z <p>Valko: </p> <hr /> <div>__NOTOC__<br /> <br /> = Spring 2019 =<br /> <br /> &lt;b&gt;Thursdays in 901 Van Vleck Hall at 2:25 PM&lt;/b&gt;, unless otherwise noted. <br /> &lt;b&gt;We usually end for questions at 3:15 PM.&lt;/b&gt;<br /> <br /> If you would like to sign up for the email list to receive seminar announcements then please send an email to <br /> [mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br /> <br /> <br /> <br /> == January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br /> <br /> Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br /> <br /> Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda&gt;0$.<br /> Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br /> <br /> == &lt;span style=&quot;color:red&quot;&gt; Wednesday, February 6 at 4:00pm in Van Vleck 911&lt;/span&gt; , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br /> <br /> Title: '''When particle systems meet PDEs'''<br /> <br /> Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br /> <br /> == February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br /> <br /> Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br /> <br /> Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br /> <br /> == February 14, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen]==<br /> <br /> Title: '''Geometry of the corner growth model'''<br /> <br /> Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah). <br /> <br /> <br /> == February 21, TBA ==<br /> == &lt;span style=&quot;color:red&quot;&gt; Wednesday, February 27 at 1:10pm&lt;/span&gt; [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br /> <br /> <br /> &lt;div style=&quot;width:520px;height:50px;border:5px solid black&quot;&gt;<br /> &lt;b&gt;&lt;span style=&quot;color:red&quot;&gt;&amp;emsp; Please note the unusual day and time. <br /> &amp;emsp; &lt;/span&gt;&lt;/b&gt;<br /> &lt;/div&gt;<br /> <br /> == March 7, TBA ==<br /> <br /> == March 14, TBA ==<br /> == March 21, Spring Break, No seminar ==<br /> <br /> == March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br /> <br /> Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br /> <br /> Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the {\it character ratio}: <br /> <br /> $$<br /> trace(\rho(g))/dim(\rho),<br />$$<br /> <br /> for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant {\it rank}. This talk will discuss the notion of rank for GLn over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br /> <br /> == April 4, TBA ==<br /> == April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Procaccia], [http://www.math.tamu.edu/index.html Texas A&amp;M] ==<br /> <br /> == April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br /> <br /> == April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br /> <br /> == April 26, Colloquium, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br /> <br /> == April 26, TBA ==<br /> == May 2, TBA ==<br /> <br /> <br /> &lt;!--<br /> ==&lt;span style=&quot;color:red&quot;&gt; Friday, August 10, 10am, B239 Van Vleck &lt;/span&gt; András Mészáros, Central European University, Budapest ==<br /> <br /> <br /> Title: '''The distribution of sandpile groups of random regular graphs'''<br /> <br /> Abstract:<br /> We study the distribution of the sandpile group of random &lt;math&gt;d&lt;/math&gt;-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the &lt;math&gt;p&lt;/math&gt;-Sylow subgroup of the sandpile group is a given &lt;math&gt;p&lt;/math&gt;-group &lt;math&gt;P&lt;/math&gt;, is proportional to &lt;math&gt;|\operatorname{Aut}(P)|^{-1}&lt;/math&gt;. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br /> <br /> Our results extends a recent theorem of Huang saying that the adjacency matrices of random &lt;math&gt;d&lt;/math&gt;-regular directed graphs are invertible with high probability to the undirected case.<br /> <br /> <br /> ==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br /> <br /> Title: '''Stochastic quantization of Yang-Mills'''<br /> <br /> Abstract:<br /> &quot;Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br /> In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br /> <br /> --&gt;<br /> <br /> == ==<br /> <br /> [[Past Seminars]]</div> Valko https://www.math.wisc.edu/wiki/index.php?title=Graduate_student_reading_seminar&diff=16689 Graduate student reading seminar 2019-01-23T17:06:13Z <p>Valko: </p> <hr /> <div>(... in probability)<br /> <br /> <br /> Email list: join-grad_prob_seminar@lists.wisc.edu<br /> <br /> ==2019 Spring==<br /> <br /> Tuesday 2:30pm, 901 Van Vleck<br /> <br /> <br /> ==2018 Fall==<br /> <br /> Tuesday 2:30pm, 901 Van Vleck<br /> <br /> <br /> The topic this semester is large deviation theory. Send me (BV) an email, if you want access to the shared Box folder with some reading material. <br /> <br /> <br /> 9/25, 10/2: Dae Han<br /> <br /> 10/9, 10/16: Kurt<br /> <br /> 10/23, 10/30: Stephen Davis<br /> <br /> 11/6, 11/13: Brandon Legried <br /> <br /> 11/20, 11/27: Shuqi Yu<br /> <br /> 12/4, 12/11: Yun Li<br /> <br /> ==2018 Spring==<br /> <br /> Tuesday 2:30pm, B135 Van Vleck<br /> <br /> <br /> Preliminary schedule:<br /> <br /> 2/20, 2/27: Yun<br /> <br /> 3/6, 3/13: Greg<br /> <br /> 3/20, 4/3: Yu<br /> <br /> 4/10, 4/17: Shuqi<br /> <br /> 4/24, 5/1: Tony<br /> <br /> ==2017 Fall==<br /> <br /> Tuesday 2:30pm, 214 Ingraham Hall<br /> <br /> <br /> Preliminary schedule: <br /> <br /> 9/26, 10/3: Hans<br /> <br /> 10/10, 10/17: Guo<br /> <br /> 10/24, 10/31: Chaoji<br /> <br /> 11/7, 11/14: Yun <br /> <br /> 11/21, 11/28: Kurt<br /> <br /> 12/5, 12/12: Christian<br /> <br /> <br /> <br /> <br /> ==2017 Spring==<br /> <br /> Tuesday 2:25pm, B211<br /> <br /> 1/31, 2/7: Fan<br /> <br /> I will talk about the Hanson-Wright inequality, which is a large deviation estimate for random variable of the form X^* A X, where X is a random vector with independent subgaussian entries and A is an arbitrary deterministic matrix. In the first talk, I will present a beautiful proof given by Mark Rudelson and Roman Vershynin. In the second talk, I will talk about some applications of this inequality.<br /> <br /> Reference: M. Rudelson and R. Vershynin, Hanson-Wright inequality and sub-gaussian concentration, Electron. Commun. Probab. Volume 18 (2013).<br /> <br /> 3/7, 3/14 : Jinsu<br /> <br /> Title : Donsker's Theorem and its application.<br /> Donsker's Theorem roughly says normalized random walk with linear interpolation on time interval [0,1] weakly converges to the Brownian motion B[0,1] in C([0,1]). It is sometimes called Donsker's invariance principle or the functional central limit theorem. I will show main ideas for the proof of this theorem tomorrow and show a couple of applications in my 2nd talk.<br /> <br /> Reference : https://www.math.utah.edu/~davar/ps-pdf-files/donsker.pdf<br /> <br /> ==2016 Fall==<br /> <br /> 9/27 Daniele<br /> <br /> Stochastic reaction networks.<br /> <br /> Stochastic reaction networks are continuous time Markov chain models used primarily in biochemistry. I will define them, prove some results that connect them to related deterministic models and introduce some open questions. <br /> <br /> 10/4 Jessica<br /> <br /> 10/11, 10/18: Dae Han<br /> <br /> 10/25, 11/1: Jinsu<br /> <br /> Coupling of Markov processes.<br /> <br /> When we have two distributions on same probability space, we can think of a pair whose marginal probability is each of two distributions.<br /> This pairing can be used to estimate the total variation distance between two distributions. This idea is called coupling method.<br /> I am going to introduce basic concepts,ideas and applications of coupling for Markov processes.<br /> <br /> Links of References<br /> <br /> http://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf<br /> <br /> http://websites.math.leidenuniv.nl/probability/lecturenotes/CouplingLectures.pdf<br /> <br /> 11/8, 11/15: Hans<br /> <br /> 11/22, 11/29: Keith<br /> <br /> Surprisingly Determinental: DPPs and some asymptotics of ASEP <br /> <br /> I'll be reading and presenting some recent papers of Alexei Borodin and a few collaborators which have uncovered certain equivalences between determinental point processes and non-determinental processes.<br /> <br /> <br /> ==2016 Spring==<br /> <br /> Tuesday, 2:25pm, B321 Van Vleck<br /> <br /> <br /> 3/29, 4/5: Fan Yang<br /> <br /> I will talk about the ergodic decomposition theorem (EDT). More specifically, given a compact metric space X and a continuous transformation T on it, the theorem shows that any T-invariant measure on X can be decomposed into a convex combination of ergodic measures. In the first talk I introduced the EDT and some related facts. In the second talk, I will talk about the conditional measures, and prove that the ergodic measures in EDT are indeed the conditional measures.<br /> <br /> <br /> 2/16 : Jinsu<br /> <br /> Lyapunov function for Markov Processes.<br /> <br /> For ODE, we can show stability of the trajectory using Lyapunov functions.<br /> <br /> There is an analogy for Markov Processes. I'd like to talk about the existence of stationary distribution with Lyapunov function.<br /> <br /> In some cases, it is also possible to show the rate of convergence to the stationary distribution.<br /> <br /> ==2015 Fall==<br /> <br /> This semester we will focus on tools and methods.<br /> <br /> [https://www.math.wisc.edu/wiki/images/a/ac/Reading_seminar_2015.pdf Seminar notes] ([https://www.dropbox.com/s/f4km7pevwfb1vbm/Reading%20seminar%202015.tex?dl=1 tex file], [https://www.dropbox.com/s/lg7kcgyf3nsukbx/Reading_seminar_2015.bib?dl=1 bib file])<br /> <br /> 9/15, 9/22: Elnur<br /> <br /> I will talk about large deviation theory and its applications. For the first talk, my plan is to introduce Gartner-Ellis theorem and show a few applications of it to finite state discrete time Markov chains.<br /> <br /> 9/29, 10/6, 10/13 :Dae Han<br /> <br /> 10/20, 10/27, 11/3: Jessica<br /> <br /> I will first present an overview of concentration of measure and concentration inequalities with a focus on the connection with related topics in analysis and geometry. Then, I will present Log-Sobolev inequalities and their connection to concentration of measure. <br /> <br /> 11/10, 11/17: Hao Kai<br /> <br /> 11/24, 12/1, 12/8, 12/15: Chris<br /> <br /> : <br /> <br /> <br /> <br /> <br /> <br /> 2016 Spring:<br /> <br /> 2/2, 2/9: Louis<br /> <br /> <br /> 2/16, 2/23: Jinsu<br /> <br /> 3/1, 3/8: Hans<br /> <br /> ==2015 Spring==<br /> <br /> <br /> 2/3, 2/10: Scott<br /> <br /> An Introduction to Entropy for Random Variables<br /> <br /> In these lectures I will introduce entropy for random variables and present some simple, finite state-space, examples to gain some intuition. We will prove the <br /> MacMillan Theorem using entropy and the law of large numbers. Then I will introduce relative entropy and prove the Markov Chain Convergence Theorem. Finally I will <br /> define entropy for a discrete time process. The lecture notes can be found at http://www.math.wisc.edu/~shottovy/EntropyLecture.pdf.<br /> <br /> 2/17, 2/24: Dae Han<br /> <br /> 3/3, 3/10: Hans<br /> <br /> 3/17, 3/24: In Gun<br /> <br /> 4/7, 4/14: Jinsu<br /> <br /> 4/21, 4/28: Chris N.<br /> <br /> <br /> <br /> <br /> <br /> <br /> ==2014 Fall==<br /> <br /> 9/23: Dave<br /> <br /> I will go over Mike Giles’ 2008 paper “Multi-level Monte Carlo path simulation.” This paper introduced a new Monte Carlo method to approximate expectations of SDEs (driven by Brownian motions) that is significantly more efficient than what was the state of the art. This work opened up a whole new field in the numerical analysis of stochastic processes as the basic idea is quite flexible and has found a variety of applications including SDEs driven by Brownian motions, Levy-driven SDEs, SPDEs, and models from biology<br /> <br /> 9/30: Benedek<br /> <br /> A very quick introduction to Stein's method. <br /> <br /> I will give a brief introduction to Stein's method, mostly based on the the first couple of sections of the following survey article:<br /> <br /> Ross, N. (2011). Fundamentals of Stein’s method. Probability Surveys, 8, 210-293. <br /> <br /> The following webpage has a huge collection of resources if you want to go deeper: https://sites.google.com/site/yvikswan/about-stein-s-method<br /> <br /> <br /> Note that the Midwest Probability Colloquium (http://www.math.northwestern.edu/mwp/) will have a tutorial program on Stein's method this year. <br /> <br /> 10/7, 10/14: Chris J.<br /> [http://www.math.wisc.edu/~janjigia/research/MartingaleProblemNotes.pdf An introduction to the (local) martingale problem.]<br /> <br /> <br /> 10/21, 10/28: Dae Han<br /> <br /> 11/4, 11/11: Elnur<br /> <br /> 11/18, 11/25: Chris N. Free Probability with an emphasis on C* and Von Neumann Algebras<br /> <br /> 12/2, 12/9: Yun Zhai<br /> <br /> ==2014 Spring==<br /> <br /> <br /> 1/28: Greg<br /> <br /> 2/04, 2/11: Scott <br /> <br /> [http://www.math.wisc.edu/~shottovy/BLT.pdf Reflected Brownian motion, Occupation time, and applications.] <br /> <br /> 2/18: Phil-- Examples of structure results in probability theory.<br /> <br /> 2/25, 3/4: Beth-- Derivative estimation for discrete time Markov chains<br /> <br /> 3/11, 3/25: Chris J [http://www.math.wisc.edu/~janjigia/research/stationarytalk.pdf Some classical results on stationary distributions of Markov processes]<br /> <br /> 4/1, 4/8: Chris N <br /> <br /> 4/15, 4/22: Yu Sun<br /> <br /> 4/29. 5/6: Diane<br /> <br /> ==2013 Fall==<br /> <br /> 9/24, 10/1: Chris<br /> [http://www.math.wisc.edu/~janjigia/research/metastabilitytalk.pdf A light introduction to metastability]<br /> <br /> 10/8, Dae Han<br /> Majoring multiplicative cascades for directed polymers in random media<br /> <br /> 10/15, 10/22: no reading seminar<br /> <br /> 10/29, 11/5: Elnur<br /> Limit fluctuations of last passage times <br /> <br /> 11/12: Yun<br /> Helffer-Sjostrand representation and Brascamp-Lieb inequality for stochastic interface models<br /> <br /> 11/19, 11/26: Yu Sun<br /> <br /> 12/3, 12/10: Jason<br /> <br /> ==2013 Spring==<br /> <br /> 2/13: Elnur <br /> <br /> Young diagrams, RSK correspondence, corner growth models, distribution of last passage times. <br /> <br /> 2/20: Elnur<br /> <br /> 2/27: Chris<br /> <br /> A brief introduction to enlargement of filtration and the Dufresne identity<br /> [http://www.math.wisc.edu/~janjigia/research/Presentation%20Notes.pdf Notes]<br /> <br /> 3/6: Chris<br /> <br /> 3/13: Dae Han<br /> <br /> An introduction to random polymers<br /> <br /> 3/20: Dae Han<br /> <br /> Directed polymers in a random environment: path localization and strong disorder<br /> <br /> 4/3: Diane<br /> <br /> Scale and Speed for honest 1 dimensional diffusions<br /> <br /> References: &lt;br&gt;<br /> Rogers &amp; Williams - Diffusions, Markov Processes and Martingales &lt;br&gt;<br /> Ito &amp; McKean - Diffusion Processes and their Sample Paths &lt;br&gt;<br /> Breiman - Probability &lt;br&gt;<br /> http://www.statslab.cam.ac.uk/~beresty/Articles/diffusions.pdf<br /> <br /> 4/10: Diane<br /> <br /> 4/17: Yun<br /> <br /> Introduction to stochastic interface models<br /> <br /> 4/24: Yun<br /> <br /> Dynamics and Gaussian equilibrium sytems<br /> <br /> 5/1: This reading seminar will be shifted because of a probability seminar.<br /> <br /> <br /> 5/8: Greg, Maso<br /> <br /> The Bethe ansatz vs. The Replica Trick. This lecture is an overview of the two <br /> approaches. See [http://arxiv.org/abs/1212.2267] for a nice overview.<br /> <br /> 5/15: Greg, Maso<br /> <br /> Rigorous use of the replica trick.</div> Valko https://www.math.wisc.edu/wiki/index.php?title=Graduate_student_reading_seminar&diff=16688 Graduate student reading seminar 2019-01-23T14:01:15Z <p>Valko: </p> <hr /> <div>(... in probability)<br /> <br /> <br /> ==2019 Spring==<br /> <br /> Tuesday 2:30pm, 901 Van Vleck<br /> <br /> <br /> ==2018 Fall==<br /> <br /> Tuesday 2:30pm, 901 Van Vleck<br /> <br /> <br /> The topic this semester is large deviation theory. Send me (BV) an email, if you want access to the shared Box folder with some reading material. <br /> <br /> <br /> 9/25, 10/2: Dae Han<br /> <br /> 10/9, 10/16: Kurt<br /> <br /> 10/23, 10/30: Stephen Davis<br /> <br /> 11/6, 11/13: Brandon Legried <br /> <br /> 11/20, 11/27: Shuqi Yu<br /> <br /> 12/4, 12/11: Yun Li<br /> <br /> ==2018 Spring==<br /> <br /> Tuesday 2:30pm, B135 Van Vleck<br /> <br /> <br /> Preliminary schedule:<br /> <br /> 2/20, 2/27: Yun<br /> <br /> 3/6, 3/13: Greg<br /> <br /> 3/20, 4/3: Yu<br /> <br /> 4/10, 4/17: Shuqi<br /> <br /> 4/24, 5/1: Tony<br /> <br /> ==2017 Fall==<br /> <br /> Tuesday 2:30pm, 214 Ingraham Hall<br /> <br /> <br /> Preliminary schedule: <br /> <br /> 9/26, 10/3: Hans<br /> <br /> 10/10, 10/17: Guo<br /> <br /> 10/24, 10/31: Chaoji<br /> <br /> 11/7, 11/14: Yun <br /> <br /> 11/21, 11/28: Kurt<br /> <br /> 12/5, 12/12: Christian<br /> <br /> <br /> <br /> <br /> ==2017 Spring==<br /> <br /> Tuesday 2:25pm, B211<br /> <br /> 1/31, 2/7: Fan<br /> <br /> I will talk about the Hanson-Wright inequality, which is a large deviation estimate for random variable of the form X^* A X, where X is a random vector with independent subgaussian entries and A is an arbitrary deterministic matrix. In the first talk, I will present a beautiful proof given by Mark Rudelson and Roman Vershynin. In the second talk, I will talk about some applications of this inequality.<br /> <br /> Reference: M. Rudelson and R. Vershynin, Hanson-Wright inequality and sub-gaussian concentration, Electron. Commun. Probab. Volume 18 (2013).<br /> <br /> 3/7, 3/14 : Jinsu<br /> <br /> Title : Donsker's Theorem and its application.<br /> Donsker's Theorem roughly says normalized random walk with linear interpolation on time interval [0,1] weakly converges to the Brownian motion B[0,1] in C([0,1]). It is sometimes called Donsker's invariance principle or the functional central limit theorem. I will show main ideas for the proof of this theorem tomorrow and show a couple of applications in my 2nd talk.<br /> <br /> Reference : https://www.math.utah.edu/~davar/ps-pdf-files/donsker.pdf<br /> <br /> ==2016 Fall==<br /> <br /> 9/27 Daniele<br /> <br /> Stochastic reaction networks.<br /> <br /> Stochastic reaction networks are continuous time Markov chain models used primarily in biochemistry. I will define them, prove some results that connect them to related deterministic models and introduce some open questions. <br /> <br /> 10/4 Jessica<br /> <br /> 10/11, 10/18: Dae Han<br /> <br /> 10/25, 11/1: Jinsu<br /> <br /> Coupling of Markov processes.<br /> <br /> When we have two distributions on same probability space, we can think of a pair whose marginal probability is each of two distributions.<br /> This pairing can be used to estimate the total variation distance between two distributions. This idea is called coupling method.<br /> I am going to introduce basic concepts,ideas and applications of coupling for Markov processes.<br /> <br /> Links of References<br /> <br /> http://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf<br /> <br /> http://websites.math.leidenuniv.nl/probability/lecturenotes/CouplingLectures.pdf<br /> <br /> 11/8, 11/15: Hans<br /> <br /> 11/22, 11/29: Keith<br /> <br /> Surprisingly Determinental: DPPs and some asymptotics of ASEP <br /> <br /> I'll be reading and presenting some recent papers of Alexei Borodin and a few collaborators which have uncovered certain equivalences between determinental point processes and non-determinental processes.<br /> <br /> <br /> ==2016 Spring==<br /> <br /> Tuesday, 2:25pm, B321 Van Vleck<br /> <br /> <br /> 3/29, 4/5: Fan Yang<br /> <br /> I will talk about the ergodic decomposition theorem (EDT). More specifically, given a compact metric space X and a continuous transformation T on it, the theorem shows that any T-invariant measure on X can be decomposed into a convex combination of ergodic measures. In the first talk I introduced the EDT and some related facts. In the second talk, I will talk about the conditional measures, and prove that the ergodic measures in EDT are indeed the conditional measures.<br /> <br /> <br /> 2/16 : Jinsu<br /> <br /> Lyapunov function for Markov Processes.<br /> <br /> For ODE, we can show stability of the trajectory using Lyapunov functions.<br /> <br /> There is an analogy for Markov Processes. I'd like to talk about the existence of stationary distribution with Lyapunov function.<br /> <br /> In some cases, it is also possible to show the rate of convergence to the stationary distribution.<br /> <br /> ==2015 Fall==<br /> <br /> This semester we will focus on tools and methods.<br /> <br /> [https://www.math.wisc.edu/wiki/images/a/ac/Reading_seminar_2015.pdf Seminar notes] ([https://www.dropbox.com/s/f4km7pevwfb1vbm/Reading%20seminar%202015.tex?dl=1 tex file], [https://www.dropbox.com/s/lg7kcgyf3nsukbx/Reading_seminar_2015.bib?dl=1 bib file])<br /> <br /> 9/15, 9/22: Elnur<br /> <br /> I will talk about large deviation theory and its applications. For the first talk, my plan is to introduce Gartner-Ellis theorem and show a few applications of it to finite state discrete time Markov chains.<br /> <br /> 9/29, 10/6, 10/13 :Dae Han<br /> <br /> 10/20, 10/27, 11/3: Jessica<br /> <br /> I will first present an overview of concentration of measure and concentration inequalities with a focus on the connection with related topics in analysis and geometry. Then, I will present Log-Sobolev inequalities and their connection to concentration of measure. <br /> <br /> 11/10, 11/17: Hao Kai<br /> <br /> 11/24, 12/1, 12/8, 12/15: Chris<br /> <br /> : <br /> <br /> <br /> <br /> <br /> <br /> 2016 Spring:<br /> <br /> 2/2, 2/9: Louis<br /> <br /> <br /> 2/16, 2/23: Jinsu<br /> <br /> 3/1, 3/8: Hans<br /> <br /> ==2015 Spring==<br /> <br /> <br /> 2/3, 2/10: Scott<br /> <br /> An Introduction to Entropy for Random Variables<br /> <br /> In these lectures I will introduce entropy for random variables and present some simple, finite state-space, examples to gain some intuition. We will prove the <br /> MacMillan Theorem using entropy and the law of large numbers. Then I will introduce relative entropy and prove the Markov Chain Convergence Theorem. Finally I will <br /> define entropy for a discrete time process. The lecture notes can be found at http://www.math.wisc.edu/~shottovy/EntropyLecture.pdf.<br /> <br /> 2/17, 2/24: Dae Han<br /> <br /> 3/3, 3/10: Hans<br /> <br /> 3/17, 3/24: In Gun<br /> <br /> 4/7, 4/14: Jinsu<br /> <br /> 4/21, 4/28: Chris N.<br /> <br /> <br /> <br /> <br /> <br /> <br /> ==2014 Fall==<br /> <br /> 9/23: Dave<br /> <br /> I will go over Mike Giles’ 2008 paper “Multi-level Monte Carlo path simulation.” This paper introduced a new Monte Carlo method to approximate expectations of SDEs (driven by Brownian motions) that is significantly more efficient than what was the state of the art. This work opened up a whole new field in the numerical analysis of stochastic processes as the basic idea is quite flexible and has found a variety of applications including SDEs driven by Brownian motions, Levy-driven SDEs, SPDEs, and models from biology<br /> <br /> 9/30: Benedek<br /> <br /> A very quick introduction to Stein's method. <br /> <br /> I will give a brief introduction to Stein's method, mostly based on the the first couple of sections of the following survey article:<br /> <br /> Ross, N. (2011). Fundamentals of Stein’s method. Probability Surveys, 8, 210-293. <br /> <br /> The following webpage has a huge collection of resources if you want to go deeper: https://sites.google.com/site/yvikswan/about-stein-s-method<br /> <br /> <br /> Note that the Midwest Probability Colloquium (http://www.math.northwestern.edu/mwp/) will have a tutorial program on Stein's method this year. <br /> <br /> 10/7, 10/14: Chris J.<br /> [http://www.math.wisc.edu/~janjigia/research/MartingaleProblemNotes.pdf An introduction to the (local) martingale problem.]<br /> <br /> <br /> 10/21, 10/28: Dae Han<br /> <br /> 11/4, 11/11: Elnur<br /> <br /> 11/18, 11/25: Chris N. Free Probability with an emphasis on C* and Von Neumann Algebras<br /> <br /> 12/2, 12/9: Yun Zhai<br /> <br /> ==2014 Spring==<br /> <br /> <br /> 1/28: Greg<br /> <br /> 2/04, 2/11: Scott <br /> <br /> [http://www.math.wisc.edu/~shottovy/BLT.pdf Reflected Brownian motion, Occupation time, and applications.] <br /> <br /> 2/18: Phil-- Examples of structure results in probability theory.<br /> <br /> 2/25, 3/4: Beth-- Derivative estimation for discrete time Markov chains<br /> <br /> 3/11, 3/25: Chris J [http://www.math.wisc.edu/~janjigia/research/stationarytalk.pdf Some classical results on stationary distributions of Markov processes]<br /> <br /> 4/1, 4/8: Chris N <br /> <br /> 4/15, 4/22: Yu Sun<br /> <br /> 4/29. 5/6: Diane<br /> <br /> ==2013 Fall==<br /> <br /> 9/24, 10/1: Chris<br /> [http://www.math.wisc.edu/~janjigia/research/metastabilitytalk.pdf A light introduction to metastability]<br /> <br /> 10/8, Dae Han<br /> Majoring multiplicative cascades for directed polymers in random media<br /> <br /> 10/15, 10/22: no reading seminar<br /> <br /> 10/29, 11/5: Elnur<br /> Limit fluctuations of last passage times <br /> <br /> 11/12: Yun<br /> Helffer-Sjostrand representation and Brascamp-Lieb inequality for stochastic interface models<br /> <br /> 11/19, 11/26: Yu Sun<br /> <br /> 12/3, 12/10: Jason<br /> <br /> ==2013 Spring==<br /> <br /> 2/13: Elnur <br /> <br /> Young diagrams, RSK correspondence, corner growth models, distribution of last passage times. <br /> <br /> 2/20: Elnur<br /> <br /> 2/27: Chris<br /> <br /> A brief introduction to enlargement of filtration and the Dufresne identity<br /> [http://www.math.wisc.edu/~janjigia/research/Presentation%20Notes.pdf Notes]<br /> <br /> 3/6: Chris<br /> <br /> 3/13: Dae Han<br /> <br /> An introduction to random polymers<br /> <br /> 3/20: Dae Han<br /> <br /> Directed polymers in a random environment: path localization and strong disorder<br /> <br /> 4/3: Diane<br /> <br /> Scale and Speed for honest 1 dimensional diffusions<br /> <br /> References: &lt;br&gt;<br /> Rogers &amp; Williams - Diffusions, Markov Processes and Martingales &lt;br&gt;<br /> Ito &amp; McKean - Diffusion Processes and their Sample Paths &lt;br&gt;<br /> Breiman - Probability &lt;br&gt;<br /> http://www.statslab.cam.ac.uk/~beresty/Articles/diffusions.pdf<br /> <br /> 4/10: Diane<br /> <br /> 4/17: Yun<br /> <br /> Introduction to stochastic interface models<br /> <br /> 4/24: Yun<br /> <br /> Dynamics and Gaussian equilibrium sytems<br /> <br /> 5/1: This reading seminar will be shifted because of a probability seminar.<br /> <br /> <br /> 5/8: Greg, Maso<br /> <br /> The Bethe ansatz vs. The Replica Trick. This lecture is an overview of the two <br /> approaches. See [http://arxiv.org/abs/1212.2267] for a nice overview.<br /> <br /> 5/15: Greg, Maso<br /> <br /> Rigorous use of the replica trick.</div> Valko https://www.math.wisc.edu/wiki/index.php?title=Graduate_student_reading_seminar&diff=16295 Graduate student reading seminar 2018-10-29T13:35:36Z <p>Valko: </p> <hr /> <div>(... in probability)<br /> <br /> <br /> ==2018 Fall==<br /> <br /> Tuesday 2:30pm, 901 Van Vleck<br /> <br /> <br /> The topic this semester is large deviation theory. Send me (BV) an email, if you want access to the shared Box folder with some reading material. <br /> <br /> <br /> 9/25, 10/2: Dae Han<br /> <br /> 10/9, 10/16: Kurt<br /> <br /> 10/23, 10/30: Stephen Davis<br /> <br /> 11/6, 11/13: Brandon Legried <br /> <br /> 11/20, 11/27: Shuqi Yu<br /> <br /> 12/4, 12/11: Yun Li<br /> <br /> ==2018 Spring==<br /> <br /> Tuesday 2:30pm, B135 Van Vleck<br /> <br /> <br /> Preliminary schedule:<br /> <br /> 2/20, 2/27: Yun<br /> <br /> 3/6, 3/13: Greg<br /> <br /> 3/20, 4/3: Yu<br /> <br /> 4/10, 4/17: Shuqi<br /> <br /> 4/24, 5/1: Tony<br /> <br /> ==2017 Fall==<br /> <br /> Tuesday 2:30pm, 214 Ingraham Hall<br /> <br /> <br /> Preliminary schedule: <br /> <br /> 9/26, 10/3: Hans<br /> <br /> 10/10, 10/17: Guo<br /> <br /> 10/24, 10/31: Chaoji<br /> <br /> 11/7, 11/14: Yun <br /> <br /> 11/21, 11/28: Kurt<br /> <br /> 12/5, 12/12: Christian<br /> <br /> <br /> <br /> <br /> ==2017 Spring==<br /> <br /> Tuesday 2:25pm, B211<br /> <br /> 1/31, 2/7: Fan<br /> <br /> I will talk about the Hanson-Wright inequality, which is a large deviation estimate for random variable of the form X^* A X, where X is a random vector with independent subgaussian entries and A is an arbitrary deterministic matrix. In the first talk, I will present a beautiful proof given by Mark Rudelson and Roman Vershynin. In the second talk, I will talk about some applications of this inequality.<br /> <br /> Reference: M. Rudelson and R. Vershynin, Hanson-Wright inequality and sub-gaussian concentration, Electron. Commun. Probab. Volume 18 (2013).<br /> <br /> 3/7, 3/14 : Jinsu<br /> <br /> Title : Donsker's Theorem and its application.<br /> Donsker's Theorem roughly says normalized random walk with linear interpolation on time interval [0,1] weakly converges to the Brownian motion B[0,1] in C([0,1]). It is sometimes called Donsker's invariance principle or the functional central limit theorem. I will show main ideas for the proof of this theorem tomorrow and show a couple of applications in my 2nd talk.<br /> <br /> Reference : https://www.math.utah.edu/~davar/ps-pdf-files/donsker.pdf<br /> <br /> ==2016 Fall==<br /> <br /> 9/27 Daniele<br /> <br /> Stochastic reaction networks.<br /> <br /> Stochastic reaction networks are continuous time Markov chain models used primarily in biochemistry. I will define them, prove some results that connect them to related deterministic models and introduce some open questions. <br /> <br /> 10/4 Jessica<br /> <br /> 10/11, 10/18: Dae Han<br /> <br /> 10/25, 11/1: Jinsu<br /> <br /> Coupling of Markov processes.<br /> <br /> When we have two distributions on same probability space, we can think of a pair whose marginal probability is each of two distributions.<br /> This pairing can be used to estimate the total variation distance between two distributions. This idea is called coupling method.<br /> I am going to introduce basic concepts,ideas and applications of coupling for Markov processes.<br /> <br /> Links of References<br /> <br /> http://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf<br /> <br /> http://websites.math.leidenuniv.nl/probability/lecturenotes/CouplingLectures.pdf<br /> <br /> 11/8, 11/15: Hans<br /> <br /> 11/22, 11/29: Keith<br /> <br /> Surprisingly Determinental: DPPs and some asymptotics of ASEP <br /> <br /> I'll be reading and presenting some recent papers of Alexei Borodin and a few collaborators which have uncovered certain equivalences between determinental point processes and non-determinental processes.<br /> <br /> <br /> ==2016 Spring==<br /> <br /> Tuesday, 2:25pm, B321 Van Vleck<br /> <br /> <br /> 3/29, 4/5: Fan Yang<br /> <br /> I will talk about the ergodic decomposition theorem (EDT). More specifically, given a compact metric space X and a continuous transformation T on it, the theorem shows that any T-invariant measure on X can be decomposed into a convex combination of ergodic measures. In the first talk I introduced the EDT and some related facts. In the second talk, I will talk about the conditional measures, and prove that the ergodic measures in EDT are indeed the conditional measures.<br /> <br /> <br /> 2/16 : Jinsu<br /> <br /> Lyapunov function for Markov Processes.<br /> <br /> For ODE, we can show stability of the trajectory using Lyapunov functions.<br /> <br /> There is an analogy for Markov Processes. I'd like to talk about the existence of stationary distribution with Lyapunov function.<br /> <br /> In some cases, it is also possible to show the rate of convergence to the stationary distribution.<br /> <br /> ==2015 Fall==<br /> <br /> This semester we will focus on tools and methods.<br /> <br /> [https://www.math.wisc.edu/wiki/images/a/ac/Reading_seminar_2015.pdf Seminar notes] ([https://www.dropbox.com/s/f4km7pevwfb1vbm/Reading%20seminar%202015.tex?dl=1 tex file], [https://www.dropbox.com/s/lg7kcgyf3nsukbx/Reading_seminar_2015.bib?dl=1 bib file])<br /> <br /> 9/15, 9/22: Elnur<br /> <br /> I will talk about large deviation theory and its applications. For the first talk, my plan is to introduce Gartner-Ellis theorem and show a few applications of it to finite state discrete time Markov chains.<br /> <br /> 9/29, 10/6, 10/13 :Dae Han<br /> <br /> 10/20, 10/27, 11/3: Jessica<br /> <br /> I will first present an overview of concentration of measure and concentration inequalities with a focus on the connection with related topics in analysis and geometry. Then, I will present Log-Sobolev inequalities and their connection to concentration of measure. <br /> <br /> 11/10, 11/17: Hao Kai<br /> <br /> 11/24, 12/1, 12/8, 12/15: Chris<br /> <br /> : <br /> <br /> <br /> <br /> <br /> <br /> 2016 Spring:<br /> <br /> 2/2, 2/9: Louis<br /> <br /> <br /> 2/16, 2/23: Jinsu<br /> <br /> 3/1, 3/8: Hans<br /> <br /> ==2015 Spring==<br /> <br /> <br /> 2/3, 2/10: Scott<br /> <br /> An Introduction to Entropy for Random Variables<br /> <br /> In these lectures I will introduce entropy for random variables and present some simple, finite state-space, examples to gain some intuition. We will prove the <br /> MacMillan Theorem using entropy and the law of large numbers. Then I will introduce relative entropy and prove the Markov Chain Convergence Theorem. Finally I will <br /> define entropy for a discrete time process. The lecture notes can be found at http://www.math.wisc.edu/~shottovy/EntropyLecture.pdf.<br /> <br /> 2/17, 2/24: Dae Han<br /> <br /> 3/3, 3/10: Hans<br /> <br /> 3/17, 3/24: In Gun<br /> <br /> 4/7, 4/14: Jinsu<br /> <br /> 4/21, 4/28: Chris N.<br /> <br /> <br /> <br /> <br /> <br /> <br /> ==2014 Fall==<br /> <br /> 9/23: Dave<br /> <br /> I will go over Mike Giles’ 2008 paper “Multi-level Monte Carlo path simulation.” This paper introduced a new Monte Carlo method to approximate expectations of SDEs (driven by Brownian motions) that is significantly more efficient than what was the state of the art. This work opened up a whole new field in the numerical analysis of stochastic processes as the basic idea is quite flexible and has found a variety of applications including SDEs driven by Brownian motions, Levy-driven SDEs, SPDEs, and models from biology<br /> <br /> 9/30: Benedek<br /> <br /> A very quick introduction to Stein's method. <br /> <br /> I will give a brief introduction to Stein's method, mostly based on the the first couple of sections of the following survey article:<br /> <br /> Ross, N. (2011). Fundamentals of Stein’s method. Probability Surveys, 8, 210-293. <br /> <br /> The following webpage has a huge collection of resources if you want to go deeper: https://sites.google.com/site/yvikswan/about-stein-s-method<br /> <br /> <br /> Note that the Midwest Probability Colloquium (http://www.math.northwestern.edu/mwp/) will have a tutorial program on Stein's method this year. <br /> <br /> 10/7, 10/14: Chris J.<br /> [http://www.math.wisc.edu/~janjigia/research/MartingaleProblemNotes.pdf An introduction to the (local) martingale problem.]<br /> <br /> <br /> 10/21, 10/28: Dae Han<br /> <br /> 11/4, 11/11: Elnur<br /> <br /> 11/18, 11/25: Chris N. Free Probability with an emphasis on C* and Von Neumann Algebras<br /> <br /> 12/2, 12/9: Yun Zhai<br /> <br /> ==2014 Spring==<br /> <br /> <br /> 1/28: Greg<br /> <br /> 2/04, 2/11: Scott <br /> <br /> [http://www.math.wisc.edu/~shottovy/BLT.pdf Reflected Brownian motion, Occupation time, and applications.] <br /> <br /> 2/18: Phil-- Examples of structure results in probability theory.<br /> <br /> 2/25, 3/4: Beth-- Derivative estimation for discrete time Markov chains<br /> <br /> 3/11, 3/25: Chris J [http://www.math.wisc.edu/~janjigia/research/stationarytalk.pdf Some classical results on stationary distributions of Markov processes]<br /> <br /> 4/1, 4/8: Chris N <br /> <br /> 4/15, 4/22: Yu Sun<br /> <br /> 4/29. 5/6: Diane<br /> <br /> ==2013 Fall==<br /> <br /> 9/24, 10/1: Chris<br /> [http://www.math.wisc.edu/~janjigia/research/metastabilitytalk.pdf A light introduction to metastability]<br /> <br /> 10/8, Dae Han<br /> Majoring multiplicative cascades for directed polymers in random media<br /> <br /> 10/15, 10/22: no reading seminar<br /> <br /> 10/29, 11/5: Elnur<br /> Limit fluctuations of last passage times <br /> <br /> 11/12: Yun<br /> Helffer-Sjostrand representation and Brascamp-Lieb inequality for stochastic interface models<br /> <br /> 11/19, 11/26: Yu Sun<br /> <br /> 12/3, 12/10: Jason<br /> <br /> ==2013 Spring==<br /> <br /> 2/13: Elnur <br /> <br /> Young diagrams, RSK correspondence, corner growth models, distribution of last passage times. <br /> <br /> 2/20: Elnur<br /> <br /> 2/27: Chris<br /> <br /> A brief introduction to enlargement of filtration and the Dufresne identity<br /> [http://www.math.wisc.edu/~janjigia/research/Presentation%20Notes.pdf Notes]<br /> <br /> 3/6: Chris<br /> <br /> 3/13: Dae Han<br /> <br /> An introduction to random polymers<br /> <br /> 3/20: Dae Han<br /> <br /> Directed polymers in a random environment: path localization and strong disorder<br /> <br /> 4/3: Diane<br /> <br /> Scale and Speed for honest 1 dimensional diffusions<br /> <br /> References: &lt;br&gt;<br /> Rogers &amp; Williams - Diffusions, Markov Processes and Martingales &lt;br&gt;<br /> Ito &amp; McKean - Diffusion Processes and their Sample Paths &lt;br&gt;<br /> Breiman - Probability &lt;br&gt;<br /> http://www.statslab.cam.ac.uk/~beresty/Articles/diffusions.pdf<br /> <br /> 4/10: Diane<br /> <br /> 4/17: Yun<br /> <br /> Introduction to stochastic interface models<br /> <br /> 4/24: Yun<br /> <br /> Dynamics and Gaussian equilibrium sytems<br /> <br /> 5/1: This reading seminar will be shifted because of a probability seminar.<br /> <br /> <br /> 5/8: Greg, Maso<br /> <br /> The Bethe ansatz vs. The Replica Trick. This lecture is an overview of the two <br /> approaches. See [http://arxiv.org/abs/1212.2267] for a nice overview.<br /> <br /> 5/15: Greg, Maso<br /> <br /> Rigorous use of the replica trick.</div> Valko https://www.math.wisc.edu/wiki/index.php?title=Graduate_student_reading_seminar&diff=15980 Graduate student reading seminar 2018-09-13T19:50:51Z <p>Valko: /* 2018 Fall */</p> <hr /> <div>==2018 Fall==<br /> <br /> Tuesday 2:30pm, 901 Van Vleck<br /> <br /> <br /> The topic this semester is large deviation theory. Send me (BV) an email, if you want access to the shared Box folder with some reading material. <br /> <br /> <br /> 9/25, 10/2: Dae Han<br /> <br /> 10/9, 10/16: Kurt<br /> <br /> 10/23, 10/30: Stephen Davis<br /> <br /> 11/6, 11/13: Brandon Legried <br /> <br /> 11/20, 11/27: Shuqi Yu<br /> <br /> 12/4, 12/11: Yun Li<br /> <br /> ==2018 Spring==<br /> <br /> Tuesday 2:30pm, B135 Van Vleck<br /> <br /> <br /> Preliminary schedule:<br /> <br /> 2/20, 2/27: Yun<br /> <br /> 3/6, 3/13: Greg<br /> <br /> 3/20, 4/3: Yu<br /> <br /> 4/10, 4/17: Shuqi<br /> <br /> 4/24, 5/1: Tony<br /> <br /> ==2017 Fall==<br /> <br /> Tuesday 2:30pm, 214 Ingraham Hall<br /> <br /> <br /> Preliminary schedule: <br /> <br /> 9/26, 10/3: Hans<br /> <br /> 10/10, 10/17: Guo<br /> <br /> 10/24, 10/31: Chaoji<br /> <br /> 11/7, 11/14: Yun <br /> <br /> 11/21, 11/28: Kurt<br /> <br /> 12/5, 12/12: Christian<br /> <br /> <br /> <br /> <br /> ==2017 Spring==<br /> <br /> Tuesday 2:25pm, B211<br /> <br /> 1/31, 2/7: Fan<br /> <br /> I will talk about the Hanson-Wright inequality, which is a large deviation estimate for random variable of the form X^* A X, where X is a random vector with independent subgaussian entries and A is an arbitrary deterministic matrix. In the first talk, I will present a beautiful proof given by Mark Rudelson and Roman Vershynin. In the second talk, I will talk about some applications of this inequality.<br /> <br /> Reference: M. Rudelson and R. Vershynin, Hanson-Wright inequality and sub-gaussian concentration, Electron. Commun. Probab. Volume 18 (2013).<br /> <br /> 3/7, 3/14 : Jinsu<br /> <br /> Title : Donsker's Theorem and its application.<br /> Donsker's Theorem roughly says normalized random walk with linear interpolation on time interval [0,1] weakly converges to the Brownian motion B[0,1] in C([0,1]). It is sometimes called Donsker's invariance principle or the functional central limit theorem. I will show main ideas for the proof of this theorem tomorrow and show a couple of applications in my 2nd talk.<br /> <br /> Reference : https://www.math.utah.edu/~davar/ps-pdf-files/donsker.pdf<br /> <br /> ==2016 Fall==<br /> <br /> 9/27 Daniele<br /> <br /> Stochastic reaction networks.<br /> <br /> Stochastic reaction networks are continuous time Markov chain models used primarily in biochemistry. I will define them, prove some results that connect them to related deterministic models and introduce some open questions. <br /> <br /> 10/4 Jessica<br /> <br /> 10/11, 10/18: Dae Han<br /> <br /> 10/25, 11/1: Jinsu<br /> <br /> Coupling of Markov processes.<br /> <br /> When we have two distributions on same probability space, we can think of a pair whose marginal probability is each of two distributions.<br /> This pairing can be used to estimate the total variation distance between two distributions. This idea is called coupling method.<br /> I am going to introduce basic concepts,ideas and applications of coupling for Markov processes.<br /> <br /> Links of References<br /> <br /> http://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf<br /> <br /> http://websites.math.leidenuniv.nl/probability/lecturenotes/CouplingLectures.pdf<br /> <br /> 11/8, 11/15: Hans<br /> <br /> 11/22, 11/29: Keith<br /> <br /> Surprisingly Determinental: DPPs and some asymptotics of ASEP <br /> <br /> I'll be reading and presenting some recent papers of Alexei Borodin and a few collaborators which have uncovered certain equivalences between determinental point processes and non-determinental processes.<br /> <br /> <br /> ==2016 Spring==<br /> <br /> Tuesday, 2:25pm, B321 Van Vleck<br /> <br /> <br /> 3/29, 4/5: Fan Yang<br /> <br /> I will talk about the ergodic decomposition theorem (EDT). More specifically, given a compact metric space X and a continuous transformation T on it, the theorem shows that any T-invariant measure on X can be decomposed into a convex combination of ergodic measures. In the first talk I introduced the EDT and some related facts. In the second talk, I will talk about the conditional measures, and prove that the ergodic measures in EDT are indeed the conditional measures.<br /> <br /> <br /> 2/16 : Jinsu<br /> <br /> Lyapunov function for Markov Processes.<br /> <br /> For ODE, we can show stability of the trajectory using Lyapunov functions.<br /> <br /> There is an analogy for Markov Processes. I'd like to talk about the existence of stationary distribution with Lyapunov function.<br /> <br /> In some cases, it is also possible to show the rate of convergence to the stationary distribution.<br /> <br /> ==2015 Fall==<br /> <br /> This semester we will focus on tools and methods.<br /> <br /> [https://www.math.wisc.edu/wiki/images/a/ac/Reading_seminar_2015.pdf Seminar notes] ([https://www.dropbox.com/s/f4km7pevwfb1vbm/Reading%20seminar%202015.tex?dl=1 tex file], [https://www.dropbox.com/s/lg7kcgyf3nsukbx/Reading_seminar_2015.bib?dl=1 bib file])<br /> <br /> 9/15, 9/22: Elnur<br /> <br /> I will talk about large deviation theory and its applications. For the first talk, my plan is to introduce Gartner-Ellis theorem and show a few applications of it to finite state discrete time Markov chains.<br /> <br /> 9/29, 10/6, 10/13 :Dae Han<br /> <br /> 10/20, 10/27, 11/3: Jessica<br /> <br /> I will first present an overview of concentration of measure and concentration inequalities with a focus on the connection with related topics in analysis and geometry. Then, I will present Log-Sobolev inequalities and their connection to concentration of measure. <br /> <br /> 11/10, 11/17: Hao Kai<br /> <br /> 11/24, 12/1, 12/8, 12/15: Chris<br /> <br /> : <br /> <br /> <br /> <br /> <br /> <br /> 2016 Spring:<br /> <br /> 2/2, 2/9: Louis<br /> <br /> <br /> 2/16, 2/23: Jinsu<br /> <br /> 3/1, 3/8: Hans<br /> <br /> ==2015 Spring==<br /> <br /> <br /> 2/3, 2/10: Scott<br /> <br /> An Introduction to Entropy for Random Variables<br /> <br /> In these lectures I will introduce entropy for random variables and present some simple, finite state-space, examples to gain some intuition. We will prove the <br /> MacMillan Theorem using entropy and the law of large numbers. Then I will introduce relative entropy and prove the Markov Chain Convergence Theorem. Finally I will <br /> define entropy for a discrete time process. The lecture notes can be found at http://www.math.wisc.edu/~shottovy/EntropyLecture.pdf.<br /> <br /> 2/17, 2/24: Dae Han<br /> <br /> 3/3, 3/10: Hans<br /> <br /> 3/17, 3/24: In Gun<br /> <br /> 4/7, 4/14: Jinsu<br /> <br /> 4/21, 4/28: Chris N.<br /> <br /> <br /> <br /> <br /> <br /> <br /> ==2014 Fall==<br /> <br /> 9/23: Dave<br /> <br /> I will go over Mike Giles’ 2008 paper “Multi-level Monte Carlo path simulation.” This paper introduced a new Monte Carlo method to approximate expectations of SDEs (driven by Brownian motions) that is significantly more efficient than what was the state of the art. This work opened up a whole new field in the numerical analysis of stochastic processes as the basic idea is quite flexible and has found a variety of applications including SDEs driven by Brownian motions, Levy-driven SDEs, SPDEs, and models from biology<br /> <br /> 9/30: Benedek<br /> <br /> A very quick introduction to Stein's method. <br /> <br /> I will give a brief introduction to Stein's method, mostly based on the the first couple of sections of the following survey article:<br /> <br /> Ross, N. (2011). Fundamentals of Stein’s method. Probability Surveys, 8, 210-293. <br /> <br /> The following webpage has a huge collection of resources if you want to go deeper: https://sites.google.com/site/yvikswan/about-stein-s-method<br /> <br /> <br /> Note that the Midwest Probability Colloquium (http://www.math.northwestern.edu/mwp/) will have a tutorial program on Stein's method this year. <br /> <br /> 10/7, 10/14: Chris J.<br /> [http://www.math.wisc.edu/~janjigia/research/MartingaleProblemNotes.pdf An introduction to the (local) martingale problem.]<br /> <br /> <br /> 10/21, 10/28: Dae Han<br /> <br /> 11/4, 11/11: Elnur<br /> <br /> 11/18, 11/25: Chris N. Free Probability with an emphasis on C* and Von Neumann Algebras<br /> <br /> 12/2, 12/9: Yun Zhai<br /> <br /> ==2014 Spring==<br /> <br /> <br /> 1/28: Greg<br /> <br /> 2/04, 2/11: Scott <br /> <br /> [http://www.math.wisc.edu/~shottovy/BLT.pdf Reflected Brownian motion, Occupation time, and applications.] <br /> <br /> 2/18: Phil-- Examples of structure results in probability theory.<br /> <br /> 2/25, 3/4: Beth-- Derivative estimation for discrete time Markov chains<br /> <br /> 3/11, 3/25: Chris J [http://www.math.wisc.edu/~janjigia/research/stationarytalk.pdf Some classical results on stationary distributions of Markov processes]<br /> <br /> 4/1, 4/8: Chris N <br /> <br /> 4/15, 4/22: Yu Sun<br /> <br /> 4/29. 5/6: Diane<br /> <br /> ==2013 Fall==<br /> <br /> 9/24, 10/1: Chris<br /> [http://www.math.wisc.edu/~janjigia/research/metastabilitytalk.pdf A light introduction to metastability]<br /> <br /> 10/8, Dae Han<br /> Majoring multiplicative cascades for directed polymers in random media<br /> <br /> 10/15, 10/22: no reading seminar<br /> <br /> 10/29, 11/5: Elnur<br /> Limit fluctuations of last passage times <br /> <br /> 11/12: Yun<br /> Helffer-Sjostrand representation and Brascamp-Lieb inequality for stochastic interface models<br /> <br /> 11/19, 11/26: Yu Sun<br /> <br /> 12/3, 12/10: Jason<br /> <br /> ==2013 Spring==<br /> <br /> 2/13: Elnur <br /> <br /> Young diagrams, RSK correspondence, corner growth models, distribution of last passage times. <br /> <br /> 2/20: Elnur<br /> <br /> 2/27: Chris<br /> <br /> A brief introduction to enlargement of filtration and the Dufresne identity<br /> [http://www.math.wisc.edu/~janjigia/research/Presentation%20Notes.pdf Notes]<br /> <br /> 3/6: Chris<br /> <br /> 3/13: Dae Han<br /> <br /> An introduction to random polymers<br /> <br /> 3/20: Dae Han<br /> <br /> Directed polymers in a random environment: path localization and strong disorder<br /> <br /> 4/3: Diane<br /> <br /> Scale and Speed for honest 1 dimensional diffusions<br /> <br /> References: &lt;br&gt;<br /> Rogers &amp; Williams - Diffusions, Markov Processes and Martingales &lt;br&gt;<br /> Ito &amp; McKean - Diffusion Processes and their Sample Paths &lt;br&gt;<br /> Breiman - Probability &lt;br&gt;<br /> http://www.statslab.cam.ac.uk/~beresty/Articles/diffusions.pdf<br /> <br /> 4/10: Diane<br /> <br /> 4/17: Yun<br /> <br /> Introduction to stochastic interface models<br /> <br /> 4/24: Yun<br /> <br /> Dynamics and Gaussian equilibrium sytems<br /> <br /> 5/1: This reading seminar will be shifted because of a probability seminar.<br /> <br /> <br /> 5/8: Greg, Maso<br /> <br /> The Bethe ansatz vs. The Replica Trick. This lecture is an overview of the two <br /> approaches. See [http://arxiv.org/abs/1212.2267] for a nice overview.<br /> <br /> 5/15: Greg, Maso<br /> <br /> Rigorous use of the replica trick.</div> Valko https://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=15972 Probability Seminar 2018-09-13T12:38:16Z <p>Valko: </p> <hr /> <div>__NOTOC__<br /> <br /> = Fall 2018 =<br /> <br /> &lt;b&gt;Thursdays in 901 Van Vleck Hall at 2:25 PM&lt;/b&gt;, unless otherwise noted. <br /> &lt;b&gt;We usually end for questions at 3:15 PM.&lt;/b&gt;<br /> <br /> If you would like to sign up for the email list to receive seminar announcements then please send an email to <br /> [mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br /> <br /> <br /> <br /> ==&lt;span style=&quot;color:red&quot;&gt; Friday, August 10, 10am, B239 Van Vleck &lt;/span&gt; András Mészáros, Central European University, Budapest ==<br /> <br /> <br /> Title: '''The distribution of sandpile groups of random regular graphs'''<br /> <br /> Abstract:<br /> We study the distribution of the sandpile group of random &lt;math&gt;d&lt;/math&gt;-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the &lt;math&gt;p&lt;/math&gt;-Sylow subgroup of the sandpile group is a given &lt;math&gt;p&lt;/math&gt;-group &lt;math&gt;P&lt;/math&gt;, is proportional to &lt;math&gt;|\operatorname{Aut}(P)|^{-1}&lt;/math&gt;. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br /> <br /> Our results extends a recent theorem of Huang saying that the adjacency matrices of random &lt;math&gt;d&lt;/math&gt;-regular directed graphs are invertible with high probability to the undirected case.<br /> <br /> &lt;!-- ==September 13, TBA == --&gt;<br /> <br /> ==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br /> <br /> Title: '''Stochastic quantization of Yang-Mills'''<br /> <br /> Abstract:<br /> &quot;Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br /> In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br /> <br /> <br /> <br /> ==September 27, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen] ==<br /> <br /> Title:'''Random walk in random environment and the Kardar-Parisi-Zhang class'''<br /> <br /> Abstract:This talk concerns a relationship between two much-studied classes of models of motion in a random medium, namely random walk in random environment (RWRE) and the Kardar-Parisi-Zhang (KPZ) universality class. Barraquand and Corwin (Columbia) discovered that in 1+1 dimensional RWRE in a dynamical beta environment the correction to the quenched large deviation principle obeys KPZ behavior. In this talk we condition the beta walk to escape at an atypical velocity and show that the resulting Doob-transformed RWRE obeys the KPZ wandering exponent 2/3. Based on joint work with Márton Balázs (Bristol) and Firas Rassoul-Agha (Utah). <br /> <br /> ==October 4, [https://people.math.osu.edu/paquette.30/ Elliot Paquette], [https://math.osu.edu/ OSU] ==<br /> <br /> ==October 11, [https://www.math.utah.edu/~janjigia/ Chris Janjigian], [https://www.math.utah.edu/ University of Utah] ==<br /> <br /> ==October 18-20, [http://sites.math.northwestern.edu/mwp/ Midwest Probability Colloquium], No Seminar ==<br /> <br /> ==October 25, [http://stat.columbia.edu/department-directory/name/promit-ghosal/ Promit Ghosal], Columbia ==<br /> <br /> ==November 1, TBA ==<br /> <br /> ==November 8, [https://cims.nyu.edu/~thomasl/ Thomas Leblé], NYU ==<br /> <br /> ==November 15, TBA ==<br /> <br /> ==November 22, [https://en.wikipedia.org/wiki/Thanksgiving Thanksgiving] Break, No Seminar ==<br /> <br /> ==November 29, TBA ==<br /> <br /> ==December 6, TBA ==<br /> <br /> <br /> == ==<br /> <br /> [[Past Seminars]]</div> Valko https://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=15971 Probability Seminar 2018-09-13T12:37:53Z <p>Valko: </p> <hr /> <div>__NOTOC__<br /> <br /> = Fall 2018 =<br /> <br /> &lt;b&gt;Thursdays in 901 Van Vleck Hall at 2:25 PM&lt;/b&gt;, unless otherwise noted. <br /> &lt;b&gt;We usually end for questions at 3:15 PM.&lt;/b&gt;<br /> <br /> If you would like to sign up for the email list to receive seminar announcements then please send an email to <br /> [mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br /> <br /> <br /> <br /> ==&lt;span style=&quot;color:red&quot;&gt; Friday, August 10, 10am, B239 Van Vleck &lt;/span&gt; András Mészáros, Central European University, Budapest ==<br /> <br /> <br /> Title: '''The distribution of sandpile groups of random regular graphs'''<br /> <br /> Abstract:<br /> We study the distribution of the sandpile group of random &lt;math&gt;d&lt;/math&gt;-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the &lt;math&gt;p&lt;/math&gt;-Sylow subgroup of the sandpile group is a given &lt;math&gt;p&lt;/math&gt;-group &lt;math&gt;P&lt;/math&gt;, is proportional to &lt;math&gt;|\operatorname{Aut}(P)|^{-1}&lt;/math&gt;. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br /> <br /> Our results extends a recent theorem of Huang saying that the adjacency matrices of random &lt;math&gt;d&lt;/math&gt;-regular directed graphs are invertible with high probability to the undirected case.<br /> <br /> &lt;!-- ==September 13, TBA == --&gt;<br /> <br /> ==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br /> <br /> Title: '''Stochastic quantization of Yang-Mills'''<br /> <br /> Abstract:<br /> &quot;Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br /> In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br /> <br /> <br /> <br /> ==September 27, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen] ==<br /> <br /> Title:'''Random walk in random environment and the Kardar-Parisi-Zhang class'''<br /> <br /> Abstract:This talk concerns a relationship between two much-studied classes of models of motion in a random medium, namely random walk in random environment (RWRE) and the Kardar-Parisi-Zhang (KPZ) universality class. Barraquand and Corwin (Columbia) discovered that in 1+1 dimensional RWRE in a dynamical beta environment the correction to the quenched large deviation principle obeys KPZ behavior. In this talk we condition the beta walk to escape at an atypical velocity and show that the resulting Doob-transformed RWRE obeys the KPZ wandering exponent 2/3. Based on joint work with Márton Balázs (Bristol) and Firas Rassoul-Agha (Utah). <br /> <br /> ==October 4, [https://people.math.osu.edu/paquette.30/ Elliot Paquette], [https://math.osu.edu/ OSU] ==<br /> <br /> ==October 11, [https://www.math.utah.edu/~janjigia/ Chris Janjigian], [https://www.math.utah.edu/ University of Utah] ==<br /> <br /> ==October 18-20, [http://sites.math.northwestern.edu/mwp/ Midwest Probability Colloquium], No Seminar ==<br /> <br /> ==October 25, [http://stat.columbia.edu/department-directory/name/promit-ghosal/ Promit Ghosal], Columbia ==<br /> <br /> ==November 1, TBA ==<br /> <br /> ==November 8, [https://cims.nyu.edu/~thomasl/ Thomas Leblé], NYU ==<br /> <br /> ==November 15, TBA ==<br /> <br /> ==November 22, [https://en.wikipedia.org/wiki/Thanksgiving Thanksgiving] Break, No Seminar ==<br /> <br /> ==November 29, TBA ==<br /> <br /> ==December 6, TBA ==<br /> <br /> <br /> == ==<br /> <br /> [[Past Seminars]]</div> Valko https://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=15881 Probability Seminar 2018-09-05T23:20:56Z <p>Valko: /* November 8, TBA */</p> <hr /> <div>__NOTOC__<br /> <br /> = Fall 2018 =<br /> <br /> &lt;b&gt;Thursdays in 901 Van Vleck Hall at 2:25 PM&lt;/b&gt;, unless otherwise noted. <br /> &lt;b&gt;We usually end for questions at 3:15 PM.&lt;/b&gt;<br /> <br /> If you would like to sign up for the email list to receive seminar announcements then please send an email to <br /> [mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br /> <br /> <br /> <br /> ==&lt;span style=&quot;color:red&quot;&gt; Friday, August 10, 10am, B239 Van Vleck &lt;/span&gt; András Mészáros, Central European University, Budapest ==<br /> <br /> <br /> Title: '''The distribution of sandpile groups of random regular graphs'''<br /> <br /> Abstract:<br /> We study the distribution of the sandpile group of random &lt;math&gt;d&lt;/math&gt;-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the &lt;math&gt;p&lt;/math&gt;-Sylow subgroup of the sandpile group is a given &lt;math&gt;p&lt;/math&gt;-group &lt;math&gt;P&lt;/math&gt;, is proportional to &lt;math&gt;|\operatorname{Aut}(P)|^{-1}&lt;/math&gt;. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br /> <br /> Our results extends a recent theorem of Huang saying that the adjacency matrices of random &lt;math&gt;d&lt;/math&gt;-regular directed graphs are invertible with high probability to the undirected case.<br /> <br /> ==September 13, TBA ==<br /> <br /> ==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br /> <br /> ==September 27, TBA ==<br /> <br /> ==October 4, [https://people.math.osu.edu/paquette.30/ Elliot Paquette], [https://math.osu.edu/ OSU] ==<br /> <br /> ==October 11, [https://www.math.utah.edu/~janjigia/ Chris Janjigian], [https://www.math.utah.edu/ University of Utah] ==<br /> <br /> ==October 18-20, [http://sites.math.northwestern.edu/mwp/ Midwest Probability Colloquium], No Seminar ==<br /> <br /> ==October 25, [http://stat.columbia.edu/department-directory/name/promit-ghosal/ Promit Ghosal], Columbia ==<br /> <br /> ==November 1, TBA ==<br /> <br /> ==November 8, [https://cims.nyu.edu/~thomasl/ Thomas Leblé], NYU ==<br /> <br /> ==November 15, TBA ==<br /> <br /> ==November 22, [https://en.wikipedia.org/wiki/Thanksgiving Thanksgiving] Break, No Seminar ==<br /> <br /> ==November 29, TBA ==<br /> <br /> ==December 6, TBA ==<br /> <br /> <br /> == ==<br /> <br /> [[Past Seminars]]</div> Valko https://www.math.wisc.edu/wiki/index.php?title=Graduate_student_reading_seminar&diff=15860 Graduate student reading seminar 2018-09-04T21:01:33Z <p>Valko: </p> <hr /> <div>==2018 Fall==<br /> <br /> Tuesday 2:30pm, 901 Van Vleck<br /> <br /> <br /> ==2018 Spring==<br /> <br /> Tuesday 2:30pm, B135 Van Vleck<br /> <br /> <br /> Preliminary schedule:<br /> <br /> 2/20, 2/27: Yun<br /> <br /> 3/6, 3/13: Greg<br /> <br /> 3/20, 4/3: Yu<br /> <br /> 4/10, 4/17: Shuqi<br /> <br /> 4/24, 5/1: Tony<br /> <br /> ==2017 Fall==<br /> <br /> Tuesday 2:30pm, 214 Ingraham Hall<br /> <br /> <br /> Preliminary schedule: <br /> <br /> 9/26, 10/3: Hans<br /> <br /> 10/10, 10/17: Guo<br /> <br /> 10/24, 10/31: Chaoji<br /> <br /> 11/7, 11/14: Yun <br /> <br /> 11/21, 11/28: Kurt<br /> <br /> 12/5, 12/12: Christian<br /> <br /> <br /> <br /> <br /> ==2017 Spring==<br /> <br /> Tuesday 2:25pm, B211<br /> <br /> 1/31, 2/7: Fan<br /> <br /> I will talk about the Hanson-Wright inequality, which is a large deviation estimate for random variable of the form X^* A X, where X is a random vector with independent subgaussian entries and A is an arbitrary deterministic matrix. In the first talk, I will present a beautiful proof given by Mark Rudelson and Roman Vershynin. In the second talk, I will talk about some applications of this inequality.<br /> <br /> Reference: M. Rudelson and R. Vershynin, Hanson-Wright inequality and sub-gaussian concentration, Electron. Commun. Probab. Volume 18 (2013).<br /> <br /> 3/7, 3/14 : Jinsu<br /> <br /> Title : Donsker's Theorem and its application.<br /> Donsker's Theorem roughly says normalized random walk with linear interpolation on time interval [0,1] weakly converges to the Brownian motion B[0,1] in C([0,1]). It is sometimes called Donsker's invariance principle or the functional central limit theorem. I will show main ideas for the proof of this theorem tomorrow and show a couple of applications in my 2nd talk.<br /> <br /> Reference : https://www.math.utah.edu/~davar/ps-pdf-files/donsker.pdf<br /> <br /> ==2016 Fall==<br /> <br /> 9/27 Daniele<br /> <br /> Stochastic reaction networks.<br /> <br /> Stochastic reaction networks are continuous time Markov chain models used primarily in biochemistry. I will define them, prove some results that connect them to related deterministic models and introduce some open questions. <br /> <br /> 10/4 Jessica<br /> <br /> 10/11, 10/18: Dae Han<br /> <br /> 10/25, 11/1: Jinsu<br /> <br /> Coupling of Markov processes.<br /> <br /> When we have two distributions on same probability space, we can think of a pair whose marginal probability is each of two distributions.<br /> This pairing can be used to estimate the total variation distance between two distributions. This idea is called coupling method.<br /> I am going to introduce basic concepts,ideas and applications of coupling for Markov processes.<br /> <br /> Links of References<br /> <br /> http://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf<br /> <br /> http://websites.math.leidenuniv.nl/probability/lecturenotes/CouplingLectures.pdf<br /> <br /> 11/8, 11/15: Hans<br /> <br /> 11/22, 11/29: Keith<br /> <br /> Surprisingly Determinental: DPPs and some asymptotics of ASEP <br /> <br /> I'll be reading and presenting some recent papers of Alexei Borodin and a few collaborators which have uncovered certain equivalences between determinental point processes and non-determinental processes.<br /> <br /> <br /> ==2016 Spring==<br /> <br /> Tuesday, 2:25pm, B321 Van Vleck<br /> <br /> <br /> 3/29, 4/5: Fan Yang<br /> <br /> I will talk about the ergodic decomposition theorem (EDT). More specifically, given a compact metric space X and a continuous transformation T on it, the theorem shows that any T-invariant measure on X can be decomposed into a convex combination of ergodic measures. In the first talk I introduced the EDT and some related facts. In the second talk, I will talk about the conditional measures, and prove that the ergodic measures in EDT are indeed the conditional measures.<br /> <br /> <br /> 2/16 : Jinsu<br /> <br /> Lyapunov function for Markov Processes.<br /> <br /> For ODE, we can show stability of the trajectory using Lyapunov functions.<br /> <br /> There is an analogy for Markov Processes. I'd like to talk about the existence of stationary distribution with Lyapunov function.<br /> <br /> In some cases, it is also possible to show the rate of convergence to the stationary distribution.<br /> <br /> ==2015 Fall==<br /> <br /> This semester we will focus on tools and methods.<br /> <br /> [https://www.math.wisc.edu/wiki/images/a/ac/Reading_seminar_2015.pdf Seminar notes] ([https://www.dropbox.com/s/f4km7pevwfb1vbm/Reading%20seminar%202015.tex?dl=1 tex file], [https://www.dropbox.com/s/lg7kcgyf3nsukbx/Reading_seminar_2015.bib?dl=1 bib file])<br /> <br /> 9/15, 9/22: Elnur<br /> <br /> I will talk about large deviation theory and its applications. For the first talk, my plan is to introduce Gartner-Ellis theorem and show a few applications of it to finite state discrete time Markov chains.<br /> <br /> 9/29, 10/6, 10/13 :Dae Han<br /> <br /> 10/20, 10/27, 11/3: Jessica<br /> <br /> I will first present an overview of concentration of measure and concentration inequalities with a focus on the connection with related topics in analysis and geometry. Then, I will present Log-Sobolev inequalities and their connection to concentration of measure. <br /> <br /> 11/10, 11/17: Hao Kai<br /> <br /> 11/24, 12/1, 12/8, 12/15: Chris<br /> <br /> : <br /> <br /> <br /> <br /> <br /> <br /> 2016 Spring:<br /> <br /> 2/2, 2/9: Louis<br /> <br /> <br /> 2/16, 2/23: Jinsu<br /> <br /> 3/1, 3/8: Hans<br /> <br /> ==2015 Spring==<br /> <br /> <br /> 2/3, 2/10: Scott<br /> <br /> An Introduction to Entropy for Random Variables<br /> <br /> In these lectures I will introduce entropy for random variables and present some simple, finite state-space, examples to gain some intuition. We will prove the <br /> MacMillan Theorem using entropy and the law of large numbers. Then I will introduce relative entropy and prove the Markov Chain Convergence Theorem. Finally I will <br /> define entropy for a discrete time process. The lecture notes can be found at http://www.math.wisc.edu/~shottovy/EntropyLecture.pdf.<br /> <br /> 2/17, 2/24: Dae Han<br /> <br /> 3/3, 3/10: Hans<br /> <br /> 3/17, 3/24: In Gun<br /> <br /> 4/7, 4/14: Jinsu<br /> <br /> 4/21, 4/28: Chris N.<br /> <br /> <br /> <br /> <br /> <br /> <br /> ==2014 Fall==<br /> <br /> 9/23: Dave<br /> <br /> I will go over Mike Giles’ 2008 paper “Multi-level Monte Carlo path simulation.” This paper introduced a new Monte Carlo method to approximate expectations of SDEs (driven by Brownian motions) that is significantly more efficient than what was the state of the art. This work opened up a whole new field in the numerical analysis of stochastic processes as the basic idea is quite flexible and has found a variety of applications including SDEs driven by Brownian motions, Levy-driven SDEs, SPDEs, and models from biology<br /> <br /> 9/30: Benedek<br /> <br /> A very quick introduction to Stein's method. <br /> <br /> I will give a brief introduction to Stein's method, mostly based on the the first couple of sections of the following survey article:<br /> <br /> Ross, N. (2011). Fundamentals of Stein’s method. Probability Surveys, 8, 210-293. <br /> <br /> The following webpage has a huge collection of resources if you want to go deeper: https://sites.google.com/site/yvikswan/about-stein-s-method<br /> <br /> <br /> Note that the Midwest Probability Colloquium (http://www.math.northwestern.edu/mwp/) will have a tutorial program on Stein's method this year. <br /> <br /> 10/7, 10/14: Chris J.<br /> [http://www.math.wisc.edu/~janjigia/research/MartingaleProblemNotes.pdf An introduction to the (local) martingale problem.]<br /> <br /> <br /> 10/21, 10/28: Dae Han<br /> <br /> 11/4, 11/11: Elnur<br /> <br /> 11/18, 11/25: Chris N. Free Probability with an emphasis on C* and Von Neumann Algebras<br /> <br /> 12/2, 12/9: Yun Zhai<br /> <br /> ==2014 Spring==<br /> <br /> <br /> 1/28: Greg<br /> <br /> 2/04, 2/11: Scott <br /> <br /> [http://www.math.wisc.edu/~shottovy/BLT.pdf Reflected Brownian motion, Occupation time, and applications.] <br /> <br /> 2/18: Phil-- Examples of structure results in probability theory.<br /> <br /> 2/25, 3/4: Beth-- Derivative estimation for discrete time Markov chains<br /> <br /> 3/11, 3/25: Chris J [http://www.math.wisc.edu/~janjigia/research/stationarytalk.pdf Some classical results on stationary distributions of Markov processes]<br /> <br /> 4/1, 4/8: Chris N <br /> <br /> 4/15, 4/22: Yu Sun<br /> <br /> 4/29. 5/6: Diane<br /> <br /> ==2013 Fall==<br /> <br /> 9/24, 10/1: Chris<br /> [http://www.math.wisc.edu/~janjigia/research/metastabilitytalk.pdf A light introduction to metastability]<br /> <br /> 10/8, Dae Han<br /> Majoring multiplicative cascades for directed polymers in random media<br /> <br /> 10/15, 10/22: no reading seminar<br /> <br /> 10/29, 11/5: Elnur<br /> Limit fluctuations of last passage times <br /> <br /> 11/12: Yun<br /> Helffer-Sjostrand representation and Brascamp-Lieb inequality for stochastic interface models<br /> <br /> 11/19, 11/26: Yu Sun<br /> <br /> 12/3, 12/10: Jason<br /> <br /> ==2013 Spring==<br /> <br /> 2/13: Elnur <br /> <br /> Young diagrams, RSK correspondence, corner growth models, distribution of last passage times. <br /> <br /> 2/20: Elnur<br /> <br /> 2/27: Chris<br /> <br /> A brief introduction to enlargement of filtration and the Dufresne identity<br /> [http://www.math.wisc.edu/~janjigia/research/Presentation%20Notes.pdf Notes]<br /> <br /> 3/6: Chris<br /> <br /> 3/13: Dae Han<br /> <br /> An introduction to random polymers<br /> <br /> 3/20: Dae Han<br /> <br /> Directed polymers in a random environment: path localization and strong disorder<br /> <br /> 4/3: Diane<br /> <br /> Scale and Speed for honest 1 dimensional diffusions<br /> <br /> References: &lt;br&gt;<br /> Rogers &amp; Williams - Diffusions, Markov Processes and Martingales &lt;br&gt;<br /> Ito &amp; McKean - Diffusion Processes and their Sample Paths &lt;br&gt;<br /> Breiman - Probability &lt;br&gt;<br /> http://www.statslab.cam.ac.uk/~beresty/Articles/diffusions.pdf<br /> <br /> 4/10: Diane<br /> <br /> 4/17: Yun<br /> <br /> Introduction to stochastic interface models<br /> <br /> 4/24: Yun<br /> <br /> Dynamics and Gaussian equilibrium sytems<br /> <br /> 5/1: This reading seminar will be shifted because of a probability seminar.<br /> <br /> <br /> 5/8: Greg, Maso<br /> <br /> The Bethe ansatz vs. The Replica Trick. This lecture is an overview of the two <br /> approaches. See [http://arxiv.org/abs/1212.2267] for a nice overview.<br /> <br /> 5/15: Greg, Maso<br /> <br /> Rigorous use of the replica trick.</div> Valko https://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=15859 Probability Seminar 2018-09-04T21:00:35Z <p>Valko: </p> <hr /> <div>__NOTOC__<br /> <br /> = Fall 2018 =<br /> <br /> &lt;b&gt;Thursdays in 901 Van Vleck Hall at 2:25 PM&lt;/b&gt;, unless otherwise noted. <br /> &lt;b&gt;We usually end for questions at 3:15 PM.&lt;/b&gt;<br /> <br /> If you would like to sign up for the email list to receive seminar announcements then please send an email to <br /> [mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br /> <br /> <br /> <br /> ==&lt;span style=&quot;color:red&quot;&gt; Friday, August 10, 10am, B239 Van Vleck &lt;/span&gt; András Mészáros, Central European University, Budapest ==<br /> <br /> <br /> Title: '''The distribution of sandpile groups of random regular graphs'''<br /> <br /> Abstract:<br /> We study the distribution of the sandpile group of random &lt;math&gt;d&lt;/math&gt;-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the &lt;math&gt;p&lt;/math&gt;-Sylow subgroup of the sandpile group is a given &lt;math&gt;p&lt;/math&gt;-group &lt;math&gt;P&lt;/math&gt;, is proportional to &lt;math&gt;|\operatorname{Aut}(P)|^{-1}&lt;/math&gt;. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br /> <br /> Our results extends a recent theorem of Huang saying that the adjacency matrices of random &lt;math&gt;d&lt;/math&gt;-regular directed graphs are invertible with high probability to the undirected case.<br /> <br /> ==September 13, TBA ==<br /> <br /> ==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br /> <br /> ==September 27, TBA ==<br /> <br /> ==October 4, [https://people.math.osu.edu/paquette.30/ Elliot Paquette], [https://math.osu.edu/ OSU] ==<br /> <br /> ==October 11, [https://www.math.utah.edu/~janjigia/ Chris Janjigian], [https://www.math.utah.edu/ University of Utah] ==<br /> <br /> ==October 18-20, [http://sites.math.northwestern.edu/mwp/ Midwest Probability Colloquium], No Seminar ==<br /> <br /> ==October 25, [http://stat.columbia.edu/department-directory/name/promit-ghosal/ Promit Ghosal], Columbia ==<br /> <br /> ==November 1, TBA ==<br /> <br /> ==November 8, TBA ==<br /> <br /> ==November 15, TBA ==<br /> <br /> ==November 22, [https://en.wikipedia.org/wiki/Thanksgiving Thanksgiving] Break, No Seminar ==<br /> <br /> ==November 29, TBA ==<br /> <br /> ==December 6, TBA ==<br /> <br /> <br /> == ==<br /> <br /> [[Past Seminars]]</div> Valko https://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=15858 Probability Seminar 2018-09-04T20:59:40Z <p>Valko: /* October 25, TBA */</p> <hr /> <div>__NOTOC__<br /> <br /> = Fall 2018 =<br /> <br /> &lt;b&gt;Thursdays in 901 Van Vleck Hall at 2:25 PM&lt;/b&gt;, unless otherwise noted. <br /> &lt;b&gt;We usually end for questions at 3:15 PM.&lt;/b&gt;<br /> <br /> If you would like to sign up for the email list to receive seminar announcements then please send an email to <br /> [mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br /> <br /> <br /> <br /> ==&lt;span style=&quot;color:red&quot;&gt; Friday, August 10, 10am, B239 Van Vleck &lt;/span&gt; András Mészáros, Central European University, Budapest ==<br /> <br /> <br /> Title: '''The distribution of sandpile groups of random regular graphs'''<br /> <br /> Abstract:<br /> We study the distribution of the sandpile group of random &lt;math&gt;d&lt;/math&gt;-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the &lt;math&gt;p&lt;/math&gt;-Sylow subgroup of the sandpile group is a given &lt;math&gt;p&lt;/math&gt;-group &lt;math&gt;P&lt;/math&gt;, is proportional to &lt;math&gt;|\operatorname{Aut}(P)|^{-1}&lt;/math&gt;. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br /> <br /> Our results extends a recent theorem of Huang saying that the adjacency matrices of random &lt;math&gt;d&lt;/math&gt;-regular directed graphs are invertible with high probability to the undirected case.<br /> <br /> ==September 13, TBA ==<br /> <br /> ==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br /> <br /> ==September 27, TBA ==<br /> <br /> ==October 4, [https://people.math.osu.edu/paquette.30/ Elliot Paquette], [https://math.osu.edu/ OSU] ==<br /> <br /> ==October 11, [https://www.math.utah.edu/~janjigia/ Chris Janjigian], [https://www.math.utah.edu/ University of Utah] ==<br /> <br /> ==October 18-20, [http://sites.math.northwestern.edu/mwp/ Midwest Probability Colloquium], No Seminar ==<br /> <br /> ==October 25, Promit Ghosal (Columbia) ==<br /> <br /> ==November 1, TBA ==<br /> <br /> ==November 8, TBA ==<br /> <br /> ==November 15, TBA ==<br /> <br /> ==November 22, [https://en.wikipedia.org/wiki/Thanksgiving Thanksgiving] Break, No Seminar ==<br /> <br /> ==November 29, TBA ==<br /> <br /> ==December 6, TBA ==<br /> <br /> <br /> == ==<br /> <br /> [[Past Seminars]]</div> Valko https://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=15857 Probability Seminar 2018-09-04T20:58:52Z <p>Valko: </p> <hr /> <div>__NOTOC__<br /> <br /> = Fall 2018 =<br /> <br /> &lt;b&gt;Thursdays in 901 Van Vleck Hall at 2:25 PM&lt;/b&gt;, unless otherwise noted. <br /> &lt;b&gt;We usually end for questions at 3:15 PM.&lt;/b&gt;<br /> <br /> If you would like to sign up for the email list to receive seminar announcements then please send an email to <br /> [mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br /> <br /> <br /> <br /> ==&lt;span style=&quot;color:red&quot;&gt; Friday, August 10, 10am, B239 Van Vleck &lt;/span&gt; András Mészáros, Central European University, Budapest ==<br /> <br /> <br /> Title: '''The distribution of sandpile groups of random regular graphs'''<br /> <br /> Abstract:<br /> We study the distribution of the sandpile group of random &lt;math&gt;d&lt;/math&gt;-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the &lt;math&gt;p&lt;/math&gt;-Sylow subgroup of the sandpile group is a given &lt;math&gt;p&lt;/math&gt;-group &lt;math&gt;P&lt;/math&gt;, is proportional to &lt;math&gt;|\operatorname{Aut}(P)|^{-1}&lt;/math&gt;. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br /> <br /> Our results extends a recent theorem of Huang saying that the adjacency matrices of random &lt;math&gt;d&lt;/math&gt;-regular directed graphs are invertible with high probability to the undirected case.<br /> <br /> ==September 13, TBA ==<br /> <br /> ==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br /> <br /> ==September 27, TBA ==<br /> <br /> ==October 4, [https://people.math.osu.edu/paquette.30/ Elliot Paquette], [https://math.osu.edu/ OSU] ==<br /> <br /> ==October 11, [https://www.math.utah.edu/~janjigia/ Chris Janjigian], [https://www.math.utah.edu/ University of Utah] ==<br /> <br /> ==October 18-20, [http://sites.math.northwestern.edu/mwp/ Midwest Probability Colloquium], No Seminar ==<br /> <br /> ==October 25, TBA ==<br /> <br /> ==November 1, TBA ==<br /> <br /> ==November 8, TBA ==<br /> <br /> ==November 15, TBA ==<br /> <br /> ==November 22, [https://en.wikipedia.org/wiki/Thanksgiving Thanksgiving] Break, No Seminar ==<br /> <br /> ==November 29, TBA ==<br /> <br /> ==December 6, TBA ==<br /> <br /> <br /> == ==<br /> <br /> [[Past Seminars]]</div> Valko https://www.math.wisc.edu/wiki/index.php?title=Probability&diff=15856 Probability 2018-09-04T20:47:03Z <p>Valko: /* Graduate Courses in Probability */</p> <hr /> <div>__NOTOC__<br /> <br /> = '''Probability at UW-Madison''' =<br /> <br /> &lt;br&gt;<br /> <br /> == Tenured and tenure-track faculty ==<br /> <br /> [http://www.math.wisc.edu/~anderson/ David Anderson] (Duke, 2005) applied probability, numerical methods, mathematical biology.<br /> <br /> [http://www.math.wisc.edu/~roch/ Sebastien Roch] (UC Berkeley, 2007) applied probability, mathematical biology, theoretical computer science.<br /> <br /> [http://www.math.wisc.edu/~seppalai/ Timo Seppäläinen] (Minnesota, 1991) motion in a random medium, random growth models, interacting particle systems, large deviation theory.<br /> <br /> Hao Shen (Princeton, 2013) stochastic partial differential equations, integrable probability<br /> <br /> [http://www.math.wisc.edu/~valko/ Benedek Valko] (Budapest, 2004) interacting particle systems, random matrices.<br /> <br /> [http://www.math.wisc.edu/~pmwood/ Philip Matchett Wood] (Rutgers, 2009) combinatorics, random matrices.<br /> <br /> <br /> == Emeriti ==<br /> <br /> [http://psoup.math.wisc.edu/kitchen.html David Griffeath] (Cornell, 1976)<br /> <br /> [http://www.math.wisc.edu/~kuelbs Jim Kuelbs] (Minnesota, 1965)<br /> <br /> [http://www.math.wisc.edu/~kurtz Tom Kurtz] (Stanford, 1967)<br /> <br /> Peter Ney (Columbia, 1961)<br /> <br /> Josh Chover (Michigan, 1952)<br /> <br /> == Graduate students ==<br /> <br /> <br /> [http://www.math.wisc.edu/~kehlert/ Kurt Ehlert] <br /> <br /> [http://www.math.wisc.edu/~kang Dae Han Kang]<br /> <br /> [https://sites.google.com/a/wisc.edu/brandon-legried/ Brandon Legried]<br /> <br /> Yun Li<br /> <br /> [http://sites.google.com/a/wisc.edu/tung-nguyen/ Tung Nguyen]<br /> <br /> [http://www.math.wisc.edu/~cyuan25/ Chaojie Yuan]<br /> <br /> <br /> <br /> == [[Probability Seminar]] ==<br /> <br /> Thursdays at 2:25pm, VV901<br /> <br /> ==[[Graduate student reading seminar]]==<br /> <br /> Email list: join-grad_prob_seminar@lists.wisc.edu<br /> <br /> Tuesdays, 2:30pm, 901 Van Vleck<br /> <br /> == [[Probability group timetable]]==<br /> <br /> == [[Undergraduate courses in probability]]==<br /> <br /> == Graduate Courses in Probability ==<br /> <br /> <br /> <br /> '''2018 Fall'''<br /> <br /> [https://www.math.wisc.edu/~anderson/733F18/733.html Math/Stat 733 Theory of Probability I]<br /> <br /> Math 735 Stochastic Analysis<br /> <br /> <br /> <br /> '''2017 Spring'''<br /> <br /> Math/Stat 734 Theory of Probability II <br /> <br /> Math 833 Topics in Probability: Random Matrix Theory</div> Valko https://www.math.wisc.edu/wiki/index.php?title=Probability&diff=15840 Probability 2018-09-04T15:21:36Z <p>Valko: </p> <hr /> <div>__NOTOC__<br /> <br /> = '''Probability at UW-Madison''' =<br /> <br /> &lt;br&gt;<br /> <br /> == Tenured and tenure-track faculty ==<br /> <br /> [http://www.math.wisc.edu/~anderson/ David Anderson] (Duke, 2005) applied probability, numerical methods, mathematical biology.<br /> <br /> [http://www.math.wisc.edu/~roch/ Sebastien Roch] (UC Berkeley, 2007) applied probability, mathematical biology, theoretical computer science.<br /> <br /> [http://www.math.wisc.edu/~seppalai/ Timo Seppäläinen] (Minnesota, 1991) motion in a random medium, random growth models, interacting particle systems, large deviation theory.<br /> <br /> Hao Shen (Princeton, 2013) stochastic partial differential equations, integrable probability<br /> <br /> [http://www.math.wisc.edu/~valko/ Benedek Valko] (Budapest, 2004) interacting particle systems, random matrices.<br /> <br /> [http://www.math.wisc.edu/~pmwood/ Philip Matchett Wood] (Rutgers, 2009) combinatorics, random matrices.<br /> <br /> <br /> == Emeriti ==<br /> <br /> [http://psoup.math.wisc.edu/kitchen.html David Griffeath] (Cornell, 1976)<br /> <br /> [http://www.math.wisc.edu/~kuelbs Jim Kuelbs] (Minnesota, 1965)<br /> <br /> [http://www.math.wisc.edu/~kurtz Tom Kurtz] (Stanford, 1967)<br /> <br /> Peter Ney (Columbia, 1961)<br /> <br /> Josh Chover (Michigan, 1952)<br /> <br /> == Graduate students ==<br /> <br /> <br /> [http://www.math.wisc.edu/~kehlert/ Kurt Ehlert] <br /> <br /> [http://www.math.wisc.edu/~kang Dae Han Kang]<br /> <br /> [https://sites.google.com/a/wisc.edu/brandon-legried/ Brandon Legried]<br /> <br /> Yun Li<br /> <br /> Chaojie Yuan<br /> <br /> <br /> == [[Probability Seminar]] ==<br /> <br /> Thursdays at 2:25pm, VV901<br /> <br /> ==[[Graduate student reading seminar]]==<br /> <br /> Email list: join-grad_prob_seminar@lists.wisc.edu<br /> <br /> Tuesdays, 2:30pm, 901 Van Vleck<br /> <br /> == [[Probability group timetable]]==<br /> <br /> == [[Undergraduate courses in probability]]==<br /> <br /> == Graduate Courses in Probability ==<br /> <br /> <br /> <br /> '''2018 Fall'''<br /> <br /> [https://www.math.wisc.edu/~anderson/733F18/733.htmll Math/Stat 733 Theory of Probability I]<br /> <br /> Math 735 Stochastic Analysis<br /> <br /> <br /> <br /> '''2017 Spring'''<br /> <br /> Math/Stat 734 Theory of Probability II <br /> <br /> Math 833 Topics in Probability: Random Matrix Theory</div> Valko https://www.math.wisc.edu/wiki/index.php?title=Probability&diff=15839 Probability 2018-09-04T15:20:06Z <p>Valko: </p> <hr /> <div>__NOTOC__<br /> <br /> = '''Probability at UW-Madison''' =<br /> <br /> &lt;br&gt;<br /> <br /> == Tenured and tenure-track faculty ==<br /> <br /> [http://www.math.wisc.edu/~anderson/ David Anderson] (Duke, 2005) applied probability, numerical methods, mathematical biology.<br /> <br /> [http://www.math.wisc.edu/~roch/ Sebastien Roch] (UC Berkeley, 2007) applied probability, mathematical biology, theoretical computer science.<br /> <br /> [http://www.math.wisc.edu/~seppalai/ Timo Seppäläinen] (Minnesota, 1991) motion in a random medium, random growth models, interacting particle systems, large deviation theory.<br /> <br /> Hao Shen (Princeton, 2013) stochastic partial differential equations, integrable probability<br /> <br /> [http://www.math.wisc.edu/~valko/ Benedek Valko] (Budapest, 2004) interacting particle systems, random matrices.<br /> <br /> [http://www.math.wisc.edu/~pmwood/ Philip Matchett Wood] (Rutgers, 2009) combinatorics, random matrices.<br /> <br /> <br /> == Emeriti ==<br /> <br /> [http://psoup.math.wisc.edu/kitchen.html David Griffeath] (Cornell, 1976)<br /> <br /> [http://www.math.wisc.edu/~kuelbs Jim Kuelbs] (Minnesota, 1965)<br /> <br /> [http://www.math.wisc.edu/~kurtz Tom Kurtz] (Stanford, 1967)<br /> <br /> Peter Ney (Columbia, 1961)<br /> <br /> Josh Chover (Michigan, 1952)<br /> <br /> == Graduate students ==<br /> <br /> <br /> [http://www.math.wisc.edu/~kehlert/ Kurt Ehlert] <br /> <br /> [http://www.math.wisc.edu/~kang Dae Han Kang]<br /> <br /> Brandon Legried<br /> <br /> Yun Li<br /> <br /> Chaojie Yuan<br /> <br /> <br /> == [[Probability Seminar]] ==<br /> <br /> Thursdays at 2:25pm, VV901<br /> <br /> ==[[Graduate student reading seminar]]==<br /> <br /> Email list: join-grad_prob_seminar@lists.wisc.edu<br /> <br /> Tuesdays, 2:30pm, 901 Van Vleck<br /> <br /> == [[Probability group timetable]]==<br /> <br /> == [[Undergraduate courses in probability]]==<br /> <br /> == Graduate Courses in Probability ==<br /> <br /> <br /> <br /> '''2018 Fall'''<br /> <br /> [https://www.math.wisc.edu/~anderson/733F18/733.htmll Math/Stat 733 Theory of Probability I]<br /> <br /> Math 735 Stochastic Analysis<br /> <br /> <br /> <br /> '''2017 Spring'''<br /> <br /> Math/Stat 734 Theory of Probability II <br /> <br /> Math 833 Topics in Probability: Random Matrix Theory</div> Valko https://www.math.wisc.edu/wiki/index.php?title=Probability&diff=15836 Probability 2018-09-04T14:29:27Z <p>Valko: /* Graduate Courses in Probability */</p> <hr /> <div>__NOTOC__<br /> <br /> = '''Probability at UW-Madison''' =<br /> <br /> &lt;br&gt;<br /> <br /> == Tenured and tenure-track faculty ==<br /> <br /> [http://www.math.wisc.edu/~anderson/ David Anderson] (Duke, 2005) applied probability, numerical methods, mathematical biology.<br /> <br /> [http://www.math.wisc.edu/~roch/ Sebastien Roch] (UC Berkeley, 2007) applied probability, mathematical biology, theoretical computer science.<br /> <br /> [http://www.math.wisc.edu/~seppalai/ Timo Seppäläinen] (Minnesota, 1991) motion in a random medium, random growth models, interacting particle systems, large deviation theory.<br /> <br /> Hao Shen (Princeton, 2013) stochastic partial differential equations, integrable probability<br /> <br /> [http://www.math.wisc.edu/~valko/ Benedek Valko] (Budapest, 2004) interacting particle systems, random matrices.<br /> <br /> [http://www.math.wisc.edu/~pmwood/ Philip Matchett Wood] (Rutgers, 2009) combinatorics, random matrices.<br /> <br /> <br /> == Emeriti ==<br /> <br /> [http://psoup.math.wisc.edu/kitchen.html David Griffeath] (Cornell, 1976)<br /> <br /> [http://www.math.wisc.edu/~kuelbs Jim Kuelbs] (Minnesota, 1965)<br /> <br /> [http://www.math.wisc.edu/~kurtz Tom Kurtz] (Stanford, 1967)<br /> <br /> Peter Ney (Columbia, 1961)<br /> <br /> Josh Chover (Michigan, 1952)<br /> <br /> == Graduate students ==<br /> <br /> <br /> [http://www.math.wisc.edu/~kehlert/ Kurt Ehlert] <br /> <br /> [http://www.math.wisc.edu/~kang Dae Han Kang]<br /> <br /> Chaojie Yuan<br /> <br /> Yun Li<br /> <br /> == [[Probability Seminar]] ==<br /> <br /> Thursdays at 2:25pm, VV901<br /> <br /> ==[[Graduate student reading seminar]]==<br /> <br /> Email list: join-grad_prob_seminar@lists.wisc.edu<br /> <br /> Tuesdays, 2:30pm, 901 Van Vleck<br /> <br /> == [[Probability group timetable]]==<br /> <br /> == [[Undergraduate courses in probability]]==<br /> <br /> == Graduate Courses in Probability ==<br /> <br /> <br /> <br /> '''2018 Fall'''<br /> <br /> [https://www.math.wisc.edu/~anderson/733F18/733.htmll Math/Stat 733 Theory of Probability I]<br /> <br /> Math 735 Stochastic Analysis<br /> <br /> <br /> <br /> '''2017 Spring'''<br /> <br /> Math/Stat 734 Theory of Probability II <br /> <br /> Math 833 Topics in Probability: Random Matrix Theory</div> Valko https://www.math.wisc.edu/wiki/index.php?title=Probability_group_timetable&diff=15835 Probability group timetable 2018-09-04T14:22:27Z <p>Valko: </p> <hr /> <div>2018 Fall<br /> <br /> <br /> {| border=&quot;2&quot;<br /> | ||Monday||Tuesday||Wednesday||Thursday||Friday<br /> |-<br /> | 9-10|| || Timo 632 (9:30) || || Timo 632 (9:30) || <br /> |- <br /> | 10-11|| || || || || <br /> |-<br /> | 11-12|| || Hao 735 || || Hao 735 ||<br /> |-<br /> | 12-1|| || || || || <br /> |-<br /> | 1-2|| || Dave 733 || || Dave 733 ||<br /> |-<br /> | 2-3|| || graduate probability seminar (2:25) || || probability seminar (2:25) ||<br /> |-<br /> | 3-4|| || || || || <br /> |-<br /> | 4-5|| Benedek OH || || Benedek OH || || colloquium<br /> |-<br /> | 5-6|| || || || ||<br /> |}<br /> <br /> <br /> &lt;!-- <br /> {| border=&quot;2&quot;<br /> | ||Monday||Tuesday||Wednesday||Thursday||Friday<br /> |-<br /> | 9-10|| Timo 431, Kurt 222|| Benedek 431, Sebastien 632, Louis 735 (9:30), Kurt CS719 || Timo 431, Kurt 222 || Benedek 431, Sebastien 632, Louis 735 (9:30), Kurt CS719 || Timo 431<br /> |-<br /> | 10-11|| Kurt 222, Hans 234 || Phil out all day, Kurt 735 || Kurt 222, Hans 234 || Kurt 735 || Phil out all day, Hans 234 <br /> |-<br /> | 11-12|| Jinsu 375, Kurt 222, Hans 846, Christian 846 || Jinsu 375, Kurt 703 || Jinsu 375, Kurt 222, Hans 846, Christian 846 || Jinsu 375, Kurt 703 || Hans 846, Christian 846<br /> |-<br /> | 12-1|| Dave 431, Jinsu 375 || Kurt 703 (12:15) || Dave 431, Jinsu 375 || Kurt 703 (12:15) || Dave 431 <br /> |-<br /> | 1-2|| || Sebastien 632, Benedek 733, Jinsu 801, Hans 234 || || Sebastien 632, Benedek 733, Jinsu 801, Hans 234 ||<br /> |-<br /> | 2-3|| Daniele 431 (2:25) || graduate probability seminar (2:25) || Daniele 431 (2:25) || probability seminar (2:25) || Daniele 431 (2:25)<br /> |-<br /> | 3-4|| || Kurt 222, Hans 234 || || Kurt 222, Hans 234 || <br /> |-<br /> | 4-5|| || || || || colloquium<br /> |-<br /> | 5-6|| || || || ||<br /> |}<br /> --&gt;<br /> <br /> &lt;!--<br /> {| border=&quot;2&quot;<br /> | ||Monday||Tuesday||Wednesday||Thursday||Friday<br /> |-<br /> | 9-10|| Phil out all day || Benedek 531 (9:30)|| || Benedek 531 (9:30) || Phil out all day<br /> |-<br /> | 10-11||Jinsu 722, Louis 431 || || Jinsu 722, Louis 431|| ||Jinsu 722, Louis 431<br /> |-<br /> | 11-12|| || Hans 820 || || Hans 820 ||<br /> |-<br /> | 12-1|| Jinsu 222, Louis 632 || ||Jinsu 222, Louis 632 || || Jinsu 222, Louis 632<br /> |-<br /> | 1-2|| Jinsu 222, Hans 851 || Benedek OH, Hans 843 || Jinsu 222, Hans 851|| Hans 843 ||Jinsu 222, Hans 851<br /> |-<br /> | 2-3|| || graduate probability seminar (2:25) || Louis (Seb) || probability seminar (2:25) ||<br /> |-<br /> | 3-4|| ||Benedek (OH (3:30) || Benedek OH || || <br /> |-<br /> | 4-5|| || || Louis (OH 4:30)|| Louis (OH 4:30)|| colloquium<br /> |-<br /> | 5-6|| || || || ||<br /> |}<br /> --&gt;</div> Valko https://www.math.wisc.edu/wiki/index.php?title=Probability_group_timetable&diff=15834 Probability group timetable 2018-09-04T14:21:57Z <p>Valko: </p> <hr /> <div>2018 Fall<br /> <br /> <br /> {| border=&quot;2&quot;<br /> | ||Monday||Tuesday||Wednesday||Thursday||Friday<br /> |-<br /> | 9-10|| || Timo 632 || || Timo 632 || <br /> |- <br /> | 10-11|| || || || || <br /> |-<br /> | 11-12|| || Hao 735 || || Hao 735 ||<br /> |-<br /> | 12-1|| || || || || <br /> |-<br /> | 1-2|| || Dave 733 || || Dave 733 ||<br /> |-<br /> | 2-3|| || graduate probability seminar (2:25) || || probability seminar (2:25) ||<br /> |-<br /> | 3-4|| || || || || <br /> |-<br /> | 4-5|| Benedek OH || || Benedek OH || || colloquium<br /> |-<br /> | 5-6|| || || || ||<br /> |}<br /> <br /> <br /> &lt;!-- <br /> {| border=&quot;2&quot;<br /> | ||Monday||Tuesday||Wednesday||Thursday||Friday<br /> |-<br /> | 9-10|| Timo 431, Kurt 222|| Benedek 431, Sebastien 632, Louis 735 (9:30), Kurt CS719 || Timo 431, Kurt 222 || Benedek 431, Sebastien 632, Louis 735 (9:30), Kurt CS719 || Timo 431<br /> |-<br /> | 10-11|| Kurt 222, Hans 234 || Phil out all day, Kurt 735 || Kurt 222, Hans 234 || Kurt 735 || Phil out all day, Hans 234 <br /> |-<br /> | 11-12|| Jinsu 375, Kurt 222, Hans 846, Christian 846 || Jinsu 375, Kurt 703 || Jinsu 375, Kurt 222, Hans 846, Christian 846 || Jinsu 375, Kurt 703 || Hans 846, Christian 846<br /> |-<br /> | 12-1|| Dave 431, Jinsu 375 || Kurt 703 (12:15) || Dave 431, Jinsu 375 || Kurt 703 (12:15) || Dave 431 <br /> |-<br /> | 1-2|| || Sebastien 632, Benedek 733, Jinsu 801, Hans 234 || || Sebastien 632, Benedek 733, Jinsu 801, Hans 234 ||<br /> |-<br /> | 2-3|| Daniele 431 (2:25) || graduate probability seminar (2:25) || Daniele 431 (2:25) || probability seminar (2:25) || Daniele 431 (2:25)<br /> |-<br /> | 3-4|| || Kurt 222, Hans 234 || || Kurt 222, Hans 234 || <br /> |-<br /> | 4-5|| || || || || colloquium<br /> |-<br /> | 5-6|| || || || ||<br /> |}<br /> --&gt;<br /> <br /> &lt;!--<br /> {| border=&quot;2&quot;<br /> | ||Monday||Tuesday||Wednesday||Thursday||Friday<br /> |-<br /> | 9-10|| Phil out all day || Benedek 531 (9:30)|| || Benedek 531 (9:30) || Phil out all day<br /> |-<br /> | 10-11||Jinsu 722, Louis 431 || || Jinsu 722, Louis 431|| ||Jinsu 722, Louis 431<br /> |-<br /> | 11-12|| || Hans 820 || || Hans 820 ||<br /> |-<br /> | 12-1|| Jinsu 222, Louis 632 || ||Jinsu 222, Louis 632 || || Jinsu 222, Louis 632<br /> |-<br /> | 1-2|| Jinsu 222, Hans 851 || Benedek OH, Hans 843 || Jinsu 222, Hans 851|| Hans 843 ||Jinsu 222, Hans 851<br /> |-<br /> | 2-3|| || graduate probability seminar (2:25) || Louis (Seb) || probability seminar (2:25) ||<br /> |-<br /> | 3-4|| ||Benedek (OH (3:30) || Benedek OH || || <br /> |-<br /> | 4-5|| || || Louis (OH 4:30)|| Louis (OH 4:30)|| colloquium<br /> |-<br /> | 5-6|| || || || ||<br /> |}<br /> --&gt;</div> Valko https://www.math.wisc.edu/wiki/index.php?title=Probability_group_timetable&diff=15833 Probability group timetable 2018-09-04T14:18:22Z <p>Valko: </p> <hr /> <div>2018 Fall<br /> <br /> <br /> {| border=&quot;2&quot;<br /> | ||Monday||Tuesday||Wednesday||Thursday||Friday<br /> |-<br /> | 9-10|| Sebastien 431 || || Sebastien 431 || || Sebastien 431 <br /> |- <br /> | 10-11|| Louis 211; Phil OH || || Louis 211 || || Louis 211; Phil OH <br /> |-<br /> | 11-12|| Daniele 632, Timo OH || ||Daniele 632, Timo OH || ||Daniele 632 <br /> |-<br /> | 12-1|| Phil 211 || || Phil 211 || || Phil 211 <br /> |-<br /> | 1-2|| || Timo 733; Dave OH || Dave OH || Timo 733 ||<br /> |-<br /> | 2-3|| Sebastien 833; Daniele 632 || graduate probability seminar (2:25); Dave OH || Sebastien 833; Daniele 632; qBio seminars (2pm) || probability seminar (2:25) || Sebastien 833; Daniele 632<br /> |-<br /> | 3-4|| Dave 221 (3:30 - 4:20) || Dave OH ||Dave 221 (3:30 - 4:20); Daniele OH || Faculty meeting time || Dave 221 (3:30 - 4:20)<br /> |-<br /> | 4-5|| || || Daniele OH || || colloquium<br /> |-<br /> | 5-6|| || || || ||<br /> |}<br /> <br /> <br /> &lt;!-- <br /> {| border=&quot;2&quot;<br /> | ||Monday||Tuesday||Wednesday||Thursday||Friday<br /> |-<br /> | 9-10|| Timo 431, Kurt 222|| Benedek 431, Sebastien 632, Louis 735 (9:30), Kurt CS719 || Timo 431, Kurt 222 || Benedek 431, Sebastien 632, Louis 735 (9:30), Kurt CS719 || Timo 431<br /> |-<br /> | 10-11|| Kurt 222, Hans 234 || Phil out all day, Kurt 735 || Kurt 222, Hans 234 || Kurt 735 || Phil out all day, Hans 234 <br /> |-<br /> | 11-12|| Jinsu 375, Kurt 222, Hans 846, Christian 846 || Jinsu 375, Kurt 703 || Jinsu 375, Kurt 222, Hans 846, Christian 846 || Jinsu 375, Kurt 703 || Hans 846, Christian 846<br /> |-<br /> | 12-1|| Dave 431, Jinsu 375 || Kurt 703 (12:15) || Dave 431, Jinsu 375 || Kurt 703 (12:15) || Dave 431 <br /> |-<br /> | 1-2|| || Sebastien 632, Benedek 733, Jinsu 801, Hans 234 || || Sebastien 632, Benedek 733, Jinsu 801, Hans 234 ||<br /> |-<br /> | 2-3|| Daniele 431 (2:25) || graduate probability seminar (2:25) || Daniele 431 (2:25) || probability seminar (2:25) || Daniele 431 (2:25)<br /> |-<br /> | 3-4|| || Kurt 222, Hans 234 || || Kurt 222, Hans 234 || <br /> |-<br /> | 4-5|| || || || || colloquium<br /> |-<br /> | 5-6|| || || || ||<br /> |}<br /> --&gt;<br /> <br /> &lt;!--<br /> {| border=&quot;2&quot;<br /> | ||Monday||Tuesday||Wednesday||Thursday||Friday<br /> |-<br /> | 9-10|| Phil out all day || Benedek 531 (9:30)|| || Benedek 531 (9:30) || Phil out all day<br /> |-<br /> | 10-11||Jinsu 722, Louis 431 || || Jinsu 722, Louis 431|| ||Jinsu 722, Louis 431<br /> |-<br /> | 11-12|| || Hans 820 || || Hans 820 ||<br /> |-<br /> | 12-1|| Jinsu 222, Louis 632 || ||Jinsu 222, Louis 632 || || Jinsu 222, Louis 632<br /> |-<br /> | 1-2|| Jinsu 222, Hans 851 || Benedek OH, Hans 843 || Jinsu 222, Hans 851|| Hans 843 ||Jinsu 222, Hans 851<br /> |-<br /> | 2-3|| || graduate probability seminar (2:25) || Louis (Seb) || probability seminar (2:25) ||<br /> |-<br /> | 3-4|| ||Benedek (OH (3:30) || Benedek OH || || <br /> |-<br /> | 4-5|| || || Louis (OH 4:30)|| Louis (OH 4:30)|| colloquium<br /> |-<br /> | 5-6|| || || || ||<br /> |}<br /> --&gt;</div> Valko https://www.math.wisc.edu/wiki/index.php?title=Probability&diff=15832 Probability 2018-09-04T14:13:29Z <p>Valko: /* Graduate Courses in Probability */</p> <hr /> <div>__NOTOC__<br /> <br /> = '''Probability at UW-Madison''' =<br /> <br /> &lt;br&gt;<br /> <br /> == Tenured and tenure-track faculty ==<br /> <br /> [http://www.math.wisc.edu/~anderson/ David Anderson] (Duke, 2005) applied probability, numerical methods, mathematical biology.<br /> <br /> [http://www.math.wisc.edu/~roch/ Sebastien Roch] (UC Berkeley, 2007) applied probability, mathematical biology, theoretical computer science.<br /> <br /> [http://www.math.wisc.edu/~seppalai/ Timo Seppäläinen] (Minnesota, 1991) motion in a random medium, random growth models, interacting particle systems, large deviation theory.<br /> <br /> Hao Shen (Princeton, 2013) stochastic partial differential equations, integrable probability<br /> <br /> [http://www.math.wisc.edu/~valko/ Benedek Valko] (Budapest, 2004) interacting particle systems, random matrices.<br /> <br /> [http://www.math.wisc.edu/~pmwood/ Philip Matchett Wood] (Rutgers, 2009) combinatorics, random matrices.<br /> <br /> <br /> == Emeriti ==<br /> <br /> [http://psoup.math.wisc.edu/kitchen.html David Griffeath] (Cornell, 1976)<br /> <br /> [http://www.math.wisc.edu/~kuelbs Jim Kuelbs] (Minnesota, 1965)<br /> <br /> [http://www.math.wisc.edu/~kurtz Tom Kurtz] (Stanford, 1967)<br /> <br /> Peter Ney (Columbia, 1961)<br /> <br /> Josh Chover (Michigan, 1952)<br /> <br /> == Graduate students ==<br /> <br /> <br /> [http://www.math.wisc.edu/~kehlert/ Kurt Ehlert] <br /> <br /> [http://www.math.wisc.edu/~kang Dae Han Kang]<br /> <br /> Chaojie Yuan<br /> <br /> Yun Li<br /> <br /> == [[Probability Seminar]] ==<br /> <br /> Thursdays at 2:25pm, VV901<br /> <br /> ==[[Graduate student reading seminar]]==<br /> <br /> Email list: join-grad_prob_seminar@lists.wisc.edu<br /> <br /> Tuesdays, 2:30pm, 901 Van Vleck<br /> <br /> == [[Probability group timetable]]==<br /> <br /> == [[Undergraduate courses in probability]]==<br /> <br /> == Graduate Courses in Probability ==<br /> <br /> <br /> <br /> '''2018 Fall'''<br /> <br /> [https://www.math.wisc.edu/~anderson/733F18/733.htmll Math/Stat 733 Theory of Probability I]<br /> <br /> Math 735: Stochastic Analysis<br /> <br /> <br /> <br /> '''2017 Spring'''<br /> <br /> Math/Stat 734 Theory of Probability II <br /> <br /> Math 833 Topics in Probability: Random Matrix Theory</div> Valko https://www.math.wisc.edu/wiki/index.php?title=Probability&diff=15831 Probability 2018-09-04T14:13:13Z <p>Valko: </p> <hr /> <div>__NOTOC__<br /> <br /> = '''Probability at UW-Madison''' =<br /> <br /> &lt;br&gt;<br /> <br /> == Tenured and tenure-track faculty ==<br /> <br /> [http://www.math.wisc.edu/~anderson/ David Anderson] (Duke, 2005) applied probability, numerical methods, mathematical biology.<br /> <br /> [http://www.math.wisc.edu/~roch/ Sebastien Roch] (UC Berkeley, 2007) applied probability, mathematical biology, theoretical computer science.<br /> <br /> [http://www.math.wisc.edu/~seppalai/ Timo Seppäläinen] (Minnesota, 1991) motion in a random medium, random growth models, interacting particle systems, large deviation theory.<br /> <br /> Hao Shen (Princeton, 2013) stochastic partial differential equations, integrable probability<br /> <br /> [http://www.math.wisc.edu/~valko/ Benedek Valko] (Budapest, 2004) interacting particle systems, random matrices.<br /> <br /> [http://www.math.wisc.edu/~pmwood/ Philip Matchett Wood] (Rutgers, 2009) combinatorics, random matrices.<br /> <br /> <br /> == Emeriti ==<br /> <br /> [http://psoup.math.wisc.edu/kitchen.html David Griffeath] (Cornell, 1976)<br /> <br /> [http://www.math.wisc.edu/~kuelbs Jim Kuelbs] (Minnesota, 1965)<br /> <br /> [http://www.math.wisc.edu/~kurtz Tom Kurtz] (Stanford, 1967)<br /> <br /> Peter Ney (Columbia, 1961)<br /> <br /> Josh Chover (Michigan, 1952)<br /> <br /> == Graduate students ==<br /> <br /> <br /> [http://www.math.wisc.edu/~kehlert/ Kurt Ehlert] <br /> <br /> [http://www.math.wisc.edu/~kang Dae Han Kang]<br /> <br /> Chaojie Yuan<br /> <br /> Yun Li<br /> <br /> == [[Probability Seminar]] ==<br /> <br /> Thursdays at 2:25pm, VV901<br /> <br /> ==[[Graduate student reading seminar]]==<br /> <br /> Email list: join-grad_prob_seminar@lists.wisc.edu<br /> <br /> Tuesdays, 2:30pm, 901 Van Vleck<br /> <br /> == [[Probability group timetable]]==<br /> <br /> == [[Undergraduate courses in probability]]==<br /> <br /> == Graduate Courses in Probability ==<br /> <br /> <br /> <br /> '''2018 Fall'''<br /> <br /> [https://www.math.wisc.edu/~anderson/733F18/733.htmll Math/Stat 733 Theory of Probability I (formerly 831)]<br /> <br /> Math 735: Stochastic Analysis<br /> <br /> <br /> <br /> '''2017 Spring'''<br /> <br /> Math/Stat 734 Theory of Probability II <br /> <br /> Math 833 Topics in Probability: Random Matrix Theory</div> Valko https://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=15110 Probability Seminar 2018-02-14T12:54:17Z <p>Valko: /* Thursday, February 15, 2018, Benedek Valkó, UW-Madison */</p> <hr /> <div>__NOTOC__<br /> <br /> = Spring 2018 =<br /> <br /> &lt;b&gt;Thursdays in 901 Van Vleck Hall at 2:25 PM&lt;/b&gt;, unless otherwise noted. <br /> &lt;b&gt;We usually end for questions at 3:15 PM.&lt;/b&gt;<br /> <br /> If you would like to sign up for the email list to receive seminar announcements then please send an email to join-probsem@lists.wisc.edu.<br /> <br /> &lt;!-- == Thursday, January 25, 2018, TBA== --&gt;<br /> <br /> == Thursday, February 1, 2018, [https://people.math.osu.edu/nguyen.1261/ Hoi Nguyen], [https://math.osu.edu/ OSU]==<br /> <br /> Title: '''A remark on long-range repulsion in spectrum'''<br /> <br /> Abstract: In this talk we will address a &quot;long-range&quot; type repulsion among the singular values of random iid matrices, as well as among the eigenvalues of random Wigner matrices. We show evidence of repulsion under arbitrary perturbation even in matrices of discrete entry distributions. In many cases our method yields nearly optimal bounds.<br /> <br /> == Thursday, February 8, 2018, [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br /> <br /> Title: '''Quantitative CLTs for random walks in random environments'''<br /> <br /> Abstract:The classical central limit theorem (CLT) states that for sums of a large number of i.i.d. random variables with finite variance, the distribution of the rescaled sum is approximately Gaussian. However, the statement of the central limit theorem doesn't give any quantitative error estimates for this approximation. Under slightly stronger moment assumptions, quantitative bounds for the CLT are given by the Berry-Esseen estimates. In this talk we will consider similar questions for CLTs for random walks in random environments (RWRE). That is, for certain models of RWRE it is known that the position of the random walk has a Gaussian limiting distribution, and we obtain quantitative error estimates on the rate of convergence to the Gaussian distribution for such RWRE. This talk is based on joint works with Sungwon Ahn and Xiaoqin Guo.<br /> <br /> == &lt;span style=&quot;color:red&quot;&gt; Friday, 4pm &lt;/span&gt; February 9, 2018, &lt;span style=&quot;color:red&quot;&gt;Van Vleck B239&lt;/span&gt; [http://www.math.cmu.edu/~wes/ Wes Pegden], [http://www.math.cmu.edu/ CMU]==<br /> <br /> <br /> &lt;div style=&quot;width:400px;height:75px;border:5px solid black&quot;&gt;<br /> &lt;b&gt;&lt;span style=&quot;color:red&quot;&gt; This is a probability-related colloquium---Please note the unusual room, day, and time! &lt;/span&gt;&lt;/b&gt;<br /> &lt;/div&gt;<br /> <br /> Title: '''The fractal nature of the Abelian Sandpile'''<br /> <br /> Abstract: The Abelian Sandpile is a simple diffusion process on the integer lattice, in which configurations of chips disperse according to a simple rule: when a vertex has at least 4 chips, it can distribute one chip to each neighbor.<br /> Introduced in the statistical physics community in the 1980s, the Abelian sandpile exhibits striking fractal behavior which long resisted rigorous mathematical analysis (or even a plausible explanation). We now have a relatively robust mathematical understanding of this fractal nature of the sandpile, which involves surprising connections between integer superharmonic functions on the lattice, discrete tilings of the plane, and Apollonian circle packings. In this talk, we will survey our work in this area, and discuss avenues of current and future research.<br /> <br /> == Thursday, February 15, 2018, Benedek Valkó, UW-Madison ==<br /> <br /> Title: '''Random matrices, operators and analytic functions'''<br /> <br /> Abstract: Many of the important results of random matrix theory deal with limits of the eigenvalues of certain random matrix ensembles. In this talk I review some recent results on limits of higher level objects' related to random matrices: the limits of random matrices viewed as operators and also limits of the corresponding characteristic functions. <br /> <br /> Joint with B. Virág (Toronto/Budapest).<br /> <br /> == Thursday, February 22, 2018, [http://pages.cs.wisc.edu/~raskutti/ Garvesh Raskutti] [https://www.stat.wisc.edu/ UW-Madison Stats] and [https://wid.wisc.edu/people/garvesh-raskutti/ WID]==<br /> <br /> Title: TBA<br /> <br /> &lt;!-- == Thursday, March 1, 2018, TBA== --&gt;<br /> <br /> == Thursday, March 8, 2018, TBA==<br /> == Thursday, March 15, 2018, [http://web.mst.edu/~huwen/ Wenqing Hu] [http://math.mst.edu/ Missouri S&amp;T]==<br /> <br /> TBa<br /> <br /> == Thursday, March 22, 2018, [http://math.mit.edu/~mustazee/ Mustazee Rahman], [http://math.mit.edu/index.php MIT]==<br /> <br /> == Thursday, March 29, 2018, Spring Break ==<br /> == Thursday, April 5, 2018, TBA==<br /> == Thursday, April 12, 2018, TBA==<br /> == Thursday, April 19, 2018, TBA==<br /> == Thursday, April 26, 2018, TBA==<br /> == Thursday, May 3, 2018,TBA==<br /> == Thursday, May 10, 2018, TBA==<br /> <br /> <br /> <br /> <br /> == ==<br /> <br /> [[Past Seminars]]</div> Valko https://www.math.wisc.edu/wiki/index.php?title=Graduate_student_reading_seminar&diff=15091 Graduate student reading seminar 2018-02-12T15:41:46Z <p>Valko: /* 2018 Spring */</p> <hr /> <div><br /> ==2018 Spring==<br /> <br /> Tuesday 2:30pm, B135 Van Vleck<br /> <br /> <br /> Preliminary schedule:<br /> <br /> 2/20, 2/27: Yun<br /> <br /> 3/6, 3/13: Greg<br /> <br /> 3/20, 4/3: Yu<br /> <br /> 4/10, 4/17: Shuqi<br /> <br /> 4/24, 5/1: Tony<br /> <br /> ==2017 Fall==<br /> <br /> Tuesday 2:30pm, 214 Ingraham Hall<br /> <br /> <br /> Preliminary schedule: <br /> <br /> 9/26, 10/3: Hans<br /> <br /> 10/10, 10/17: Guo<br /> <br /> 10/24, 10/31: Chaoji<br /> <br /> 11/7, 11/14: Yun <br /> <br /> 11/21, 11/28: Kurt<br /> <br /> 12/5, 12/12: Christian<br /> <br /> <br /> <br /> <br /> ==2017 Spring==<br /> <br /> Tuesday 2:25pm, B211<br /> <br /> 1/31, 2/7: Fan<br /> <br /> I will talk about the Hanson-Wright inequality, which is a large deviation estimate for random variable of the form X^* A X, where X is a random vector with independent subgaussian entries and A is an arbitrary deterministic matrix. In the first talk, I will present a beautiful proof given by Mark Rudelson and Roman Vershynin. In the second talk, I will talk about some applications of this inequality.<br /> <br /> Reference: M. Rudelson and R. Vershynin, Hanson-Wright inequality and sub-gaussian concentration, Electron. Commun. Probab. Volume 18 (2013).<br /> <br /> 3/7, 3/14 : Jinsu<br /> <br /> Title : Donsker's Theorem and its application.<br /> Donsker's Theorem roughly says normalized random walk with linear interpolation on time interval [0,1] weakly converges to the Brownian motion B[0,1] in C([0,1]). It is sometimes called Donsker's invariance principle or the functional central limit theorem. I will show main ideas for the proof of this theorem tomorrow and show a couple of applications in my 2nd talk.<br /> <br /> Reference : https://www.math.utah.edu/~davar/ps-pdf-files/donsker.pdf<br /> <br /> ==2016 Fall==<br /> <br /> 9/27 Daniele<br /> <br /> Stochastic reaction networks.<br /> <br /> Stochastic reaction networks are continuous time Markov chain models used primarily in biochemistry. I will define them, prove some results that connect them to related deterministic models and introduce some open questions. <br /> <br /> 10/4 Jessica<br /> <br /> 10/11, 10/18: Dae Han<br /> <br /> 10/25, 11/1: Jinsu<br /> <br /> Coupling of Markov processes.<br /> <br /> When we have two distributions on same probability space, we can think of a pair whose marginal probability is each of two distributions.<br /> This pairing can be used to estimate the total variation distance between two distributions. This idea is called coupling method.<br /> I am going to introduce basic concepts,ideas and applications of coupling for Markov processes.<br /> <br /> Links of References<br /> <br /> http://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf<br /> <br /> http://websites.math.leidenuniv.nl/probability/lecturenotes/CouplingLectures.pdf<br /> <br /> 11/8, 11/15: Hans<br /> <br /> 11/22, 11/29: Keith<br /> <br /> Surprisingly Determinental: DPPs and some asymptotics of ASEP <br /> <br /> I'll be reading and presenting some recent papers of Alexei Borodin and a few collaborators which have uncovered certain equivalences between determinental point processes and non-determinental processes.<br /> <br /> <br /> ==2016 Spring==<br /> <br /> Tuesday, 2:25pm, B321 Van Vleck<br /> <br /> <br /> 3/29, 4/5: Fan Yang<br /> <br /> I will talk about the ergodic decomposition theorem (EDT). More specifically, given a compact metric space X and a continuous transformation T on it, the theorem shows that any T-invariant measure on X can be decomposed into a convex combination of ergodic measures. In the first talk I introduced the EDT and some related facts. In the second talk, I will talk about the conditional measures, and prove that the ergodic measures in EDT are indeed the conditional measures.<br /> <br /> <br /> 2/16 : Jinsu<br /> <br /> Lyapunov function for Markov Processes.<br /> <br /> For ODE, we can show stability of the trajectory using Lyapunov functions.<br /> <br /> There is an analogy for Markov Processes. I'd like to talk about the existence of stationary distribution with Lyapunov function.<br /> <br /> In some cases, it is also possible to show the rate of convergence to the stationary distribution.<br /> <br /> ==2015 Fall==<br /> <br /> This semester we will focus on tools and methods.<br /> <br /> [https://www.math.wisc.edu/wiki/images/a/ac/Reading_seminar_2015.pdf Seminar notes] ([https://www.dropbox.com/s/f4km7pevwfb1vbm/Reading%20seminar%202015.tex?dl=1 tex file], [https://www.dropbox.com/s/lg7kcgyf3nsukbx/Reading_seminar_2015.bib?dl=1 bib file])<br /> <br /> 9/15, 9/22: Elnur<br /> <br /> I will talk about large deviation theory and its applications. For the first talk, my plan is to introduce Gartner-Ellis theorem and show a few applications of it to finite state discrete time Markov chains.<br /> <br /> 9/29, 10/6, 10/13 :Dae Han<br /> <br /> 10/20, 10/27, 11/3: Jessica<br /> <br /> I will first present an overview of concentration of measure and concentration inequalities with a focus on the connection with related topics in analysis and geometry. Then, I will present Log-Sobolev inequalities and their connection to concentration of measure. <br /> <br /> 11/10, 11/17: Hao Kai<br /> <br /> 11/24, 12/1, 12/8, 12/15: Chris<br /> <br /> : <br /> <br /> <br /> <br /> <br /> <br /> 2016 Spring:<br /> <br /> 2/2, 2/9: Louis<br /> <br /> <br /> 2/16, 2/23: Jinsu<br /> <br /> 3/1, 3/8: Hans<br /> <br /> ==2015 Spring==<br /> <br /> <br /> 2/3, 2/10: Scott<br /> <br /> An Introduction to Entropy for Random Variables<br /> <br /> In these lectures I will introduce entropy for random variables and present some simple, finite state-space, examples to gain some intuition. We will prove the <br /> MacMillan Theorem using entropy and the law of large numbers. Then I will introduce relative entropy and prove the Markov Chain Convergence Theorem. Finally I will <br /> define entropy for a discrete time process. The lecture notes can be found at http://www.math.wisc.edu/~shottovy/EntropyLecture.pdf.<br /> <br /> 2/17, 2/24: Dae Han<br /> <br /> 3/3, 3/10: Hans<br /> <br /> 3/17, 3/24: In Gun<br /> <br /> 4/7, 4/14: Jinsu<br /> <br /> 4/21, 4/28: Chris N.<br /> <br /> <br /> <br /> <br /> <br /> <br /> ==2014 Fall==<br /> <br /> 9/23: Dave<br /> <br /> I will go over Mike Giles’ 2008 paper “Multi-level Monte Carlo path simulation.” This paper introduced a new Monte Carlo method to approximate expectations of SDEs (driven by Brownian motions) that is significantly more efficient than what was the state of the art. This work opened up a whole new field in the numerical analysis of stochastic processes as the basic idea is quite flexible and has found a variety of applications including SDEs driven by Brownian motions, Levy-driven SDEs, SPDEs, and models from biology<br /> <br /> 9/30: Benedek<br /> <br /> A very quick introduction to Stein's method. <br /> <br /> I will give a brief introduction to Stein's method, mostly based on the the first couple of sections of the following survey article:<br /> <br /> Ross, N. (2011). Fundamentals of Stein’s method. Probability Surveys, 8, 210-293. <br /> <br /> The following webpage has a huge collection of resources if you want to go deeper: https://sites.google.com/site/yvikswan/about-stein-s-method<br /> <br /> <br /> Note that the Midwest Probability Colloquium (http://www.math.northwestern.edu/mwp/) will have a tutorial program on Stein's method this year. <br /> <br /> 10/7, 10/14: Chris J.<br /> [http://www.math.wisc.edu/~janjigia/research/MartingaleProblemNotes.pdf An introduction to the (local) martingale problem.]<br /> <br /> <br /> 10/21, 10/28: Dae Han<br /> <br /> 11/4, 11/11: Elnur<br /> <br /> 11/18, 11/25: Chris N. Free Probability with an emphasis on C* and Von Neumann Algebras<br /> <br /> 12/2, 12/9: Yun Zhai<br /> <br /> ==2014 Spring==<br /> <br /> <br /> 1/28: Greg<br /> <br /> 2/04, 2/11: Scott <br /> <br /> [http://www.math.wisc.edu/~shottovy/BLT.pdf Reflected Brownian motion, Occupation time, and applications.] <br /> <br /> 2/18: Phil-- Examples of structure results in probability theory.<br /> <br /> 2/25, 3/4: Beth-- Derivative estimation for discrete time Markov chains<br /> <br /> 3/11, 3/25: Chris J [http://www.math.wisc.edu/~janjigia/research/stationarytalk.pdf Some classical results on stationary distributions of Markov processes]<br /> <br /> 4/1, 4/8: Chris N <br /> <br /> 4/15, 4/22: Yu Sun<br /> <br /> 4/29. 5/6: Diane<br /> <br /> ==2013 Fall==<br /> <br /> 9/24, 10/1: Chris<br /> [http://www.math.wisc.edu/~janjigia/research/metastabilitytalk.pdf A light introduction to metastability]<br /> <br /> 10/8, Dae Han<br /> Majoring multiplicative cascades for directed polymers in random media<br /> <br /> 10/15, 10/22: no reading seminar<br /> <br /> 10/29, 11/5: Elnur<br /> Limit fluctuations of last passage times <br /> <br /> 11/12: Yun<br /> Helffer-Sjostrand representation and Brascamp-Lieb inequality for stochastic interface models<br /> <br /> 11/19, 11/26: Yu Sun<br /> <br /> 12/3, 12/10: Jason<br /> <br /> ==2013 Spring==<br /> <br /> 2/13: Elnur <br /> <br /> Young diagrams, RSK correspondence, corner growth models, distribution of last passage times. <br /> <br /> 2/20: Elnur<br /> <br /> 2/27: Chris<br /> <br /> A brief introduction to enlargement of filtration and the Dufresne identity<br /> [http://www.math.wisc.edu/~janjigia/research/Presentation%20Notes.pdf Notes]<br /> <br /> 3/6: Chris<br /> <br /> 3/13: Dae Han<br /> <br /> An introduction to random polymers<br /> <br /> 3/20: Dae Han<br /> <br /> Directed polymers in a random environment: path localization and strong disorder<br /> <br /> 4/3: Diane<br /> <br /> Scale and Speed for honest 1 dimensional diffusions<br /> <br /> References: &lt;br&gt;<br /> Rogers &amp; Williams - Diffusions, Markov Processes and Martingales &lt;br&gt;<br /> Ito &amp; McKean - Diffusion Processes and their Sample Paths &lt;br&gt;<br /> Breiman - Probability &lt;br&gt;<br /> http://www.statslab.cam.ac.uk/~beresty/Articles/diffusions.pdf<br /> <br /> 4/10: Diane<br /> <br /> 4/17: Yun<br /> <br /> Introduction to stochastic interface models<br /> <br /> 4/24: Yun<br /> <br /> Dynamics and Gaussian equilibrium sytems<br /> <br /> 5/1: This reading seminar will be shifted because of a probability seminar.<br /> <br /> <br /> 5/8: Greg, Maso<br /> <br /> The Bethe ansatz vs. The Replica Trick. This lecture is an overview of the two <br /> approaches. See [http://arxiv.org/abs/1212.2267] for a nice overview.<br /> <br /> 5/15: Greg, Maso<br /> <br /> Rigorous use of the replica trick.</div> Valko https://www.math.wisc.edu/wiki/index.php?title=Problem_Solver%27s_Toolbox&diff=15080 Problem Solver's Toolbox 2018-02-11T20:38:18Z <p>Valko: </p> <hr /> <div>The goal of this page is to collect simple problem solving strategies and tools. We hope that students interested in the Wisconsin Math Talent Search would find the described ideas useful. <br /> This page and the discussed topics can be used as a starting point for future exploration.<br /> <br /> <br /> == General ideas ==<br /> <br /> <br /> There is no universal recipe for math problems that would work every time, that's what makes math fun! There are however a number of general strategies that could be useful in most cases, here is a short list of them. (Many of these ideas were popularized by the Hungarian born mathematician George Pólya in his book [https://en.wikipedia.org/wiki/How_to_Solve_It How to Solve It].)<br /> * Make sure that you understand the problem. <br /> * If possible, draw a figure. <br /> * Can you connect the problem to a problem you have solved before? <br /> * If you have to show something for all numbers (or a large number) then try to check the statement for small values first.<br /> * Can you solve the problem in a special case first? Can you solve a modified version of the problem first? <br /> * Is there some symmetry in the problem that you can exploit? <br /> * Is it possible to work backward? <br /> * Does it help to consider an extreme case of the problem?<br /> * Is it possible to generalize the problem? (Sometimes the generalized is easier to solve.)<br /> <br /> == Modular arithmetic ==<br /> <br /> <br /> When we have to divide two integers, they don't always divide evenly, and there is a quotient and a remainder. For example when we divide 10 by 3 we get a remainder of 1.<br /> It turns out that these remainders behave very well under addition, subtraction, and multiplication. We say two numbers are the same &quot;modulo &lt;math&gt;m&lt;/math&gt;&quot; if they have the same remainder when divided by &lt;math&gt;m&lt;/math&gt;. If &lt;math&gt;a&lt;/math&gt; and &lt;math&gt;x&lt;/math&gt; are the same modulo &lt;math&gt;m&lt;/math&gt;, and &lt;math&gt;b&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; are the same modulo &lt;math&gt;m&lt;/math&gt;, then &lt;math&gt;a+b&lt;/math&gt; and &lt;math&gt;x+y&lt;/math&gt; are the same modulo &lt;math&gt;m&lt;/math&gt;, and similarly for subtraction and multiplication. <br /> <br /> For example, 5 is the same as 1 modulo 4, and hence &lt;math&gt;5\cdot 5 \cdot 5 \cdot 5=5^4&lt;/math&gt; is the same as &lt;math&gt;1\cdot 1\cdot 1\cdot 1=1&lt;/math&gt; modulo &lt;math&gt;4&lt;/math&gt;. Same way you can show that &lt;math&gt;5^{1000}&lt;/math&gt; has a remainder of 1 when we divide it by 4.<br /> <br /> Modular arithmetic often makes calculation much simpler. For example, see [https://www.math.wisc.edu/talent/sites/default/files/Talent16-2q.pdf 2016-17 Set #2 Problem 3].<br /> <br /> See [http://artofproblemsolving.com/wiki/index.php?title=Modular_arithmetic/Introduction Art of Problem Solving's introduction to modular arithmetic] for more information.<br /> <br /> == Mathematical induction ==<br /> <br /> Suppose that you want to prove a statement for all positive integers, for example that for each positive integer &lt;math&gt;n&lt;/math&gt; the following is true: &lt;math display=&quot;block&quot;&gt;1\cdot 2+2\cdot 3+3\cdot 4+\cdots+n\cdot (n-1)=\frac{n(n+1)(n+2)}{3}.\qquad\qquad(*) &lt;/math&gt;<br /> Mathematical induction provides a tool for doing this. You need to show the following two things:<br /> # (Base case) The statement is true for &lt;math&gt;n=1&lt;/math&gt;. <br /> # (Induction step) If the statement is true for &lt;math&gt;n&lt;/math&gt; then it must be true for &lt;math&gt;n+1&lt;/math&gt; as well.<br /> <br /> If we can show both of these parts, then it follows that the statement is true for all positive integer &lt;math&gt;n&lt;/math&gt;. Why? The first part (the base case) shows that the statement is true for &lt;math&gt;n=1&lt;/math&gt;. But then by the second part (the induction step) the statement must be true for &lt;math&gt;n=2&lt;/math&gt; as well. Using the second part again and again we see that the statement is true for &lt;math&gt;n=3, 4, 5, \cdots&lt;/math&gt; and repeating this sufficiently times we can prove that the statement is true for any fixed value of &lt;math&gt;n&lt;/math&gt;. <br /> <br /> Often the idea of induction is demonstrated as a version of Domino effect'. Imagine that you have an infinite row of dominos numbered with the positive integers, where if &lt;math&gt;n&lt;/math&gt;th domino falls then the next one will fall as well (this is the induction step). If we make the first domino fall (this is the base case) then eventually all other dominos will fall as well. <br /> <br /> * Try to use induction to show the identity &lt;math&gt;(*)&lt;/math&gt; above for all positive integer &lt;math&gt;n&lt;/math&gt;.<br /> * You can also use induction to show a statement for all integers &lt;math&gt;n\ge 5&lt;/math&gt;. Then for your base case you have to show that the statement is true for &lt;math&gt;n=5&lt;/math&gt;. (The induction step is the same.)<br /> <br /> See this page from [https://www.mathsisfun.com/algebra/mathematical-induction.html Math Is Fun] for some simple applications of induction.<br /> <br /> == Proof by contradiction ==<br /> <br /> This is a commonly used problem solving method. Suppose that you have to prove a certain statement. Now pretend that the statement is not true and try to derive (as a consequence) a false statement. The found false statement shows that your assumption about the original statement was incorrect: thus the original statement must be true. <br /> <br /> Here is a simple example: we will prove that the product of three consecutive positive integers cannot be a prime number. Assume the opposite: that means that there is a positive integer &lt;math&gt;n&lt;/math&gt; so that &lt;math&gt;n(n+1)(n+2)&lt;/math&gt; is a prime. But among three consecutive integers we will always have a multiple of 2, and also a multiple of 3. Thus the product of the three numbers must be divisible by both 2 and 3, and hence &lt;math&gt;n(n+1)(n+2)&lt;/math&gt; cannot be a prime. This contradicts our assumption that &lt;math&gt;n(n+1)(n+2)&lt;/math&gt; is a prime, which shows that our assumption had to be incorrect. <br /> <br /> Proof by contradiction can be used for example in [https://www.math.wisc.edu/talent/sites/default/files/Talent16-1q.pdf 2016-17 Set #1 Problem 4].<br /> <br /> == Pigeonhole Principle ==<br /> <br /> The Pigeonhole Principle is one of the simplest tools in mathematics, but it can be very powerful. Suppose that &lt;math&gt;n&lt;m&lt;/math&gt; are positive integers, and we have &lt;math&gt;m&lt;/math&gt; objects and &lt;math&gt;n&lt;/math&gt; boxes. The Pigeonhole Principle states that If we place each of the &lt;math&gt;m&lt;/math&gt; objects into one of the &lt;math&gt;n&lt;/math&gt; boxes then there must be at least one box with at least two objects in it. <br /> The statement can be proved by contradiction: if we can find an arrangement of objects so that each box has less than two objects in it, then each box would contain at most one object, and hence we had at most &lt;math&gt;n&lt;/math&gt; objects all together. This is a contradiction, which means that the original statement must be correct. <br /> <br /> The Pigeonhole Principle is often used in the following, more general form. Suppose that &lt;math&gt;n, m, k&lt;/math&gt; are positive integers with &lt;math&gt;n k&lt; m &lt;/math&gt;. If we place each of &lt;math&gt;m&lt;/math&gt; objects into one of &lt;math&gt;n&lt;/math&gt; boxes then there must be at least one box with at least &lt;math&gt;k+1&lt;/math&gt; objects in it. Try to prove this version by contradiction.<br /> <br /> Here is a simple application: if we roll a die 13 times then there must be a number that appears at least three times. Here each die roll correspond to an object, each of the 6 possible outcomes correspond to a possible box. Since &lt;math&gt;2\cdot 6&lt;13&lt;/math&gt;, we must have a box with at least &lt;math&gt;2+1=3&lt;/math&gt; objects. In other words: there will be number that appears at least three times. <br /> <br /> Pigeonhole Principle can be used for example in [https://www.math.wisc.edu/talent/sites/default/files/T14-1q_0_0.pdf 2014-15 Set #1 Problem 4].<br /> <br /> == Angles in the circle ==<br /> <br /> The following theorems are often useful when working with geometry problems. [[File:Thales_thm.jpg|250px|thumb|right|An illustration of Thales' Theorem. O is the center of the circle.]] <br /> <br /> '''Thales' Theorem''' <br /> <br /> Suppose that the distinct points &lt;math&gt;A, B, C&lt;/math&gt; are all on a circle, and &lt;math&gt;AB&lt;/math&gt; is a diameter of of the circle. Then the angle &lt;math&gt;\ang ACB&lt;/math&gt; is &lt;math&gt;90^{\text{o}}&lt;/math&gt;. In other words: the triangle &lt;math&gt;\triangle ABC&lt;/math&gt; is a right triangle with hypotenuse &lt;math&gt;AB&lt;/math&gt;. <br /> <br /> The theorem can be proved with a little bit of angle-chasing'. Denote the center of the circle by &lt;math&gt;O&lt;/math&gt;. Then &lt;math&gt;AO, BO, CO&lt;/math&gt; are all radii of the circle, so they have the same length. Thus &lt;math&gt;\triangle AOC&lt;/math&gt; and &lt;math&gt;\triangle BOC&lt;/math&gt; are both isosceles triangles. Now try labeling the various angles in the picture and you should quickly arrive to a proof. (You can find the worked out proof at the [https://en.wikipedia.org/wiki/Thales%27_theorem wiki page] of the theorem, but it is more fun if you figure it out on your own!)<br /> <br /> The converse of Thales's theorem states that if &lt;math&gt;\triangle ABC&lt;/math&gt; is a right triangle with hypotenuse &lt;math&gt;AB&lt;/math&gt; then we can draw a circle with a center that is the midpoint of &lt;math&gt;AB&lt;/math&gt; that passes through &lt;math&gt;A, B, C&lt;/math&gt;.<br /> <br /> <br /> The Inscribed Angle Theorem below is a generalization of Thales' Theorem. <br /> <br /> <br /> '''The Inscribed Angle Theorem'''<br /> <br /> Suppose that the distinct points &lt;math&gt;A, B, C&lt;/math&gt; are all on a circle and let &lt;math&gt;O&lt;/math&gt; be the center of the circle. Then depending on the position of these points we have the following statements:<br /> <br /> * If &lt;math&gt;O&lt;/math&gt; is on the line &lt;math&gt;AB&lt;/math&gt; then &lt;math&gt;\angle ACB=90^{\text{o}}&lt;/math&gt;. (This is just Thales' theorem again.)<br /> * If &lt;math&gt;O&lt;/math&gt; and &lt;math&gt;C&lt;/math&gt; are both on the same side of the line &lt;math&gt;AB&lt;/math&gt; then the inscribed angle &lt;math&gt;\angle ACB&lt;/math&gt; is half of &lt;math&gt;360^{\text{o}}&lt;/math&gt; minus the central angle &lt;math&gt;\angle AOB&lt;/math&gt;: <br /> &lt;math display=&quot;block&quot;&gt; 2 \angle ACB= \angle AOB.&lt;/math&gt;<br /> * If &lt;math&gt;O&lt;/math&gt; and &lt;math&gt;C&lt;/math&gt; are on the opposite sides of the line &lt;math&gt;AB&lt;/math&gt; then the inscribed angle &lt;math&gt;\angle ACB&lt;/math&gt; is half of the central angle &lt;math&gt;\angle AOB&lt;/math&gt;: <br /> &lt;math display=&quot;block&quot;&gt; 2 \angle ACB= 360^{\text{o}}-\angle AOB.&lt;/math&gt;<br /> <br /> If we measure the central angle &lt;math&gt;\angle AOB&lt;/math&gt; the right way' then we don't need to separate the three cases. In the first case the central angle is just &lt;math&gt;180^{\text{o}}&lt;/math&gt;, and the inscribed angle is exactly the half of that. In the third case if we define the central angle to be &lt;math&gt;360^{\text{o}}-\angle AOB&lt;/math&gt; then again we get that the inscribed angle is half of the central angle. <br /> <br /> <br /> The theorem can be proved with angle-chasing, using the same idea that was described for Thales' theorem. See the [https://en.wikipedia.org/wiki/Inscribed_angle wiki page] for the proof (but first try to do it on your own!).<br /> <br /> <br /> '''Applications to cyclic quadrilaterals'''<br /> <br /> The following statements (and their converses) are useful applications of the Inscribed Angle theorem.<br /> <br /> <br /> 1. Suppose that the points &lt;math&gt;A, B, C, D&lt;/math&gt; form a cyclic quadrilateral, this means that we can draw a circle going through the four points. &lt;math&gt;AB&lt;/math&gt; divides the circle into two arcs. If the points &lt;math&gt;C&lt;/math&gt; and &lt;math&gt;D&lt;/math&gt; are in the same arc (meaning that they are on the same side of &lt;math&gt;AB&lt;/math&gt;) then <br /> &lt;math display=&quot;block&quot;&gt; \angle ACB= \angle ADB.&lt;/math&gt;<br /> The converse of this statement is also true: if &lt;math&gt;A, B, C, D&lt;/math&gt; are distinct points, the points &lt;math&gt;C, D&lt;/math&gt; are on the same side of the line &lt;math&gt;AB&lt;/math&gt; and &lt;math&gt;\angle ACB= \angle ADB<br /> &lt;/math&gt; then we can draw a circle around &lt;math&gt;A, B, C, D&lt;/math&gt;, in other words &lt;math&gt;ABCD&lt;/math&gt; is a cyclic quadrilateral.<br /> <br /> 2. Suppose that &lt;math&gt;ABCD&lt;/math&gt; is a cyclic quadrilateral. Then the sum of any two opposite angles is equal to &lt;math&gt;180^{\text{o}}&lt;/math&gt;. This means that <br /> &lt;math display=&quot;block&quot;&gt; \angle ABC+\angle CDA= 180^{\text{o}}, \quad \text{and}\quad \angle BCD+\angle DAB= 180^{\text{o}}. \qquad\qquad (**)&lt;/math&gt;<br /> <br /> The converse of the previous statement is also true: suppose that &lt;math&gt;ABCD&lt;/math&gt; is a quadrilateral with angles satisfying the equations &lt;math&gt;(**)&lt;/math&gt;. Then &lt;math&gt;ABCD&lt;/math&gt; is a cyclic quadrilateral: we can draw a circle that passes through the four points.<br /> <br /> The Inscribed Angle Theorem and the statements about cyclic quadrilaterals can be used for example in [https://www.math.wisc.edu/talent/sites/default/files/Talent15-4q.pdf 2015-16 Set #4 Problem 5].</div> Valko https://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=15062 Probability Seminar 2018-02-08T15:16:23Z <p>Valko: /* Thursday, February 15, 2018, Benedek Valkó, UW-Madison */</p> <hr /> <div>__NOTOC__<br /> <br /> = Spring 2018 =<br /> <br /> &lt;b&gt;Thursdays in 901 Van Vleck Hall at 2:25 PM&lt;/b&gt;, unless otherwise noted. <br /> &lt;b&gt;We usually end for questions at 3:15 PM.&lt;/b&gt;<br /> <br /> If you would like to sign up for the email list to receive seminar announcements then please send an email to join-probsem@lists.wisc.edu.<br /> <br /> &lt;!-- == Thursday, January 25, 2018, TBA== --&gt;<br /> <br /> == Thursday, February 1, 2018, [https://people.math.osu.edu/nguyen.1261/ Hoi Nguyen], [https://math.osu.edu/ OSU]==<br /> <br /> Title: '''A remark on long-range repulsion in spectrum'''<br /> <br /> Abstract: In this talk we will address a &quot;long-range&quot; type repulsion among the singular values of random iid matrices, as well as among the eigenvalues of random Wigner matrices. We show evidence of repulsion under arbitrary perturbation even in matrices of discrete entry distributions. In many cases our method yields nearly optimal bounds.<br /> <br /> == Thursday, February 8, 2018, [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br /> <br /> Title: '''Quantitative CLTs for random walks in random environments'''<br /> <br /> Abstract:The classical central limit theorem (CLT) states that for sums of a large number of i.i.d. random variables with finite variance, the distribution of the rescaled sum is approximately Gaussian. However, the statement of the central limit theorem doesn't give any quantitative error estimates for this approximation. Under slightly stronger moment assumptions, quantitative bounds for the CLT are given by the Berry-Esseen estimates. In this talk we will consider similar questions for CLTs for random walks in random environments (RWRE). That is, for certain models of RWRE it is known that the position of the random walk has a Gaussian limiting distribution, and we obtain quantitative error estimates on the rate of convergence to the Gaussian distribution for such RWRE. This talk is based on joint works with Sungwon Ahn and Xiaoqin Guo.<br /> <br /> == &lt;span style=&quot;color:red&quot;&gt; Friday, 4pm &lt;/span&gt; February 9, 2018, &lt;span style=&quot;color:red&quot;&gt;Van Vleck B239&lt;/span&gt; [http://www.math.cmu.edu/~wes/ Wes Pegden], [http://www.math.cmu.edu/ CMU]==<br /> <br /> <br /> &lt;div style=&quot;width:400px;height:75px;border:5px solid black&quot;&gt;<br /> &lt;b&gt;&lt;span style=&quot;color:red&quot;&gt; This is a probability-related colloquium---Please note the unusual room, day, and time! &lt;/span&gt;&lt;/b&gt;<br /> &lt;/div&gt;<br /> <br /> Title: '''The fractal nature of the Abelian Sandpile'''<br /> <br /> Abstract: The Abelian Sandpile is a simple diffusion process on the integer lattice, in which configurations of chips disperse according to a simple rule: when a vertex has at least 4 chips, it can distribute one chip to each neighbor.<br /> Introduced in the statistical physics community in the 1980s, the Abelian sandpile exhibits striking fractal behavior which long resisted rigorous mathematical analysis (or even a plausible explanation). We now have a relatively robust mathematical understanding of this fractal nature of the sandpile, which involves surprising connections between integer superharmonic functions on the lattice, discrete tilings of the plane, and Apollonian circle packings. In this talk, we will survey our work in this area, and discuss avenues of current and future research.<br /> <br /> == Thursday, February 15, 2018, Benedek Valkó, UW-Madison ==<br /> <br /> Title: '''Random matrices, operators and analytic functions'''<br /> <br /> Abstract: Many of the important results of random matrix theory deals with limits of the eigenvalues of certain random matrix ensembles. In this talk I review some recent results on limits of higher level objects' related to random matrices: the limits of random matrices viewed as operators and also limits of the corresponding characteristic functions. <br /> <br /> Joint with B. Virág (Toronto/Budapest).<br /> <br /> == Thursday, February 22, 2018, [http://pages.cs.wisc.edu/~raskutti/ Garvesh Raskutti] [https://www.stat.wisc.edu/ UW-Madison Stats] and [https://wid.wisc.edu/people/garvesh-raskutti/ WID]==<br /> <br /> Title: TBA<br /> <br /> &lt;!-- == Thursday, March 1, 2018, TBA== --&gt;<br /> <br /> == Thursday, March 8, 2018, TBA==<br /> == Thursday, March 15, 2018, [http://web.mst.edu/~huwen/ Wenqing Hu] [http://math.mst.edu/ Missouri S&amp;T]==<br /> <br /> TBa<br /> <br /> == Thursday, March 22, 2018, [http://math.mit.edu/~mustazee/ Mustazee Rahman], [http://math.mit.edu/index.php MIT]==<br /> <br /> == Thursday, March 29, 2018, Spring Break ==<br /> == Thursday, April 5, 2018, TBA==<br /> == Thursday, April 12, 2018, TBA==<br /> == Thursday, April 19, 2018, TBA==<br /> == Thursday, April 26, 2018, TBA==<br /> == Thursday, May 3, 2018,TBA==<br /> == Thursday, May 10, 2018, TBA==<br /> <br /> <br /> <br /> <br /> == ==<br /> <br /> [[Past Seminars]]</div> Valko https://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=15061 Probability Seminar 2018-02-08T15:15:47Z <p>Valko: /* Thursday, February 15, 2018, Benedek Valkó, UW-Madison */</p> <hr /> <div>__NOTOC__<br /> <br /> = Spring 2018 =<br /> <br /> &lt;b&gt;Thursdays in 901 Van Vleck Hall at 2:25 PM&lt;/b&gt;, unless otherwise noted. <br /> &lt;b&gt;We usually end for questions at 3:15 PM.&lt;/b&gt;<br /> <br /> If you would like to sign up for the email list to receive seminar announcements then please send an email to join-probsem@lists.wisc.edu.<br /> <br /> &lt;!-- == Thursday, January 25, 2018, TBA== --&gt;<br /> <br /> == Thursday, February 1, 2018, [https://people.math.osu.edu/nguyen.1261/ Hoi Nguyen], [https://math.osu.edu/ OSU]==<br /> <br /> Title: '''A remark on long-range repulsion in spectrum'''<br /> <br /> Abstract: In this talk we will address a &quot;long-range&quot; type repulsion among the singular values of random iid matrices, as well as among the eigenvalues of random Wigner matrices. We show evidence of repulsion under arbitrary perturbation even in matrices of discrete entry distributions. In many cases our method yields nearly optimal bounds.<br /> <br /> == Thursday, February 8, 2018, [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br /> <br /> Title: '''Quantitative CLTs for random walks in random environments'''<br /> <br /> Abstract:The classical central limit theorem (CLT) states that for sums of a large number of i.i.d. random variables with finite variance, the distribution of the rescaled sum is approximately Gaussian. However, the statement of the central limit theorem doesn't give any quantitative error estimates for this approximation. Under slightly stronger moment assumptions, quantitative bounds for the CLT are given by the Berry-Esseen estimates. In this talk we will consider similar questions for CLTs for random walks in random environments (RWRE). That is, for certain models of RWRE it is known that the position of the random walk has a Gaussian limiting distribution, and we obtain quantitative error estimates on the rate of convergence to the Gaussian distribution for such RWRE. This talk is based on joint works with Sungwon Ahn and Xiaoqin Guo.<br /> <br /> == &lt;span style=&quot;color:red&quot;&gt; Friday, 4pm &lt;/span&gt; February 9, 2018, &lt;span style=&quot;color:red&quot;&gt;Van Vleck B239&lt;/span&gt; [http://www.math.cmu.edu/~wes/ Wes Pegden], [http://www.math.cmu.edu/ CMU]==<br /> <br /> <br /> &lt;div style=&quot;width:400px;height:75px;border:5px solid black&quot;&gt;<br /> &lt;b&gt;&lt;span style=&quot;color:red&quot;&gt; This is a probability-related colloquium---Please note the unusual room, day, and time! &lt;/span&gt;&lt;/b&gt;<br /> &lt;/div&gt;<br /> <br /> Title: '''The fractal nature of the Abelian Sandpile'''<br /> <br /> Abstract: The Abelian Sandpile is a simple diffusion process on the integer lattice, in which configurations of chips disperse according to a simple rule: when a vertex has at least 4 chips, it can distribute one chip to each neighbor.<br /> Introduced in the statistical physics community in the 1980s, the Abelian sandpile exhibits striking fractal behavior which long resisted rigorous mathematical analysis (or even a plausible explanation). We now have a relatively robust mathematical understanding of this fractal nature of the sandpile, which involves surprising connections between integer superharmonic functions on the lattice, discrete tilings of the plane, and Apollonian circle packings. In this talk, we will survey our work in this area, and discuss avenues of current and future research.<br /> <br /> == Thursday, February 15, 2018, Benedek Valkó, UW-Madison ==<br /> <br /> Title: '''Random matrices, operators and analytic functions'''<br /> <br /> Abstract: Many of the important results of random matrix theory deals with limits of the eigenvalues of certain random matrix ensembles. In this talk I review some recent results on limits of `higher level objects' related to random matrices: the limits of random matrices viewed as operators and also limits of the corresponding characteristic functions. <br /> <br /> (Joint with B. Virág (Toronto/Budapest))<br /> <br /> == Thursday, February 22, 2018, [http://pages.cs.wisc.edu/~raskutti/ Garvesh Raskutti] [https://www.stat.wisc.edu/ UW-Madison Stats] and [https://wid.wisc.edu/people/garvesh-raskutti/ WID]==<br /> <br /> Title: TBA<br /> <br /> &lt;!-- == Thursday, March 1, 2018, TBA== --&gt;<br /> <br /> == Thursday, March 8, 2018, TBA==<br /> == Thursday, March 15, 2018, [http://web.mst.edu/~huwen/ Wenqing Hu] [http://math.mst.edu/ Missouri S&amp;T]==<br /> <br /> TBa<br /> <br /> == Thursday, March 22, 2018, [http://math.mit.edu/~mustazee/ Mustazee Rahman], [http://math.mit.edu/index.php MIT]==<br /> <br /> == Thursday, March 29, 2018, Spring Break ==<br /> == Thursday, April 5, 2018, TBA==<br /> == Thursday, April 12, 2018, TBA==<br /> == Thursday, April 19, 2018, TBA==<br /> == Thursday, April 26, 2018, TBA==<br /> == Thursday, May 3, 2018,TBA==<br /> == Thursday, May 10, 2018, TBA==<br /> <br /> <br /> <br /> <br /> == ==<br /> <br /> [[Past Seminars]]</div> Valko https://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=15060 Probability Seminar 2018-02-08T15:15:08Z <p>Valko: /* Thursday, February 15, 2018, Benedek Valkó (UW-Madison) */</p> <hr /> <div>__NOTOC__<br /> <br /> = Spring 2018 =<br /> <br /> &lt;b&gt;Thursdays in 901 Van Vleck Hall at 2:25 PM&lt;/b&gt;, unless otherwise noted. <br /> &lt;b&gt;We usually end for questions at 3:15 PM.&lt;/b&gt;<br /> <br /> If you would like to sign up for the email list to receive seminar announcements then please send an email to join-probsem@lists.wisc.edu.<br /> <br /> &lt;!-- == Thursday, January 25, 2018, TBA== --&gt;<br /> <br /> == Thursday, February 1, 2018, [https://people.math.osu.edu/nguyen.1261/ Hoi Nguyen], [https://math.osu.edu/ OSU]==<br /> <br /> Title: '''A remark on long-range repulsion in spectrum'''<br /> <br /> Abstract: In this talk we will address a &quot;long-range&quot; type repulsion among the singular values of random iid matrices, as well as among the eigenvalues of random Wigner matrices. We show evidence of repulsion under arbitrary perturbation even in matrices of discrete entry distributions. In many cases our method yields nearly optimal bounds.<br /> <br /> == Thursday, February 8, 2018, [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br /> <br /> Title: '''Quantitative CLTs for random walks in random environments'''<br /> <br /> Abstract:The classical central limit theorem (CLT) states that for sums of a large number of i.i.d. random variables with finite variance, the distribution of the rescaled sum is approximately Gaussian. However, the statement of the central limit theorem doesn't give any quantitative error estimates for this approximation. Under slightly stronger moment assumptions, quantitative bounds for the CLT are given by the Berry-Esseen estimates. In this talk we will consider similar questions for CLTs for random walks in random environments (RWRE). That is, for certain models of RWRE it is known that the position of the random walk has a Gaussian limiting distribution, and we obtain quantitative error estimates on the rate of convergence to the Gaussian distribution for such RWRE. This talk is based on joint works with Sungwon Ahn and Xiaoqin Guo.<br /> <br /> == &lt;span style=&quot;color:red&quot;&gt; Friday, 4pm &lt;/span&gt; February 9, 2018, &lt;span style=&quot;color:red&quot;&gt;Van Vleck B239&lt;/span&gt; [http://www.math.cmu.edu/~wes/ Wes Pegden], [http://www.math.cmu.edu/ CMU]==<br /> <br /> <br /> &lt;div style=&quot;width:400px;height:75px;border:5px solid black&quot;&gt;<br /> &lt;b&gt;&lt;span style=&quot;color:red&quot;&gt; This is a probability-related colloquium---Please note the unusual room, day, and time! &lt;/span&gt;&lt;/b&gt;<br /> &lt;/div&gt;<br /> <br /> Title: '''The fractal nature of the Abelian Sandpile'''<br /> <br /> Abstract: The Abelian Sandpile is a simple diffusion process on the integer lattice, in which configurations of chips disperse according to a simple rule: when a vertex has at least 4 chips, it can distribute one chip to each neighbor.<br /> Introduced in the statistical physics community in the 1980s, the Abelian sandpile exhibits striking fractal behavior which long resisted rigorous mathematical analysis (or even a plausible explanation). We now have a relatively robust mathematical understanding of this fractal nature of the sandpile, which involves surprising connections between integer superharmonic functions on the lattice, discrete tilings of the plane, and Apollonian circle packings. In this talk, we will survey our work in this area, and discuss avenues of current and future research.<br /> <br /> == Thursday, February 15, 2018, Benedek Valkó, UW-Madison ==<br /> <br /> Title: '''Random matrices, operators and analytic functions'''<br /> <br /> Abstract: Many of the important results of random matrix theory deals with limits of the eigenvalues of certain random matrix ensembles. In this talk I review some recent results on limits of random matrices viewed as operators and also limits of the corresponding characteristic functions. <br /> <br /> (Joint with B. Virág (Toronto/Budapest))<br /> <br /> == Thursday, February 22, 2018, [http://pages.cs.wisc.edu/~raskutti/ Garvesh Raskutti] [https://www.stat.wisc.edu/ UW-Madison Stats] and [https://wid.wisc.edu/people/garvesh-raskutti/ WID]==<br /> <br /> Title: TBA<br /> <br /> &lt;!-- == Thursday, March 1, 2018, TBA== --&gt;<br /> <br /> == Thursday, March 8, 2018, TBA==<br /> == Thursday, March 15, 2018, [http://web.mst.edu/~huwen/ Wenqing Hu] [http://math.mst.edu/ Missouri S&amp;T]==<br /> <br /> TBa<br /> <br /> == Thursday, March 22, 2018, [http://math.mit.edu/~mustazee/ Mustazee Rahman], [http://math.mit.edu/index.php MIT]==<br /> <br /> == Thursday, March 29, 2018, Spring Break ==<br /> == Thursday, April 5, 2018, TBA==<br /> == Thursday, April 12, 2018, TBA==<br /> == Thursday, April 19, 2018, TBA==<br /> == Thursday, April 26, 2018, TBA==<br /> == Thursday, May 3, 2018,TBA==<br /> == Thursday, May 10, 2018, TBA==<br /> <br /> <br /> <br /> <br /> == ==<br /> <br /> [[Past Seminars]]</div> Valko https://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=15059 Probability Seminar 2018-02-08T15:14:47Z <p>Valko: /* Thursday, February 15, 2018, Benedek Valk\'o */</p> <hr /> <div>__NOTOC__<br /> <br /> = Spring 2018 =<br /> <br /> &lt;b&gt;Thursdays in 901 Van Vleck Hall at 2:25 PM&lt;/b&gt;, unless otherwise noted. <br /> &lt;b&gt;We usually end for questions at 3:15 PM.&lt;/b&gt;<br /> <br /> If you would like to sign up for the email list to receive seminar announcements then please send an email to join-probsem@lists.wisc.edu.<br /> <br /> &lt;!-- == Thursday, January 25, 2018, TBA== --&gt;<br /> <br /> == Thursday, February 1, 2018, [https://people.math.osu.edu/nguyen.1261/ Hoi Nguyen], [https://math.osu.edu/ OSU]==<br /> <br /> Title: '''A remark on long-range repulsion in spectrum'''<br /> <br /> Abstract: In this talk we will address a &quot;long-range&quot; type repulsion among the singular values of random iid matrices, as well as among the eigenvalues of random Wigner matrices. We show evidence of repulsion under arbitrary perturbation even in matrices of discrete entry distributions. In many cases our method yields nearly optimal bounds.<br /> <br /> == Thursday, February 8, 2018, [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br /> <br /> Title: '''Quantitative CLTs for random walks in random environments'''<br /> <br /> Abstract:The classical central limit theorem (CLT) states that for sums of a large number of i.i.d. random variables with finite variance, the distribution of the rescaled sum is approximately Gaussian. However, the statement of the central limit theorem doesn't give any quantitative error estimates for this approximation. Under slightly stronger moment assumptions, quantitative bounds for the CLT are given by the Berry-Esseen estimates. In this talk we will consider similar questions for CLTs for random walks in random environments (RWRE). That is, for certain models of RWRE it is known that the position of the random walk has a Gaussian limiting distribution, and we obtain quantitative error estimates on the rate of convergence to the Gaussian distribution for such RWRE. This talk is based on joint works with Sungwon Ahn and Xiaoqin Guo.<br /> <br /> == &lt;span style=&quot;color:red&quot;&gt; Friday, 4pm &lt;/span&gt; February 9, 2018, &lt;span style=&quot;color:red&quot;&gt;Van Vleck B239&lt;/span&gt; [http://www.math.cmu.edu/~wes/ Wes Pegden], [http://www.math.cmu.edu/ CMU]==<br /> <br /> <br /> &lt;div style=&quot;width:400px;height:75px;border:5px solid black&quot;&gt;<br /> &lt;b&gt;&lt;span style=&quot;color:red&quot;&gt; This is a probability-related colloquium---Please note the unusual room, day, and time! &lt;/span&gt;&lt;/b&gt;<br /> &lt;/div&gt;<br /> <br /> Title: '''The fractal nature of the Abelian Sandpile'''<br /> <br /> Abstract: The Abelian Sandpile is a simple diffusion process on the integer lattice, in which configurations of chips disperse according to a simple rule: when a vertex has at least 4 chips, it can distribute one chip to each neighbor.<br /> Introduced in the statistical physics community in the 1980s, the Abelian sandpile exhibits striking fractal behavior which long resisted rigorous mathematical analysis (or even a plausible explanation). We now have a relatively robust mathematical understanding of this fractal nature of the sandpile, which involves surprising connections between integer superharmonic functions on the lattice, discrete tilings of the plane, and Apollonian circle packings. In this talk, we will survey our work in this area, and discuss avenues of current and future research.<br /> <br /> == Thursday, February 15, 2018, Benedek Valkó (UW-Madison) ==<br /> <br /> Title: Random matrices, operators and analytic functions<br /> <br /> Abstract: Many of the important results of random matrix theory deals with limits of the eigenvalues of certain random matrix ensembles. In this talk I review some recent results on limits of random matrices viewed as operators and also limits of the corresponding characteristic functions. <br /> <br /> (Joint with B. Virág (Toronto/Budapest))<br /> <br /> == Thursday, February 22, 2018, [http://pages.cs.wisc.edu/~raskutti/ Garvesh Raskutti] [https://www.stat.wisc.edu/ UW-Madison Stats] and [https://wid.wisc.edu/people/garvesh-raskutti/ WID]==<br /> <br /> Title: TBA<br /> <br /> &lt;!-- == Thursday, March 1, 2018, TBA== --&gt;<br /> <br /> == Thursday, March 8, 2018, TBA==<br /> == Thursday, March 15, 2018, [http://web.mst.edu/~huwen/ Wenqing Hu] [http://math.mst.edu/ Missouri S&amp;T]==<br /> <br /> TBa<br /> <br /> == Thursday, March 22, 2018, [http://math.mit.edu/~mustazee/ Mustazee Rahman], [http://math.mit.edu/index.php MIT]==<br /> <br /> == Thursday, March 29, 2018, Spring Break ==<br /> == Thursday, April 5, 2018, TBA==<br /> == Thursday, April 12, 2018, TBA==<br /> == Thursday, April 19, 2018, TBA==<br /> == Thursday, April 26, 2018, TBA==<br /> == Thursday, May 3, 2018,TBA==<br /> == Thursday, May 10, 2018, TBA==<br /> <br /> <br /> <br /> <br /> == ==<br /> <br /> [[Past Seminars]]</div> Valko