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−  == Fall 2011 ==  +  == Spring 2012 == 
   
   
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−  == Thursday, September 15, Jun Yin, University of Wisconsin  Madison ==
 
−  '''Some recent results on random matrices with almost independent entries.'''
 
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−  In this talk, we are going to introduce some recent work on a large class of random matrices, whose entries are (almost) independent. For example, the Wigner matrix, generalized Wigner matrix, Band random matrix, Covariance matrix and Sparse random matrix. We mainly focus on the local statistics of the eigenvalues and eigenvectors of these random matrix ensembles. We will also introduce some applications of these results and some longstanding open questions.
 
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−  == Thursday, September 22, Philip Matchett Wood, University of Wisconsin  Madison ==
 
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−  '''Survey of the Circular Law'''
 
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−  What do the eigenvalues of a random matrix look like? This talk will focus on large square matrices where the entries are independent, identically distributed random variables. In the most basic case, the distribution of the eigenvalues in the complex plane (suitably scaled) approaches the uniform distribution on the unit disk, which is called the circular law. We will discuss some of the methods that have been used to prove the circular law, including recent work that has extended the circular law to the most general situation, and we will also discuss generalizations to situations where the eigenvalue distributions are stable, but noncircular.
 
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−  == Thursday, September 29, Antonio Auffinger, University of Chicago ==
 
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−  '''A simplified proof of the relation between scaling exponents in first passage percolation'''
 
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−  In first passage percolation, we place i.i.d. nonnegative weights on the nearestneighbor edges of Z^d and study the induced random metric. A longstanding conjecture gives a relation between two "scaling exponents": one describes the variance of the distance between two points and the other describes the transversal fluctuations of optimizing paths between the same points. In a recent breakthrough work, Sourav Chatterjee proved a version of this conjecture using a strong definition of the exponents. I will discuss work I just completed with Michael Damron, in which we introduce a new and intuitive idea that replaces Chatterjee's main argument and gives an alternative proof of the scaling relation. One advantage of our argument is that it does not require a nontrivial technical assumption of Chatterjee on the weight distribution.
 
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−  == Tuesday, October 4, 2:30 PM, VV901, Gregorio Moreno Flores, University of Wisconsin  Madison ==
 
−  <span style="color:#FF0000">'''UNUSUAL TIME'''<span style="color:#009000">
 
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−  '''Airy process and the polymer end point distribution'''
 
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−  We give an explicit formula for the joint density of the max and argmax of the Airy process minus a parabola. The argmax has a universal distribution which governs the rescaled endpoint of directed polymers in 1+1 dimensions.
 
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−  == Thursday, October 20, Kay Kirkpatrick, University of Illinois at UrbanaChampaign ==
 
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−  '''BoseEinstein condensation and a phase transition for the nonlinear Schrodinger equation'''
 
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−  Near absolute zero, a gas of quantum particles can condense into an unusual state of matter, called BoseEinstein condensation (BEC), that behaves like a giant quantum particle. The rigorous connection has recently been made between the physics of the microscopic dynamics and the mathematics of the macroscopic model, the cubic nonlinear Schrodinger equation (NLS). I'll discuss work with Sourav Chatterjee about a phase transition for invariant measures of the discrete focusing NLS. Using techniques from probability theory, we show that the thermodynamics of the NLS are exactly solvable in dimensions three and higher. There are a number of consequences of this result, including a prediction for experimentalists to look for a new spatially localized phase of BEC.
 
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−  == Monday, October 31, 2:30pm, VV B341, Ankit Gupta, Ecole Polytechnique, Centre de Mathematiques Appliqees ==
 
−  <span style="color:#FF0000">'''UNUSUAL TIME AND PLACE'''<span style="color:#009000">
 
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−  '''Modeling adaptive dynamics for structured populations with functional traits'''
 
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−  We develop the framework of adaptive dynamics for populations that are structured by age and functional traits. The functional trait of an individual may express itself differently during the life of an individual according to her age and a random parameter that is chosen at birth to capture the environmental stochasticity. The population evolves through birth, death and selection mechanisms. At each birth, the new individual may be a clone of its parent or a mutant. Starting from an individual based model we use averaging techniques to take the large population and rare mutation limit under a wellchosen timescale separation. This gives us the Trait Substitution Sequence process that describes the adaptive dynamics in our setting. Assuming small mutation steps we also derive the Canonical Equation which expresses the evolution of advantageous traits as a functionvalued ordinary differential equation.
 
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−  This is joint work with J.A.J Metz (Leiden University) and V.C. Tran (University of Lille).
 
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−  == Friday, November 4, 2:30pm, VV B341, Michael Cranston, University of California, Irvine ==
 
−  <span style="color:#FF0000">'''UNUSUAL TIME AND PLACE'''<span style="color:#009000">
 
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−  '''Overlaps and Pathwise Localization in the Anderson Polymer Model'''
 
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−  We consider large time behavior of typical paths under the Anderson poly
 
−  mer measure. We establish that the polymer measure gives a macroscopic mass to a typical path and this mass grows to 1 in the limit as a particular parameter tends to infinity. This gives a rigourous approach to the polymer localization. The localization is measured by considering the overlap between two independent samples drawn from the polymer measure.
 
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−  == Thursday, November 10, David Anderson, University of Wisconsin  Madison ==
 
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−  '''Computational methods for stochastic models in biology'''
 
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−  I will focus on computational methods for continuous time Markov chains, which includes the large class of stochastically modeled biochemical reaction networks and population processes. I will show how different computational methods can be understood and analyzed by using different representations for the processes. Topics discussed will be a subset of: approximation techniques, variance reduction methods, parameter sensitivities.
 
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−  == Wednesday, December 7, 2:30pm, B313, Alexander Fish, University of Wisconsin  Madison ==
 
−  <span style="color:#FF0000">'''UNUSUAL TIME AND PLACE'''<span style="color:#009000">
 
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−  '''Fast Matched Filter algorithm and applications'''
 
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−  We will explain the mathematical model of a wireless communication.
 
−  One of the main problems is finding (time,frequency) shift of a signal in a noisy environment which is caused by time asynchronization of a sender with a receiver and by a nonzero speed of a sender w.r.t a receiver.
 
−  A classical solution (Matched Filter Algorithm) of a discrete analog of the problem uses a pseudorandom waveform S(t) of the length p and gives rise to the complexity p^2 log(p) operations (using fast Fourier transform).
 
−  We will explain how to use techniques from group representation theory to construct waveforms S(t) which enable us to introduce a fast matched filter algorithm, called the "flag algorithm", which solves (time,frequency) shift problem in O(p*log (p)) operations.
 
−  We will discuss applications to radars, GPS system, and Mobile Communication.
 
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−  This is a joint work with S. Gurevich (Mathematics, UW Madison), R. Hadani
 
−  (Mathematics, UT Austin), A. Sayeed (Electrical Engineering, UW Madison),
 
−  and O. Schwartz (Computer Science, UC Berkeley).
 
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−  == Spring 2012 ==
 
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−  Planned talks:
 
   
 == Thursday, January 26, Timo Seppalainen, University of Wisconsin  Madison ==   == Thursday, January 26, Timo Seppalainen, University of Wisconsin  Madison == 