https://www.math.wisc.edu/wiki/api.php?action=feedcontributions&user=Emrah&feedformat=atomUW-Math Wiki - User contributions [en]2020-11-29T23:56:15ZUser contributionsMediaWiki 1.30.1https://www.math.wisc.edu/wiki/index.php?title=Graduate_student_reading_seminar&diff=10199Graduate student reading seminar2015-09-14T03:34:59Z<p>Emrah: /* 2015 Fall */</p>
<hr />
<div>==2015 Fall==<br />
<br />
This semester we will focus on tools and methods.<br />
<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 Dae Han<br />
<br />
10/13, 10/20: Jessica<br />
<br />
10/27, 11/3: Hao Kai<br />
<br />
11/10, 11/17: Chris<br />
<br />
11/24, 12/1: Louis<br />
<br />
12/8, 12/15: Jinsu<br />
<br />
<br />
<br />
2016 Spring:<br />
<br />
1/26, 2/2: Hans<br />
<br />
2/9, 2/16: Fan<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 />
==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-Sjostrand 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: <br><br />
Rogers & Williams - Diffusions, Markov Processes and Martingales <br><br />
Ito & McKean - Diffusion Processes and their Sample Paths <br><br />
Breiman - Probability <br><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>Emrahhttps://www.math.wisc.edu/wiki/index.php?title=Probability_group_timetable&diff=10026Probability group timetable2015-08-26T23:56:14Z<p>Emrah: </p>
<hr />
<div>{| border="2"<br />
| ||Monday||Tuesday||Wednesday||Thursday||Friday<br />
|-<br />
| 9-10|| || Kurt 221, Chiara 221, Jun 431|| || Kurt 221, Chiara 221 , Jun 431 || <br />
|-<br />
| 10-11|| Kurt 721, Chiara 721, Jinsu 221, Elnur 171 || Kurt 221 , Jun 431|| Kurt 721, Chiara 721, Jinsu 221, Elnur 171 || Kurt 221 , Jun 431|| Kurt 721, Chiara 721, Jinsu 221, Elnur 171<br />
|-<br />
| 11-12|| Chiara 221, Jinsu 221 || Chiara 703 || Chiara 221, Jinsu 221 || Chiara 703 || Chiara 221, Jinsu 221<br />
|-<br />
| 12-1|| Dave 605, Sebastien 632, Kurt 714, Louis 431 || || Dave 605, Sebastien 632, Kurt 714, Louis 431 || || Dave 605, Sebastien 632, Kurt 721, Louis 431 <br />
|-<br />
| 1-2|| || Kurt 733, Chiara 733, Phil 733 || || Kurt 733, Chiara 733, Phil 733 || <br />
|-<br />
| 2-3|| Benedek 431 (2:25) || reading seminar (2:25pm) || Benedek 431 (2:25) || probability seminar (2:25pm) || Benedek 431 (2:25)<br />
|-<br />
| 3-4|| Benedek OH (3:30), Kurt 221 (3:30), PDE Seminar (3:30) || Benedek OH (3:30), Phil OH || Benedek OH (3:30), Kurt 221 (3:30), Phil OH || || Kurt 221 (3:30)<br />
|-<br />
| 4-5|| Kurt 221 (4:20) || Analysis Seminar || Kurt 221 (4:20), Louis OH (4:30) || || Colloquium, Kurt 221 (4:20)<br />
|-<br />
| 5-6|| || Jun 431 || Jun 431 || ||<br />
|}</div>Emrahhttps://www.math.wisc.edu/wiki/index.php?title=Past_Probability_Seminars_Spring_2020&diff=9560Past Probability Seminars Spring 20202015-03-26T17:09:05Z<p>Emrah: /* Thursday, April 9, Elnur Emrah, UW-Madison */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2015 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<br />
<b><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 />
<!-- [[File:probsem.jpg]] --><br />
</b><br />
<br />
= =<br />
<br />
== Thursday, January 15, [http://www.stat.berkeley.edu/~racz/ Miklos Racz], [http://statistics.berkeley.edu/ UC-Berkeley Stats] ==<br />
<br />
<br />
Title: Testing for high-dimensional geometry in random graphs<br />
<br />
Abstract: I will talk about a random geometric graph model, where connections between vertices depend on distances between latent d-dimensional labels; we are particularly interested in the high-dimensional case when d is large. Upon observing a graph, we want to tell if it was generated from this geometric model, or from an Erdos-Renyi random graph. We show that there exists a computationally efficient procedure to do this which is almost optimal (in an information-theoretic sense). The key insight is based on a new statistic which we call "signed triangles". To prove optimality we use a bound on the total variation distance between Wishart matrices and the Gaussian Orthogonal Ensemble. This is joint work with Sebastien Bubeck, Jian Ding, and Ronen Eldan.<br />
<br />
== Thursday, January 22, No Seminar ==<br />
<br />
== Thursday, January 29, [http://www.math.umn.edu/~arnab/ Arnab Sen], [http://www.math.umn.edu/ University of Minnesota] ==<br />
<br />
Title: '''Double Roots of Random Littlewood Polynomials'''<br />
<br />
Abstract:<br />
We consider random polynomials whose coefficients are independent and uniform on {-1,1}. We will show that the probability that such a polynomial of degree n has a double root is o(n^{-2}) when n+1 is not divisible by 4 and is of the order n^{-2} otherwise. We will also discuss extensions to random polynomials with more general coefficient distributions. <br />
<br />
This is joint work with Ron Peled and Ofer Zeitouni.<br />
<br />
== Thursday, February 5, No seminar this week ==<br />
<br />
== Thursday, February 12, No Seminar this week==<br />
<br />
<br />
<!--<br />
== Wednesday, <span style="color:red">February 11</span>, [http://www.math.wisc.edu/~stechmann/ Sam Stechmann], [http://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
<span style="color:red">Please note the unusual time and room.<br />
</span><br />
<br />
<br />
Title: Stochastic Models for Rainfall: Extreme Events and Critical Phenomena<br />
<br />
<br />
Abstract:<br />
In recent years, tropical rainfall statistics have been shown to conform to paradigms of critical phenomena and statistical physics. In this talk, stochastic models will be presented as prototypes for understanding the atmospheric dynamics that leads to these statistics and extreme events. Key nonlinear ingredients in the models include either stochastic jump processes or thresholds (Heaviside functions). First, both exact solutions and simple numerics are used to verify that a suite of observed rainfall statistics is reproduced by the models, including power-law distributions and long-range correlations. Second, we prove that a stochastic trigger, which is a time-evolving indicator of whether it is raining or not, will converge to a deterministic threshold in an appropriate limit. Finally, we discuss the connections among these rainfall models, stochastic PDEs, and traditional models for critical phenomena.<br />
---><br />
<br />
== Thursday, February 19, [http://www.math.purdue.edu/people/bio/guo297 Xiaoqin Guo], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
Title: Quenched invariance principle for random walks in time-dependent random environment<br />
<br />
Abstract: In this talk we discuss random walks in a time-dependent zero-drift random environment in <math>Z^d</math>. We prove a quenched invariance principle under an appropriate moment condition. The proof is based on the use of a maximum principle for parabolic difference operators. This is a joint work with Jean-Dominique Deuschel and Alejandro Ramirez.<br />
<br />
== Thursday, February 26, [http://wwwf.imperial.ac.uk/~dcrisan/ Dan Crisan], [http://www.imperial.ac.uk/natural-sciences/departments/mathematics/ Imperial College London] ==<br />
<br />
Title: '''Smoothness properties of randomly perturbed semigroups with application to nonlinear filtering'''<br />
<br />
Abstract:<br />
In this talk I will discuss sharp gradient bounds for perturbed diffusion semigroups. In contrast with existing results, the perturbation is here random and the bounds obtained are pathwise. Our approach builds on the classical work of Kusuoka and Stroock and extends their program developed for the heat semi-group to solutions of stochastic partial differential equations. The work is motivated by and applied to nonlinear filtering. The analysis allows us to derive pathwise gradient bounds for the un-normalised conditional distribution of a partially observed signal. The estimates we derive have sharp small time asymptotics<br />
<br />
This is joint work with Terry Lyons (Oxford) and Christian Literrer (Ecole Polytechnique) and is based on the paper<br />
<br />
D Crisan, C Litterer, T Lyons, Kusuoka–Stroock gradient bounds for the solution of the filtering equation, Journal of Functional Analysis, 2105<br />
<br />
== Wednesday, <span style="color:red">March 4</span>, [http://www.math.wisc.edu/~stechmann/ Sam Stechmann], [http://www.math.wisc.edu/ UW-Madison], <span style="color:red"> 2:25pm Van Vleck B113</span> ==<br />
<br />
<span style="color:red">Please note the unusual time and room.<br />
</span><br />
<br />
<br />
Title: Stochastic Models for Rainfall: Extreme Events and Critical Phenomena<br />
<br />
<br />
Abstract:<br />
In recent years, tropical rainfall statistics have been shown to conform to paradigms of critical phenomena and statistical physics. In this talk, stochastic models will be presented as prototypes for understanding the atmospheric dynamics that leads to these statistics and extreme events. Key nonlinear ingredients in the models include either stochastic jump processes or thresholds (Heaviside functions). First, both exact solutions and simple numerics are used to verify that a suite of observed rainfall statistics is reproduced by the models, including power-law distributions and long-range correlations. Second, we prove that a stochastic trigger, which is a time-evolving indicator of whether it is raining or not, will converge to a deterministic threshold in an appropriate limit. Finally, we discuss the connections among these rainfall models, stochastic PDEs, and traditional models for critical phenomena.<br />
<br />
== Thursday, March 12, [http://www.ima.umn.edu/~ohadfeld/Website/index.html Ohad Feldheim], [http://www.ima.umn.edu/ IMA] ==<br />
<br />
<br />
Title: '''The 3-states AF-Potts model in high dimension'''<br />
<br />
Abstract: <br />
<!--<br />
Take a bounded odd domain of the bipartite graph $\mathbb{Z}^d$. Color the boundary of the set by $0$, then<br />
color the rest of the domain at random with the colors $\{0,\dots,q-1\}$, penalizing every<br />
configuration with proportion to the number of improper edges at a given rate $\beta>0$ (the "inverse temperature").<br />
Q: "What is the structure of such a coloring?"<br />
<br />
This model is called the $q$-states Potts antiferromagnet(AF), a classical spin glass model in statistical mechanics.<br />
The $2$-states case is the famous Ising model which is relatively well understood.<br />
The $3$-states case in high dimension has been studies for $\beta=\infty$,<br />
when the model reduces to a uniformly chosen proper three coloring of the domain.<br />
Several words, by Galvin, Kahn, Peled, Randall and Sorkin established the structure of the model <br />
showing long-range correlations and phase coexistence. In this work, we generalize this result to positive temperature, <br />
showing that for large enough $\beta$ (low enough temperature)<br />
the rigid structure persists. This is the first rigorous result for $\beta<\infty$.<br />
<br />
In the talk, assuming no acquaintance with the model, we shall give the physical background, introduce all the<br />
relevant definitions and shed some light on how such results are proved using only combinatorial methods.<br />
Joint work with Yinon Spinka.<br />
--><br />
Take a bounded odd domain of the bipartite graph <math>\mathbb{Z}^d</math>. Color the<br />
boundary of the set by <math>0</math>, then<br />
color the rest of the domain at random with the colors <math>\{0,\dots,q-1\}</math>,<br />
penalizing every<br />
configuration with proportion to the number of improper edges at a given rate<br />
<math>\beta>0</math> (the "inverse temperature").<br />
Q: "What is the structure of such a coloring?"<br />
<br />
This model is called the <math>q</math>-states Potts antiferromagnet(AF), a classical spin<br />
glass model in statistical mechanics.<br />
The <math>2</math>-states case is the famous Ising model which is relatively well<br />
understood.<br />
The <math>3</math>-states case in high dimension has been studies for <math>\beta=\infty</math>,<br />
when the model reduces to a uniformly chosen proper three coloring of the<br />
domain.<br />
Several words, by Galvin, Kahn, Peled, Randall and Sorkin established the<br />
structure of the model<br />
showing long-range correlations and phase coexistence. In this work, we<br />
generalize this result to positive temperature,<br />
showing that for large enough <math>\beta</math> (low enough temperature)<br />
the rigid structure persists. This is the first rigorous result for<br />
<math>\beta<\infty</math>.<br />
<br />
In the talk, assuming no acquaintance with the model, we shall give the<br />
physical background, introduce all the<br />
relevant definitions and shed some light on how such results are proved using<br />
only combinatorial methods.<br />
Joint work with Yinon Spinka.<br />
<br />
== Thursday, March 19, [http://www.cmc.edu/pages/faculty/MHuber/ Mark Huber], [http://www.cmc.edu/math/ Claremont McKenna Math] ==<br />
<br />
Title: Understanding relative error in Monte Carlo simulations<br />
<br />
Abstract: The problem of estimating the probability <math>p</math> of heads on an unfair coin has been around for centuries, and has inspired numerous advances in probability such as the Strong Law of Large Numbers and the Central Limit Theorem. In this talk, I'll consider a new twist: given an estimate <math>\hat p</math>, suppose we want to understand the behavior of the relative error <math>(\hat p - p)/p</math>. In classic estimators, the values that the relative error can take on depends on the value of <math>p</math>. I will present a new estimate with the remarkable property that the distribution of the relative error does not depend in any way on the value of <math>p</math>. Moreover, this new estimate is very fast: it takes a number of coin flips that is very close to the theoretical minimum. Time permitting, I will also discuss new ways to use concentration results for estimating the mean of random variables where normal approximations do not apply.<br />
<br />
== Thursday, March 26, [http://mathsci.kaist.ac.kr/~jioon/ Ji Oon Lee], [http://www.kaist.edu/html/en/index.html KAIST] ==<br />
<br />
Title: Tracy-Widom Distribution for Sample Covariance Matrices with General Population<br />
<br />
Abstract:<br />
Consider the sample covariance matrix <math>(\Sigma^{1/2} X)(\Sigma^{1/2} X)^*</math>, where the sample <math>X</math> is an <math>M \times N</math> random matrix whose entries are real independent random variables with variance <math>1/N</math> and <math>\Sigma</math> is an <math>M \times M</math> positive-definite deterministic diagonal matrix. We show that the fluctuation of its rescaled largest eigenvalue is given by the type-1 Tracy-Widom distribution. This is a joint work with Kevin Schnelli.<br />
<br />
== Thursday, April 2, No Seminar, Spring Break ==<br />
<br />
<br />
<br />
<br />
== Thursday, April 9, [http://www.math.wisc.edu/~emrah/ Elnur Emrah], [http://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: The shape functions of certain exactly solvable inhomogeneous planar corner growth models<br />
<br />
Abstract: I will talk about two kinds of inhomogeneous corner growth models with independent waiting times {W(i, j): i, j positive integers}: (1) W(i, j) is distributed exponentially with parameter <math>a_i+b_j</math> for each i, j.(2) W(i, j) is distributed geometrically with fail parameter <math>a_ib_j</math> for each i, j. These generalize exactly-solvable i.i.d. models with exponential or geometric waiting times. The parameters (a_n) and (b_n) are random with a joint distribution that is stationary with respect to the nonnegative shifts and ergodic (separately) with respect to the positive shifts of the indices. Then the shape functions of models (1) and (2) satisfy variational formulas in terms of the marginal distributions of (a_n) and (b_n). For certain choices of these marginal distributions, we still get closed-form expressions for the shape function as in the i.i.d. models.<br />
<br />
== Thursday, April 16, [http://www.math.wisc.edu/~shottovy/ Scott Hottovy], [http://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''An SDE approximation for stochastic differential delay equations with colored state-dependent noise'''<br />
<br />
Abstract: TBA<br />
<br />
== Thursday, April 23, [http://people.math.osu.edu/nguyen.1261/ Hoi Nguyen], [http://math.osu.edu/ Ohio State University] ==<br />
<br />
Title: On eigenvalue repulsion of random matrices<br />
<br />
Abstract:<br />
<br />
I will address certain repulsion behavior of roots of random polynomials and of eigenvalues of Wigner matrices, and their applications. Among other things, we show a Wegner-type estimate for the number of eigenvalues inside an extremely small interval for quite general matrix ensembles.<br />
<br />
== Thursday, April 30, TBA ==<br />
<br />
Title: TBA<br />
<br />
Abstract:<br />
<br />
<br />
== Thursday, May 7, TBA ==<br />
<br />
Title: TBA<br />
<br />
Abstract:<br />
<br />
<br />
<br />
<br />
<br />
<br />
<!--<br />
== Thursday, December 11, TBA ==<br />
<br />
Title: TBA<br />
<br />
Abstract:<br />
--><br />
<br />
<br />
<br />
<!--<br />
<br />
== Thursday, September 11, <span style="color:red">Van Vleck B105,</span> [http://www.math.wisc.edu/~mmwood/ Melanie Matchett Wood], UW-Madison ==<br />
<br />
Please note the non-standard room.<br />
<br />
Title: '''The distribution of sandpile groups of random graphs'''<br />
<br />
Abstract:<br><br />
The sandpile group is an abelian group associated to a graph, given as<br />
the cokernel of the graph Laplacian. An Erdős–Rényi random graph<br />
then gives some distribution of random abelian groups. We will give<br />
an introduction to various models of random finite abelian groups<br />
arising in number theory and the connections to the distribution<br />
conjectured by Payne et. al. for sandpile groups. We will talk about<br />
the moments of random finite abelian groups, and how in practice these<br />
are often more accessible than the distributions themselves, but<br />
frustratingly are not a priori guaranteed to determine the<br />
distribution. In this case however, we have found the moments of the<br />
sandpile groups of random graphs, and proved they determine the<br />
measure, and have proven Payne's conjecture.<br />
<br />
== Thursday, September 18, [http://www.math.purdue.edu/~peterson/ Jonathon Peterson], [http://www.math.purdue.edu/ Purdue University] ==<br />
<br />
Title: '''Hydrodynamic limits for directed traps and systems of independent RWRE'''<br />
<br />
Abstract:<br />
<br />
We study the evolution of a system of independent random walks in a common random environment (RWRE). Previously a hydrodynamic limit was proved in the case where the environment is such that the random walks are ballistic (i.e., transient with non-zero speed <math>v_0 \neq 0)</math>. In this case it was shown that the asymptotic particle density is simply translated deterministically by the speed $v_0$. In this talk we will consider the more difficult case of RWRE that are transient but with $v_0=0$. Under the appropriate space-time scaling, we prove a hydrodynamic limit for the system of random walks. The statement of the hydrodynamic limit that we prove is non-standard in that the evolution of the asymptotic particle density is given by the solution of a random rather than a deterministic PDE. The randomness in the PDE comes from the fact that under the hydrodynamic scaling the effect of the environment does not ``average out'' and so the specific instance of the environment chosen actually matters.<br />
<br />
The proof of the hydrodynamic limit for the system of RWRE will be accomplished by coupling the system of RWRE with a simpler model of a system of particles in an environment of ``directed traps.'' This talk is based on joint work with Milton Jara.<br />
<br />
== Thursday, September 25, [http://math.colorado.edu/~seor3821/ Sean O'Rourke], [http://www.colorado.edu/math/ University of Colorado Boulder] ==<br />
<br />
Title: '''Singular values and vectors under random perturbation'''<br />
<br />
Abstract:<br />
Computing the singular values and singular vectors of a large matrix is a basic task in high dimensional data analysis with many applications in computer science and statistics. In practice, however, data is often perturbed by noise. A natural question is the following. How much does a small perturbation to the matrix change the singular values and vectors? <br />
<br />
Classical (deterministic) theorems, such as those by Davis-Kahan, Wedin, and Weyl, give tight estimates for the worst-case scenario. In this talk, I will consider the case when the perturbation is random. In this setting, better estimates can be achieved when our matrix has low rank. This talk is based on joint work with Van Vu and Ke Wang.<br />
<br />
== Thursday, October 2, [http://www.math.wisc.edu/~jyin/jun-yin.html Jun Yin], [http://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Anisotropic local laws for random matrices'''<br />
<br />
Abstract:<br />
In this talk, we introduce a new method of deriving local laws of random matrices. As applications, we will show the local laws and some universality results on general sample covariance matrices: TXX^*T^* (where $T$ is non-square deterministic matrix), and deformed Wigner matrix: H+A (where A is deterministic symmetric matrix). Note: here $TT^*$ and $A$ could be full rank matrices.<br />
<br />
== Thursday, October 9, No seminar due to [http://www.math.northwestern.edu/mwp/ Midwest Probability Colloquium] ==<br />
<br />
No seminar due to [http://www.math.northwestern.edu/mwp/ Midwest Probability Colloquium].<br />
<br />
<br />
== Thursday, October 16, [http://www.math.utah.edu/~firas/ Firas Rassoul-Agha], [http://www.math.utah.edu/ University of Utah]==<br />
<br />
Title: '''The growth model: Busemann functions, shape, geodesics, and other stories'''<br />
<br />
Abstract:<br />
We consider the directed last-passage percolation model on the planar integer lattice with nearest-neighbor steps and general i.i.d. weights on the vertices, outside the class of exactly solvable models. Stationary cocycles are constructed for this percolation model from queueing fixed points. These cocycles define solutions to variational formulas that characterize limit shapes and yield new results for Busemann functions, geodesics and the competition interface. This is joint work with Nicos Georgiou and Timo Seppalainen.<br />
<br />
<br />
== Thursday, November 6, Vadim Gorin, [http://www-math.mit.edu/people/profile.php?pid=1415 MIT] ==<br />
<br />
Title: '''Multilevel Dyson Brownian Motion and its edge limits.'''<br />
<br />
Abstract: The GUE Tracy-Widom distribution is known to govern the large-time asymptotics for a variety of<br />
interacting particle systems on one side, and the asymptotic behavior for largest eigenvalues of<br />
random Hermitian matrices on the other side. In my talk I will explain some reasons for this<br />
connection between two seemingly unrelated classes of stochastic systems, and how this relation can<br />
be extended to general beta random matrices. A multilevel extension of the Dyson Brownian Motion<br />
will be the central object in the discussion.<br />
<br />
(Based on joint papers with Misha Shkolnikov.)<br />
<br />
==<span style="color:red"> Friday</span>, November 7, [http://tchumley.public.iastate.edu/ Tim Chumley], [http://www.math.iastate.edu/ Iowa State University] ==<br />
<br />
<span style="color:darkgreen">Please note the unusual day.</span><br />
<br />
Title: '''Random billiards and diffusion'''<br />
<br />
Abstract: We introduce a class of random dynamical systems derived from billiard maps and study a certain Markov chain derived from them. We then discuss the interplay between the billiard geometry and stochastic properties of the random system. The main results presented investigate the affect of billiard geometry on a diffusion process obtained from an appropriate scaling limit of the Markov chain.<br />
<br />
== Thursday, November 13, [http://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], [http://www.math.wisc.edu/ UW-Madison]==<br />
<br />
Title: '''Variational formulas for directed polymer and percolation models'''<br />
<br />
Abstract:<br />
Explicit formulas for subadditive limits of polymer and percolation models in probability and statistical mechanics have been difficult to find. We describe variational formulas for these limits and their connections with other features of the models such as Busemann functions and Kardar-Parisi-Zhang (KPZ) fluctuation exponents.<br />
<br />
<br />
<br />
== <span style="color:red">Monday</span>, December 1, [http://www.ma.utexas.edu/users/jneeman/index.html Joe Neeman], [http://www.ma.utexas.edu/ UT-Austin], <span style="color:red">4pm, Room B239 Van Vleck Hall</span>==<br />
<br />
<span style="color:darkgreen">Please note the unusual time and room.</span><br />
<br />
Title: '''Some phase transitions in the stochastic block model'''<br />
<br />
Abstract: The stochastic block model is a random graph model that was originally 30 years ago to study community detection in networks. To generate a random graph from this model, begin with two classes of vertices and then connect each pair of vertices independently at random, with probability p if they are in the same class and probability q otherwise. Some questions come to mind: can we reconstruct the classes if we only observe the graph? What if we only want to partially reconstruct the classes? How different is this model from an Erdos-Renyi graph anyway? The answers to these questions depend on p and q, and we will say exactly how.<br />
<br />
== Thursday, December 4, Arjun Krishnan, [http://www.fields.utoronto.ca/ Fields Institute] ==<br />
<br />
Title: '''Variational formula for the time-constant of first-passage percolation'''<br />
<br />
Abstract:<br />
Consider first-passage percolation with positive, stationary-ergodic<br />
weights on the square lattice in d-dimensions. Let <math>T(x)</math> be the<br />
first-passage time from the origin to <math>x</math> in <math>Z^d</math>. The convergence of<br />
<math>T([nx])/n</math> to the time constant as <math>n</math> tends to infinity is a consequence<br />
of the subadditive ergodic theorem. This convergence can be viewed as<br />
a problem of homogenization for a discrete Hamilton-Jacobi-Bellman<br />
(HJB) equation. By borrowing several tools from the continuum theory<br />
of stochastic homogenization for HJB equations, we derive an exact<br />
variational formula (duality principle) for the time-constant. Under a<br />
symmetry assumption, we will use the variational formula to construct<br />
an explicit iteration that produces the limit shape.<br />
<br />
<br />
--><br />
<br />
== ==<br />
<br />
<br />
<br />
[[Past Seminars]]</div>Emrahhttps://www.math.wisc.edu/wiki/index.php?title=Past_Probability_Seminars_Spring_2020&diff=9559Past Probability Seminars Spring 20202015-03-26T16:58:50Z<p>Emrah: /* Thursday, April 9, Elnur Emrah, UW-Madison */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2015 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<br />
<b><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 />
<!-- [[File:probsem.jpg]] --><br />
</b><br />
<br />
= =<br />
<br />
== Thursday, January 15, [http://www.stat.berkeley.edu/~racz/ Miklos Racz], [http://statistics.berkeley.edu/ UC-Berkeley Stats] ==<br />
<br />
<br />
Title: Testing for high-dimensional geometry in random graphs<br />
<br />
Abstract: I will talk about a random geometric graph model, where connections between vertices depend on distances between latent d-dimensional labels; we are particularly interested in the high-dimensional case when d is large. Upon observing a graph, we want to tell if it was generated from this geometric model, or from an Erdos-Renyi random graph. We show that there exists a computationally efficient procedure to do this which is almost optimal (in an information-theoretic sense). The key insight is based on a new statistic which we call "signed triangles". To prove optimality we use a bound on the total variation distance between Wishart matrices and the Gaussian Orthogonal Ensemble. This is joint work with Sebastien Bubeck, Jian Ding, and Ronen Eldan.<br />
<br />
== Thursday, January 22, No Seminar ==<br />
<br />
== Thursday, January 29, [http://www.math.umn.edu/~arnab/ Arnab Sen], [http://www.math.umn.edu/ University of Minnesota] ==<br />
<br />
Title: '''Double Roots of Random Littlewood Polynomials'''<br />
<br />
Abstract:<br />
We consider random polynomials whose coefficients are independent and uniform on {-1,1}. We will show that the probability that such a polynomial of degree n has a double root is o(n^{-2}) when n+1 is not divisible by 4 and is of the order n^{-2} otherwise. We will also discuss extensions to random polynomials with more general coefficient distributions. <br />
<br />
This is joint work with Ron Peled and Ofer Zeitouni.<br />
<br />
== Thursday, February 5, No seminar this week ==<br />
<br />
== Thursday, February 12, No Seminar this week==<br />
<br />
<br />
<!--<br />
== Wednesday, <span style="color:red">February 11</span>, [http://www.math.wisc.edu/~stechmann/ Sam Stechmann], [http://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
<span style="color:red">Please note the unusual time and room.<br />
</span><br />
<br />
<br />
Title: Stochastic Models for Rainfall: Extreme Events and Critical Phenomena<br />
<br />
<br />
Abstract:<br />
In recent years, tropical rainfall statistics have been shown to conform to paradigms of critical phenomena and statistical physics. In this talk, stochastic models will be presented as prototypes for understanding the atmospheric dynamics that leads to these statistics and extreme events. Key nonlinear ingredients in the models include either stochastic jump processes or thresholds (Heaviside functions). First, both exact solutions and simple numerics are used to verify that a suite of observed rainfall statistics is reproduced by the models, including power-law distributions and long-range correlations. Second, we prove that a stochastic trigger, which is a time-evolving indicator of whether it is raining or not, will converge to a deterministic threshold in an appropriate limit. Finally, we discuss the connections among these rainfall models, stochastic PDEs, and traditional models for critical phenomena.<br />
---><br />
<br />
== Thursday, February 19, [http://www.math.purdue.edu/people/bio/guo297 Xiaoqin Guo], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
Title: Quenched invariance principle for random walks in time-dependent random environment<br />
<br />
Abstract: In this talk we discuss random walks in a time-dependent zero-drift random environment in <math>Z^d</math>. We prove a quenched invariance principle under an appropriate moment condition. The proof is based on the use of a maximum principle for parabolic difference operators. This is a joint work with Jean-Dominique Deuschel and Alejandro Ramirez.<br />
<br />
== Thursday, February 26, [http://wwwf.imperial.ac.uk/~dcrisan/ Dan Crisan], [http://www.imperial.ac.uk/natural-sciences/departments/mathematics/ Imperial College London] ==<br />
<br />
Title: '''Smoothness properties of randomly perturbed semigroups with application to nonlinear filtering'''<br />
<br />
Abstract:<br />
In this talk I will discuss sharp gradient bounds for perturbed diffusion semigroups. In contrast with existing results, the perturbation is here random and the bounds obtained are pathwise. Our approach builds on the classical work of Kusuoka and Stroock and extends their program developed for the heat semi-group to solutions of stochastic partial differential equations. The work is motivated by and applied to nonlinear filtering. The analysis allows us to derive pathwise gradient bounds for the un-normalised conditional distribution of a partially observed signal. The estimates we derive have sharp small time asymptotics<br />
<br />
This is joint work with Terry Lyons (Oxford) and Christian Literrer (Ecole Polytechnique) and is based on the paper<br />
<br />
D Crisan, C Litterer, T Lyons, Kusuoka–Stroock gradient bounds for the solution of the filtering equation, Journal of Functional Analysis, 2105<br />
<br />
== Wednesday, <span style="color:red">March 4</span>, [http://www.math.wisc.edu/~stechmann/ Sam Stechmann], [http://www.math.wisc.edu/ UW-Madison], <span style="color:red"> 2:25pm Van Vleck B113</span> ==<br />
<br />
<span style="color:red">Please note the unusual time and room.<br />
</span><br />
<br />
<br />
Title: Stochastic Models for Rainfall: Extreme Events and Critical Phenomena<br />
<br />
<br />
Abstract:<br />
In recent years, tropical rainfall statistics have been shown to conform to paradigms of critical phenomena and statistical physics. In this talk, stochastic models will be presented as prototypes for understanding the atmospheric dynamics that leads to these statistics and extreme events. Key nonlinear ingredients in the models include either stochastic jump processes or thresholds (Heaviside functions). First, both exact solutions and simple numerics are used to verify that a suite of observed rainfall statistics is reproduced by the models, including power-law distributions and long-range correlations. Second, we prove that a stochastic trigger, which is a time-evolving indicator of whether it is raining or not, will converge to a deterministic threshold in an appropriate limit. Finally, we discuss the connections among these rainfall models, stochastic PDEs, and traditional models for critical phenomena.<br />
<br />
== Thursday, March 12, [http://www.ima.umn.edu/~ohadfeld/Website/index.html Ohad Feldheim], [http://www.ima.umn.edu/ IMA] ==<br />
<br />
<br />
Title: '''The 3-states AF-Potts model in high dimension'''<br />
<br />
Abstract: <br />
<!--<br />
Take a bounded odd domain of the bipartite graph $\mathbb{Z}^d$. Color the boundary of the set by $0$, then<br />
color the rest of the domain at random with the colors $\{0,\dots,q-1\}$, penalizing every<br />
configuration with proportion to the number of improper edges at a given rate $\beta>0$ (the "inverse temperature").<br />
Q: "What is the structure of such a coloring?"<br />
<br />
This model is called the $q$-states Potts antiferromagnet(AF), a classical spin glass model in statistical mechanics.<br />
The $2$-states case is the famous Ising model which is relatively well understood.<br />
The $3$-states case in high dimension has been studies for $\beta=\infty$,<br />
when the model reduces to a uniformly chosen proper three coloring of the domain.<br />
Several words, by Galvin, Kahn, Peled, Randall and Sorkin established the structure of the model <br />
showing long-range correlations and phase coexistence. In this work, we generalize this result to positive temperature, <br />
showing that for large enough $\beta$ (low enough temperature)<br />
the rigid structure persists. This is the first rigorous result for $\beta<\infty$.<br />
<br />
In the talk, assuming no acquaintance with the model, we shall give the physical background, introduce all the<br />
relevant definitions and shed some light on how such results are proved using only combinatorial methods.<br />
Joint work with Yinon Spinka.<br />
--><br />
Take a bounded odd domain of the bipartite graph <math>\mathbb{Z}^d</math>. Color the<br />
boundary of the set by <math>0</math>, then<br />
color the rest of the domain at random with the colors <math>\{0,\dots,q-1\}</math>,<br />
penalizing every<br />
configuration with proportion to the number of improper edges at a given rate<br />
<math>\beta>0</math> (the "inverse temperature").<br />
Q: "What is the structure of such a coloring?"<br />
<br />
This model is called the <math>q</math>-states Potts antiferromagnet(AF), a classical spin<br />
glass model in statistical mechanics.<br />
The <math>2</math>-states case is the famous Ising model which is relatively well<br />
understood.<br />
The <math>3</math>-states case in high dimension has been studies for <math>\beta=\infty</math>,<br />
when the model reduces to a uniformly chosen proper three coloring of the<br />
domain.<br />
Several words, by Galvin, Kahn, Peled, Randall and Sorkin established the<br />
structure of the model<br />
showing long-range correlations and phase coexistence. In this work, we<br />
generalize this result to positive temperature,<br />
showing that for large enough <math>\beta</math> (low enough temperature)<br />
the rigid structure persists. This is the first rigorous result for<br />
<math>\beta<\infty</math>.<br />
<br />
In the talk, assuming no acquaintance with the model, we shall give the<br />
physical background, introduce all the<br />
relevant definitions and shed some light on how such results are proved using<br />
only combinatorial methods.<br />
Joint work with Yinon Spinka.<br />
<br />
== Thursday, March 19, [http://www.cmc.edu/pages/faculty/MHuber/ Mark Huber], [http://www.cmc.edu/math/ Claremont McKenna Math] ==<br />
<br />
Title: Understanding relative error in Monte Carlo simulations<br />
<br />
Abstract: The problem of estimating the probability <math>p</math> of heads on an unfair coin has been around for centuries, and has inspired numerous advances in probability such as the Strong Law of Large Numbers and the Central Limit Theorem. In this talk, I'll consider a new twist: given an estimate <math>\hat p</math>, suppose we want to understand the behavior of the relative error <math>(\hat p - p)/p</math>. In classic estimators, the values that the relative error can take on depends on the value of <math>p</math>. I will present a new estimate with the remarkable property that the distribution of the relative error does not depend in any way on the value of <math>p</math>. Moreover, this new estimate is very fast: it takes a number of coin flips that is very close to the theoretical minimum. Time permitting, I will also discuss new ways to use concentration results for estimating the mean of random variables where normal approximations do not apply.<br />
<br />
== Thursday, March 26, [http://mathsci.kaist.ac.kr/~jioon/ Ji Oon Lee], [http://www.kaist.edu/html/en/index.html KAIST] ==<br />
<br />
Title: Tracy-Widom Distribution for Sample Covariance Matrices with General Population<br />
<br />
Abstract:<br />
Consider the sample covariance matrix <math>(\Sigma^{1/2} X)(\Sigma^{1/2} X)^*</math>, where the sample <math>X</math> is an <math>M \times N</math> random matrix whose entries are real independent random variables with variance <math>1/N</math> and <math>\Sigma</math> is an <math>M \times M</math> positive-definite deterministic diagonal matrix. We show that the fluctuation of its rescaled largest eigenvalue is given by the type-1 Tracy-Widom distribution. This is a joint work with Kevin Schnelli.<br />
<br />
== Thursday, April 2, No Seminar, Spring Break ==<br />
<br />
<br />
<br />
<br />
== Thursday, April 9, [http://www.math.wisc.edu/~emrah/ Elnur Emrah], [http://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: The shape functions of certain exactly solvable inhomogeneous planar corner growth models<br />
<br />
Abstract: I will talk about two kinds of inhomogeneous corner growth models with independent waiting times {W(i, j): i, j positive integers}: (1) W(i, j) is distributed exponentially with parameter a_i+b_j for each i, j. (2) W(i, j) is distributed geometrically with fail parameter a_ib_j for each i, j. These generalize exactly-solvable i.i.d. models with exponential or geometric waiting times. The parameters (a_n) and (b_n) are random with a joint distribution that is stationary with respect to the nonnegative shifts and ergodic (separately) with respect to the positive shifts of the indices. Then the shape functions of models (1) and (2) satisfy variational formulas in terms of the marginal distributions of (a_n) and (b_n). For certain choices of these marginal distributions, we still get closed-form expressions for the shape function as in the i.i.d. models.<br />
<br />
== Thursday, April 16, [http://www.math.wisc.edu/~shottovy/ Scott Hottovy], [http://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''An SDE approximation for stochastic differential delay equations with colored state-dependent noise'''<br />
<br />
Abstract: TBA<br />
<br />
== Thursday, April 23, [http://people.math.osu.edu/nguyen.1261/ Hoi Nguyen], [http://math.osu.edu/ Ohio State University] ==<br />
<br />
Title: On eigenvalue repulsion of random matrices<br />
<br />
Abstract:<br />
<br />
I will address certain repulsion behavior of roots of random polynomials and of eigenvalues of Wigner matrices, and their applications. Among other things, we show a Wegner-type estimate for the number of eigenvalues inside an extremely small interval for quite general matrix ensembles.<br />
<br />
== Thursday, April 30, TBA ==<br />
<br />
Title: TBA<br />
<br />
Abstract:<br />
<br />
<br />
== Thursday, May 7, TBA ==<br />
<br />
Title: TBA<br />
<br />
Abstract:<br />
<br />
<br />
<br />
<br />
<br />
<br />
<!--<br />
== Thursday, December 11, TBA ==<br />
<br />
Title: TBA<br />
<br />
Abstract:<br />
--><br />
<br />
<br />
<br />
<!--<br />
<br />
== Thursday, September 11, <span style="color:red">Van Vleck B105,</span> [http://www.math.wisc.edu/~mmwood/ Melanie Matchett Wood], UW-Madison ==<br />
<br />
Please note the non-standard room.<br />
<br />
Title: '''The distribution of sandpile groups of random graphs'''<br />
<br />
Abstract:<br><br />
The sandpile group is an abelian group associated to a graph, given as<br />
the cokernel of the graph Laplacian. An Erdős–Rényi random graph<br />
then gives some distribution of random abelian groups. We will give<br />
an introduction to various models of random finite abelian groups<br />
arising in number theory and the connections to the distribution<br />
conjectured by Payne et. al. for sandpile groups. We will talk about<br />
the moments of random finite abelian groups, and how in practice these<br />
are often more accessible than the distributions themselves, but<br />
frustratingly are not a priori guaranteed to determine the<br />
distribution. In this case however, we have found the moments of the<br />
sandpile groups of random graphs, and proved they determine the<br />
measure, and have proven Payne's conjecture.<br />
<br />
== Thursday, September 18, [http://www.math.purdue.edu/~peterson/ Jonathon Peterson], [http://www.math.purdue.edu/ Purdue University] ==<br />
<br />
Title: '''Hydrodynamic limits for directed traps and systems of independent RWRE'''<br />
<br />
Abstract:<br />
<br />
We study the evolution of a system of independent random walks in a common random environment (RWRE). Previously a hydrodynamic limit was proved in the case where the environment is such that the random walks are ballistic (i.e., transient with non-zero speed <math>v_0 \neq 0)</math>. In this case it was shown that the asymptotic particle density is simply translated deterministically by the speed $v_0$. In this talk we will consider the more difficult case of RWRE that are transient but with $v_0=0$. Under the appropriate space-time scaling, we prove a hydrodynamic limit for the system of random walks. The statement of the hydrodynamic limit that we prove is non-standard in that the evolution of the asymptotic particle density is given by the solution of a random rather than a deterministic PDE. The randomness in the PDE comes from the fact that under the hydrodynamic scaling the effect of the environment does not ``average out'' and so the specific instance of the environment chosen actually matters.<br />
<br />
The proof of the hydrodynamic limit for the system of RWRE will be accomplished by coupling the system of RWRE with a simpler model of a system of particles in an environment of ``directed traps.'' This talk is based on joint work with Milton Jara.<br />
<br />
== Thursday, September 25, [http://math.colorado.edu/~seor3821/ Sean O'Rourke], [http://www.colorado.edu/math/ University of Colorado Boulder] ==<br />
<br />
Title: '''Singular values and vectors under random perturbation'''<br />
<br />
Abstract:<br />
Computing the singular values and singular vectors of a large matrix is a basic task in high dimensional data analysis with many applications in computer science and statistics. In practice, however, data is often perturbed by noise. A natural question is the following. How much does a small perturbation to the matrix change the singular values and vectors? <br />
<br />
Classical (deterministic) theorems, such as those by Davis-Kahan, Wedin, and Weyl, give tight estimates for the worst-case scenario. In this talk, I will consider the case when the perturbation is random. In this setting, better estimates can be achieved when our matrix has low rank. This talk is based on joint work with Van Vu and Ke Wang.<br />
<br />
== Thursday, October 2, [http://www.math.wisc.edu/~jyin/jun-yin.html Jun Yin], [http://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Anisotropic local laws for random matrices'''<br />
<br />
Abstract:<br />
In this talk, we introduce a new method of deriving local laws of random matrices. As applications, we will show the local laws and some universality results on general sample covariance matrices: TXX^*T^* (where $T$ is non-square deterministic matrix), and deformed Wigner matrix: H+A (where A is deterministic symmetric matrix). Note: here $TT^*$ and $A$ could be full rank matrices.<br />
<br />
== Thursday, October 9, No seminar due to [http://www.math.northwestern.edu/mwp/ Midwest Probability Colloquium] ==<br />
<br />
No seminar due to [http://www.math.northwestern.edu/mwp/ Midwest Probability Colloquium].<br />
<br />
<br />
== Thursday, October 16, [http://www.math.utah.edu/~firas/ Firas Rassoul-Agha], [http://www.math.utah.edu/ University of Utah]==<br />
<br />
Title: '''The growth model: Busemann functions, shape, geodesics, and other stories'''<br />
<br />
Abstract:<br />
We consider the directed last-passage percolation model on the planar integer lattice with nearest-neighbor steps and general i.i.d. weights on the vertices, outside the class of exactly solvable models. Stationary cocycles are constructed for this percolation model from queueing fixed points. These cocycles define solutions to variational formulas that characterize limit shapes and yield new results for Busemann functions, geodesics and the competition interface. This is joint work with Nicos Georgiou and Timo Seppalainen.<br />
<br />
<br />
== Thursday, November 6, Vadim Gorin, [http://www-math.mit.edu/people/profile.php?pid=1415 MIT] ==<br />
<br />
Title: '''Multilevel Dyson Brownian Motion and its edge limits.'''<br />
<br />
Abstract: The GUE Tracy-Widom distribution is known to govern the large-time asymptotics for a variety of<br />
interacting particle systems on one side, and the asymptotic behavior for largest eigenvalues of<br />
random Hermitian matrices on the other side. In my talk I will explain some reasons for this<br />
connection between two seemingly unrelated classes of stochastic systems, and how this relation can<br />
be extended to general beta random matrices. A multilevel extension of the Dyson Brownian Motion<br />
will be the central object in the discussion.<br />
<br />
(Based on joint papers with Misha Shkolnikov.)<br />
<br />
==<span style="color:red"> Friday</span>, November 7, [http://tchumley.public.iastate.edu/ Tim Chumley], [http://www.math.iastate.edu/ Iowa State University] ==<br />
<br />
<span style="color:darkgreen">Please note the unusual day.</span><br />
<br />
Title: '''Random billiards and diffusion'''<br />
<br />
Abstract: We introduce a class of random dynamical systems derived from billiard maps and study a certain Markov chain derived from them. We then discuss the interplay between the billiard geometry and stochastic properties of the random system. The main results presented investigate the affect of billiard geometry on a diffusion process obtained from an appropriate scaling limit of the Markov chain.<br />
<br />
== Thursday, November 13, [http://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], [http://www.math.wisc.edu/ UW-Madison]==<br />
<br />
Title: '''Variational formulas for directed polymer and percolation models'''<br />
<br />
Abstract:<br />
Explicit formulas for subadditive limits of polymer and percolation models in probability and statistical mechanics have been difficult to find. We describe variational formulas for these limits and their connections with other features of the models such as Busemann functions and Kardar-Parisi-Zhang (KPZ) fluctuation exponents.<br />
<br />
<br />
<br />
== <span style="color:red">Monday</span>, December 1, [http://www.ma.utexas.edu/users/jneeman/index.html Joe Neeman], [http://www.ma.utexas.edu/ UT-Austin], <span style="color:red">4pm, Room B239 Van Vleck Hall</span>==<br />
<br />
<span style="color:darkgreen">Please note the unusual time and room.</span><br />
<br />
Title: '''Some phase transitions in the stochastic block model'''<br />
<br />
Abstract: The stochastic block model is a random graph model that was originally 30 years ago to study community detection in networks. To generate a random graph from this model, begin with two classes of vertices and then connect each pair of vertices independently at random, with probability p if they are in the same class and probability q otherwise. Some questions come to mind: can we reconstruct the classes if we only observe the graph? What if we only want to partially reconstruct the classes? How different is this model from an Erdos-Renyi graph anyway? The answers to these questions depend on p and q, and we will say exactly how.<br />
<br />
== Thursday, December 4, Arjun Krishnan, [http://www.fields.utoronto.ca/ Fields Institute] ==<br />
<br />
Title: '''Variational formula for the time-constant of first-passage percolation'''<br />
<br />
Abstract:<br />
Consider first-passage percolation with positive, stationary-ergodic<br />
weights on the square lattice in d-dimensions. Let <math>T(x)</math> be the<br />
first-passage time from the origin to <math>x</math> in <math>Z^d</math>. The convergence of<br />
<math>T([nx])/n</math> to the time constant as <math>n</math> tends to infinity is a consequence<br />
of the subadditive ergodic theorem. This convergence can be viewed as<br />
a problem of homogenization for a discrete Hamilton-Jacobi-Bellman<br />
(HJB) equation. By borrowing several tools from the continuum theory<br />
of stochastic homogenization for HJB equations, we derive an exact<br />
variational formula (duality principle) for the time-constant. Under a<br />
symmetry assumption, we will use the variational formula to construct<br />
an explicit iteration that produces the limit shape.<br />
<br />
<br />
--><br />
<br />
== ==<br />
<br />
<br />
<br />
[[Past Seminars]]</div>Emrahhttps://www.math.wisc.edu/wiki/index.php?title=Probability_group_timetable&diff=8225Probability group timetable2014-09-09T21:48:56Z<p>Emrah: </p>
<hr />
<div>{| border="2"<br />
| ||Monday||Tuesday||Wednesday||Thursday||Friday<br />
|-<br />
| 9-10|| Timo 431 (8:50) || Benedek 632 (9:30)<br> Phil away || Timo 431 (8:50)<br> Phil away || Benedek 632 (9:30) <br>Scott (9:30), Phil away || Timo 431 (8:50)<br />
|-<br />
| 10-11|| Timo 735 (9:55) <br> Elnur 234 (9:55) || Phil away || Timo 735 (9:55)<br> Phil away <br> Elnur 234 (9:55)|| Scott (9:30)<br> Phil away || Timo 735 (9:55) <br />
|-<br />
| 11-12|| Timo OH <br> Elnur 234 ||Benedek OH<br> Phil away || Timo OH <br> Phil away <br> Elnur 234 || Benedek OH <br> Phil away || Elnur OH<br />
|-<br />
| 12-1||Sebastien 632 (12:05) || Phil away ||Sebastien 632 (12:05) <br> Phil away || Phil away ||Sebastien 632 (12:05)<br />
|-<br />
| 1-2||Sebastien 833 (1:20) <br> Chris 833 <br> Elnur 833|| Benedek OH <br> Timo 733 <br> Jun 733 <br> Elnur 234||Sebastien 833 (1:20) <br> Chris 833 <br> Elnur 833 || Timo 733 <br> Jun 733 <br> Elnur 234||Sebastien 833 (1:20) <br />
|-<br />
| 2-3|| || reading seminar (2:25pm) || || probability seminar (2:25pm) ||<br />
|-<br />
| 3-4|| || || || ||<br />
|-<br />
| 4-5|| || Elnur OH || || || Colloquium<br />
|-<br />
| 5-6|| || Jun OH <br> Elnur OH|| Jun OH || ||<br />
|}</div>Emrahhttps://www.math.wisc.edu/wiki/index.php?title=Probability_group_timetable&diff=8149Probability group timetable2014-09-03T14:08:20Z<p>Emrah: </p>
<hr />
<div>{| border="2"<br />
| ||Monday||Tuesday||Wednesday||Thursday||Friday<br />
|-<br />
| 9-10|| Timo 431 (8:50) || Benedek 632 (9:30)<br> Phil away || Timo 431 (8:50)<br> Phil away || Benedek 632 (9:30) <br>Scott (9:30), Phil away || Timo 431 (8:50)<br />
|-<br />
| 10-11|| Timo 735 (9:55) <br> Elnur 234 (9:55) || Phil away || Timo 735 (9:55)<br> Phil away <br> Elnur 234 (9:55)|| Scott (9:30)<br> Phil away || Timo 735 (9:55) <br />
|-<br />
| 11-12|| Timo OH <br> Elnur 234 ||Benedek OH<br> Phil away || Timo OH <br> Phil away <br> Elnur 234 || Benedek OH <br> Phil away || Elnur OH<br />
|-<br />
| 12-1||Sebastien 632 (12:05) || Phil away ||Sebastien 632 (12:05) <br> Phil away || Phil away ||Sebastien 632 (12:05)<br />
|-<br />
| 1-2||Sebastien 833 (1:20) <br> Chris 833 <br> Elnur 833|| Benedek OH <br> Timo 733 <br> Jun 733 <br> Elnur 234||Sebastien 833 (1:20) <br> Chris 833 <br> Elnur 833 || Timo 733 <br> Jun 733 <br> Elnur 234||Sebastien 833 (1:20) <br />
|-<br />
| 2-3|| || reading seminar (2:25pm) || || probability seminar (2:25pm) ||<br />
|-<br />
| 3-4|| || || || ||<br />
|-<br />
| 4-5|| || || || Elnur OH || Colloquium<br />
|-<br />
| 5-6|| || Jun OH || Jun OH || Elnur OH ||<br />
|}</div>Emrahhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_student_reading_seminar&diff=6133Graduate student reading seminar2013-10-22T19:30:58Z<p>Emrah: /* Schedule */</p>
<hr />
<div>Time and place: Wednesday 3:30PM-4:30PM, B115<br />
<br />
==Schedule==<br />
<br />
We will meet on Tuesdays at 2:25pm. <br />
<br />
The first meeting is on 9/10 in room B203. We will discuss the schedule and the format.<br />
<br />
<br />
Schedule:<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 />
<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: <br><br />
Rogers & Williams - Diffusions, Markov Processes and Martingales <br><br />
Ito & McKean - Diffusion Processes and their Sample Paths <br><br />
Breiman - Probability <br><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>Emrahhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_student_reading_seminar&diff=6131Graduate student reading seminar2013-10-21T04:43:51Z<p>Emrah: /* Schedule */</p>
<hr />
<div>Time and place: Wednesday 3:30PM-4:30PM, B115<br />
<br />
==Schedule==<br />
<br />
We will meet on Tuesdays at 2:25pm. <br />
<br />
The first meeting is on 9/10 in room B203. We will discuss the schedule and the format.<br />
<br />
<br />
Schedule:<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: Elnur<br />
Limit fluctuations of last passage times <br />
<br />
11/5, 11/12: Yun<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: <br><br />
Rogers & Williams - Diffusions, Markov Processes and Martingales <br><br />
Ito & McKean - Diffusion Processes and their Sample Paths <br><br />
Breiman - Probability <br><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>Emrahhttps://www.math.wisc.edu/wiki/index.php?title=Probability_group_timetable&diff=5715Probability group timetable2013-08-29T07:20:42Z<p>Emrah: </p>
<hr />
<div>{| border="2"<br />
| ||Monday||Tuesday||Wednesday||Thursday||Friday<br />
|-<br />
| 9-10|| 234 Greg 7:45, Phil out|| Benedek (OH) 9:30, 340 Beth, 132 Diane, 703 Elnur 9:30 || 234 Greg 7:45 ||Benedek (OH) 9:30, 340 Beth, 132 Diane, 703 Elnur 9:30|| 234 Greg 7:45, Phil Out<br />
|-<br />
| 10-11|| 431 Timo, Greg, 331 Sebastien 9:55, Phil out || 222 Chris || 431 Timo, Greg, 331 Sebastien 9:55 || 222 Chris || 431 Timo, Greg, 331 Sebastien 9:55 , Phil Out<br />
|-<br />
| 11-12||873 Beth, Phil Out|| 431 Benedek 11, 827 Chris, 827 Elnur ||873 Beth || 431 Benedek 11, 827 Chris, 827 Elnur || 873 Beth, Phil Out<br />
|-<br />
| 12-1|| 632 Sebastien 12:05, 340 Elnur || 340 Beth || 632 Sebastien 12:05, 340 Elnur|| 340 Beth || 632 Sebastien 12:05<br />
|-<br />
| 1-2|| 431 Timo 1:20pm || 605 Dave, 733 Sebastien 1:00, 340 Beth, 222 Chris, 340 Elnur || 431 Timo 1:20pm || 605 Dave, 733 Sebastien 1:00, 340 Beth, 222 Chris, 340 Elnur || 431 Timo 1:20pm<br />
|-<br />
| 2-3|| 222 Chris 2:25pm, 340 Elnur 2:25|| Grad student probability seminar || 222 Chris 2:25pm, 340 Elnur 2:25||Probability seminar (2:25) || 222 Chris 2:25pm <br />
|-<br />
| 3-4|| || || || ||<br />
|-<br />
| 4-5|| || || || || Colloquium<br />
|-<br />
| 5-6|| || || || ||<br />
|}</div>Emrahhttps://www.math.wisc.edu/wiki/index.php?title=Probability_group_timetable&diff=5714Probability group timetable2013-08-29T07:20:16Z<p>Emrah: </p>
<hr />
<div>{| border="2"<br />
| ||Monday||Tuesday||Wednesday||Thursday||Friday<br />
|-<br />
| 9-10|| 234 Greg 7:45, Phil out|| Benedek (OH) 9:30, 340 Beth, 132 Diane, 703 Elnur 9:30 || 234 Greg 7:45 ||Benedek (OH) 9:30, 340 Beth, 132 Diane 703 Elnur 9:30|| 234 Greg 7:45, Phil Out<br />
|-<br />
| 10-11|| 431 Timo, Greg, 331 Sebastien 9:55, Phil out || 222 Chris || 431 Timo, Greg, 331 Sebastien 9:55 || 222 Chris || 431 Timo, Greg, 331 Sebastien 9:55 , Phil Out<br />
|-<br />
| 11-12||873 Beth, Phil Out|| 431 Benedek 11, 827 Chris, 827 Elnur ||873 Beth || 431 Benedek 11, 827 Chris, 827 Elnur || 873 Beth, Phil Out<br />
|-<br />
| 12-1|| 632 Sebastien 12:05, 340 Elnur || 340 Beth || 632 Sebastien 12:05, 340 Elnur|| 340 Beth || 632 Sebastien 12:05<br />
|-<br />
| 1-2|| 431 Timo 1:20pm || 605 Dave, 733 Sebastien 1:00, 340 Beth, 222 Chris, 340 Elnur || 431 Timo 1:20pm || 605 Dave, 733 Sebastien 1:00, 340 Beth, 222 Chris, 340 Elnur || 431 Timo 1:20pm<br />
|-<br />
| 2-3|| 222 Chris 2:25pm, 340 Elnur 2:25|| Grad student probability seminar || 222 Chris 2:25pm, 340 Elnur 2:25||Probability seminar (2:25) || 222 Chris 2:25pm <br />
|-<br />
| 3-4|| || || || ||<br />
|-<br />
| 4-5|| || || || || Colloquium<br />
|-<br />
| 5-6|| || || || ||<br />
|}</div>Emrahhttps://www.math.wisc.edu/wiki/index.php?title=Probability_group_timetable&diff=5713Probability group timetable2013-08-29T07:17:30Z<p>Emrah: </p>
<hr />
<div>{| border="2"<br />
| ||Monday||Tuesday||Wednesday||Thursday||Friday<br />
|-<br />
| 9-10|| 234 Greg 7:45, Phil out|| Benedek (OH) 9:30, 340 Beth, 132 Diane, 703 Elnur 9:30 || 234 Greg 7:45 ||Benedek (OH) 9:30, 340 Beth, 132 Diane 703 Elnur 9:30|| 234 Greg 7:45, Phil Out<br />
|-<br />
| 10-11|| 431 Timo, Greg, 331 Sebastien 9:55, Phil out || 222 Chris || 431 Timo, Greg, 331 Sebastien 9:55 || 222 Chris || 431 Timo, Greg, 331 Sebastien 9:55 , Phil Out<br />
|-<br />
| 11-12||873 Beth, Phil Out|| 431 Benedek 11, 827 Chris, 827 Elnur ||873 Beth || 431 Benedek 11, 827 Chris, 827 Elnur || 873 Beth, Phil Out<br />
|-<br />
| 12-1|| 632 Sebastien 12:05, 340 Elnur || 340 Beth || 632 Sebastien 12:05, 340 Elnur|| 340 Beth || 632 Sebastien 12:05<br />
|-<br />
| 1-2|| 431 Timo 1:20pm || 605 Dave, 733 Sebastien 1:00, 340 Beth, 222 Chris, 340 Elnur || 431 Timo 1:20pm || 605 Dave, 733 Sebastien 1:00, 340 Beth, 222 Chris, 340 Elnur || 431 Timo 1:20pm<br />
|-<br />
| 2-3|| 222 Chris 2:25pm, 340 Elnur 2:25|| Grad student probability seminar || 222 Chris 2:25pm 340 Elnur 2:25||Probability seminar (2:25) || 222 Chris 2:25pm <br />
|-<br />
| 3-4|| || || || ||<br />
|-<br />
| 4-5|| || || || || Colloquium<br />
|-<br />
| 5-6|| || || || ||<br />
|}</div>Emrahhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_student_reading_seminar&diff=5017Graduate student reading seminar2013-02-06T14:36:45Z<p>Emrah: </p>
<hr />
<div>Time and place: Wednesday 3:30PM-4:30PM, B115<br />
<br />
==Schedule==<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 />
3/6: Chris<br />
<br />
3/13: Dae Han<br />
<br />
3/20: Dae Han<br />
<br />
4/3: Diane<br />
<br />
4/10: Diane<br />
<br />
4/17: Yun<br />
<br />
4/24: Yun<br />
<br />
5/1: Greg, Maso<br />
<br />
5/8: Greg, Maso</div>Emrahhttps://www.math.wisc.edu/wiki/index.php?title=Probability_group_timetable&diff=4937Probability group timetable2013-01-23T19:36:15Z<p>Emrah: </p>
<hr />
<div>{| border="2"<br />
| ||Monday||Tuesday||Wednesday||Thursday||Friday<br />
|-<br />
| 9-10|| Dave (office hours), 732 Chris 8:50 ||431 (Benedek), 211 (Phil) 9:30|| 732 Chris 8:50 ||431 (Benedek), 211 (Phil) 9:30|| 732 Chris 8:50<br />
|-<br />
| 10-11|| 828 Chris 9:55 || Diane (office hours), 213 Yun 9:55, Phil (OH) || 828 Chris 9:55 || 213 Yun 9:55, Phil (OH) || 828 Chris 9:55<br />
|-<br />
| 11-12|| Diane, (office hours) 421 Chris 11 ||832 (Dave), 341 (Diane), 421 (Chris) 11 , Benedek (office hours 11-12), Phil (OH)|| 421 Chris 11 || 832 (Dave) 11, 421 Chris 11, Phil (OH) || 421 Chris 11 <br />
|-<br />
| 12-1|| 632 (Jun) 12:05, 234 (Elnur) || 341 (Diane) 12:05, 421 Chris 12:05 || 632 (Jun) 12:05, 234 (Elnur) || OH (Chris) 12 || 632 (Jun) 12:05, 234 (Elnur) <br />
|-<br />
| 1-2|| 632 (Jun) 1:20 || OH (Chris) 1, 741 Yun 1:00 || 632 (Jun) 1:20 || OH (Chris) 1, 741 Yun 1:00 || 632 (Jun) 1:20<br />
|-<br />
| 2-3|| 833 (Timo, Chris, Diane,Yun) (2:25) || || 833 (Timo, Chris, Diane,Yun) (2:25), Dave (office hours: 2:30-3:30) ||Probability seminar (2:25) || 833 (Timo, Chris, Diane,Yun) (2:25) <br />
|-<br />
| 3-4|| || || || ||<br />
|-<br />
| 4-5|| || || || ||<br />
|-<br />
| 5-6|| || || || ||<br />
|}</div>Emrahhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_student_reading_seminar&diff=4403Graduate student reading seminar2012-09-24T21:58:11Z<p>Emrah: </p>
<hr />
<div>Time and place: Monday 2:25PM-3:30PM, B329<br />
<br />
We decided to start with SLE (Sctochastic Loewner evolution). We will use Greg Lawler's [http://math.uchicago.edu/~lawler/utah.pdf Park City notes].<br />
<br />
[http://math.arizona.edu/~tgk/529/syl.html A course on SLE]<br />
<br />
[http://www2.math.tu-berlin.de/smcp/index.php?id=85 Another (mini) course on SLE]<br />
<br />
September 17: read Lecture 1 from the notes<br />
<br />
September 24: read Lecture 2 from the notes and work on the problems</div>Emrahhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_student_reading_seminar&diff=4398Graduate student reading seminar2012-09-24T21:06:53Z<p>Emrah: </p>
<hr />
<div>Time and place: Monday 2:25PM-3:30PM, B329<br />
<br />
We decided to start with SLE (Sctochastic Loewner evolution). We will use Greg Lawler's [http://math.uchicago.edu/~lawler/utah.pdf Park City notes].<br />
<br />
A link to a course on SLE: http://math.arizona.edu/~tgk/529/syl.html<br />
<br />
September 17: read Lecture 1 from the notes<br />
<br />
September 24: read Lecture 2 from the notes and work on the problems</div>Emrah