Difference between revisions of "Probability Seminar"

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== Thursday, January 19, TBA  ==
== Thursday, January 26, [http://mathematics.stanford.edu/people/name/erik-bates/ Erik Bates], [http://mathematics.stanford.edu/ Stanford]  ==
Title: '''The endpoint distribution of directed polymers'''
Abstract: On the d-dimensional integer lattice, directed polymers are paths of a random walk in random environment, except that the environment updates at each time step.  The result is a statistical mechanical system, whose qualitative behavior is governed by a temperature parameter and the law of the environment.  Historically, the phase transitions have been best understood by whether or not the path’s endpoint localizes.  While the endpoint is no longer a Markov process as in a random walk, its quenched distribution is.  The key difficulty is that the space of measures is too large for one to expect convergence results.  By adapting methods recently used by Mukherjee and Varadhan, we develop a compactification theory to resolve the issue.  In this talk, we will discuss this intriguing abstraction, as well as new concrete theorems it allows us to prove for directed polymers.
This talk is based on joint work with Sourav Chatterjee.
== Thursday, 2/2/2017, TBA ==
== Thursday, February 9, TBA ==
== Thursday, 2/16/2017, TBA ==
== Thursday, February 23, [http://www.math.wisc.edu/~jeanluc/ Jean-Luc Thiffeault], [http://www.math.wisc.edu/ UW-Madison] ==
'''Title:'''  Heat Exchange and Exit Times
A heat exchanger can be modeled as a closed domain containing an incompressible fluid.  The fluid has some temperature distribution obeying the advection-diffusion equation, with zero temperature boundary conditions at the walls.  The goal is then to start from some initial positive heat distribution, and to flux it through the walls as fast as possible.  Even for a steady flow, this is a time-dependent problem, which can be hard to optimize.  Instead, we consider the mean exit time of Brownian particles starting from inside the domain.  A flow favorable to heat exchange should lower the exit time, and so we minimize some norm of the exit time over incompressible flows (drifts) with a given energy.  This is a simpler, time-independent optimization problem, which we then proceed to solve analytically in some limits, and numerically otherwise.
== Thursday, March 2, No Seminar this week ==
The talk by [http://people.maths.ox.ac.uk/woolley/ Thomas Woolley], [https://www.maths.ox.ac.uk/ Oxford] has been moved to April 6 (see below).
<!-- == Thursday, 3/9/2017, TBA ==-->
== Thursday, March 16, [http://www-users.math.umn.edu/~wkchen/ Wei-Kuo Chen], [http://math.umn.edu/ Minnesota] ==
Title: '''Energy landscape of mean-field spin glasses'''
The Sherrington-Kirkpatirck (SK) model is a mean-field spin glass introduced by theoretical physicists in order to explain the strange behavior of certain alloy, such as CuMn. Despite of its seemingly simple formulation, it was conjectured to possess a number of fruitful properties. This talk will be focused on the energy landscape of the SK model. First, we will present a formula for the maximal energy in Parisi’s formulation. Second, we will give a description of the energy landscape by showing that near any given energy level between zero and maximal energy, there exist exponentially many equidistant spin configurations. Based on joint works with Auffinger, Handschy, and Lerman.
== Thursday, March 23, Spring Break ==
== <span style="color:red"> Wednesday, March 29, 1:00pm, </span> [http://homepages.cae.wisc.edu/~loh/index.html Po-Ling Loh], [http://www.engr.wisc.edu/department/electrical-computer-engineering/ UW-Madison] ==
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<b><span style="color:red"> Please note the unusual day and time </span>
Title: '''Confidence sets for the source of a diffusion in regular trees'''
Abstract: We study the problem of identifying the source of a diffusion spreading over a regular tree. When the degree of each node is at least three, we show that it is possible to construct confidence sets for the diffusion source with size independent of the number of infected nodes. Our estimators are motivated by analogous results in the literature concerning identification of the root node in preferential attachment and uniform attachment trees. At the core of our proofs is a probabilistic analysis of Polya urns corresponding to the number of uninfected neighbors in specific subtrees of the infection tree. We also describe extensions of our results to diffusions spreading over Galton-Watson trees. This is joint work with Justin Khim (UPenn).
== Thursday, April 6, [http://people.maths.ox.ac.uk/woolley/ Thomas Woolley], [https://www.maths.ox.ac.uk/ Oxford] ==
Title: '''Power spectra of stochastic reaction-diffusion equations on
stochastically growing domains'''
Abstract: Being able to create and sustain robust, spatial-temporal
inhomogeneity is an important concept in developmental biology.
Generally, the mathematical treatments of these biological systems have
used continuum hypotheses of the reacting populations, which ignores any
sources of intrinsic stochastic effects. We address this concern by
developing analytical Fourier methods which allow us to probe the
probabilistic framework. Further, a novel description of domain growth
is produced, which is able to rigorously link the mean-field and stochastic
descriptions. Finally, through combining all of these ideas, it is shown
that the description of diffusion on a growing domain is non-unique and,
due to these distinct descriptions, diffusion is able to support
patterning without the addition of further kinetics.
== Thursday, 4/13/2017, [https://www.math.toronto.edu/cms/dauvergne-duncan/ Duncan Dauvergne], [https://www.math.toronto.edu/cms/ Toronto] ==
Title: '''The local limit of random sorting networks'''
Abstract: A sorting network is a shortest path from the identity to the reverse permutation in the Cayley graph of <math>S_n</math> generated by adjacent transpositions. Remarkable conjectures about the global scaling limit of a uniformly random sorting network have been made based on strong empirical evidence. For example, trajectories of the individual elements 1, 2, … n appear to converge to sine curves.
One approach to proving these conjectures is to first show the existence of a local limit of random sorting networks, and then use this to piece together global information. In this talk, I will discuss this local limit, as well as progress that has been made towards understanding the global limit as a consequence of local properties.
== Thursday, April 20, [https://www.math.wisc.edu/~jskim/ Jinsu Kim], [https://www.math.wisc.edu/ UW-Madison] ==
Title : '''Sufficient Conditions for Ergodicity of Stochastic Reaction Networks and Mixing Times'''
Reaction networks are graphical configurations that can be used to describe biological interaction networks. If the abundances of the constituent species of the system are low, we can model the dynamics of species counts in a jump by jump fashion as a continuous time Markov chain. In this talk, we will mainly focus on which conditions of the graph imply ergodicity (existence of a stationary distribution) for the associated continuous time Markov chain. I will also present results related to their mixing times, which give the time required for the distribution of the continuous time Markov chain to get close to the stationary distribution.
<!-- == Thursday, 4/27/2017, TBA ==-->
== <span style="color:red"> Wednesday, May 3, 1:00pm, </span> [http://www.math.wisc.edu/~qinli/ Qin Li], [http://www.math.wisc.edu/ UW-Madison] ==
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<b><span style="color:red"> Please note the unusual day and time </span>
Title: '''Stability of stationary inverse transport equation in diffusion scaling'''
Abstract: We consider the inverse problem of reconstructing the optical parameters for stationary radiative transfer equation (RTE) from velocity-averaged measurement. The RTE describes the dynamics of the distribution for photon particles. It often contains multiple scales characterized by the magnitude of a dimensionless parameter -- the Knudsen number (Kn). In the diffusive scaling (Kn << 1), the stationary RTE is well approximated by an elliptic equation in the forward setting. However, the inverse problem for the elliptic equation is acknowledged to be severely ill-posed as compared to the well-posedness of inverse transport equation, which raises the question of how uniqueness being lost as Kn \to 0. We tackle this problem by examining the stability of inverse problem with varying Kn. We show that, the discrepancy in two measurements is amplified in the reconstructed parameters at the order of Kn^p (p = 1 or 2), and as a result leads to ill-posedness in the zero limit of Kn. Our results apply to both continuous and discrete settings. Some numerical tests are performed in the end to validate these theoretical findings.
<!--== Thursday, 5/4/2017, TBA ==-->
== Thursday, May 11,<span style="color:red"> 11:00 am </span> [http://cims.nyu.edu/~nica/ Mihai Nica], [http://cims.nyu.edu/ Courant NYU] ==
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<b><span style="color:red"> Please note the unusual time </span>
Title: '''Intermediate disorder limits for multi-layer random polymers'''
The intermediate disorder regime is a scaling limit for disordered systems where the inverse temperature is critically scaled to zero as the size of the system grows to infinity. For a random polymer given by a single random walk, Alberts, Khanin and Quastel proved that under intermediate disorder scaling the polymer partition function converges to the solution to the stochastic heat equation. In this talk, I consider polymers made up of multiple non-intersecting walkers and consider the same type of limit. The limiting object now is the multi-layer extension of the stochastic heat equation introduced by O'Connell and Warren. This result proves a conjecture about the KPZ line ensemble, which I will also introduce. Part of this talk is based on joint work with Ivan Corwin.
== Thursday, September 8, Daniele Cappelletti, [http://www.math.wisc.edu UW-Madison] ==
Title: '''Reaction networks: comparison between deterministic and stochastic models'''
Abstract: Mathematical models for chemical reaction networks are widely used in biochemistry, as well as in other fields. The original aim of the models is to predict the dynamics of a collection of reactants that undergo chemical transformations. There exist two standard modeling regimes: a deterministic and a stochastic one. These regimes are chosen case by case in accordance to what is believed to be more appropriate. It is natural to wonder whether the dynamics of the two different models are linked, and whether properties of one model can shed light on the behavior of the other one. Some connections between the two modelling regimes have been known for forty years, and new ones have been pointed out recently. However, many open questions remain, and the issue is still largely unexplored.
== <span style="color:red"> Friday</span>, September 16, <span style="color:red"> 11 am </span> [http://www.baruch.cuny.edu/math/elenak/ Elena Kosygina], [http://www.baruch.cuny.edu/ Baruch College] and the [http://www.gc.cuny.edu/Page-Elements/Academics-Research-Centers-Initiatives/Doctoral-Programs/Mathematics CUNY Graduate Center] ==
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<b><span style="color:red"> Please note the unusual day and time </span></b>
The talk will be in Van Vleck 910 as usual.
Title: '''Homogenization of viscous Hamilton-Jacobi equations: a remark and an application.'''
Abstract: It has been pointed out in the seminal work of P.-L. Lions, G. Papanicolaou, and S.R.S. Varadhan that for the first order
Hamilton-Jacobi (HJ) equation, homogenization starting with affine initial data should imply homogenization for general uniformly
continuous initial data. The argument utilized the properties of the HJ semi-group, in particular, the finite speed of propagation. The
last property is lost for viscous HJ equations. We remark that the above mentioned implication holds under natural conditions for both
viscous and non-viscous Hamilton-Jacobi equations. As an application of our result, we show homogenization in a stationary ergodic  setting for a special class of viscous HJ equations with a non-convex Hamiltonian in one space dimension.
This is a joint work with Andrea Davini, Sapienza Università di Roma.
== Thursday, September 22,  [http://www.math.wisc.edu/~pmwood/ Philip Matchett Wood], [https://www.math.wisc.edu/ UW-Madison] ==
Title:  '''Low-degree factors of random polynomials'''
Abstract: We study the probability that a monic polynomial with integer coefficients has a low-degree factor over the integers.
It is known that certain models are very likely to produce random polynomials that are irreducible, and our project
can be viewed as part of a general program of testing whether this is a universal behavior exhibited by many random
polynomial models. Interestingly, though the question comes from algebra and number theory, we primarily use tools
from combinatorics, including additive combinatorics, and probability theory. We prove for a variety of models that it
is very unlikely for a random polynomial with integer coefficients to have a low-degree factor—suggesting that this is, in
fact, a universal behavior. For example, we show that the characteristic polynomial of random matrix with independent
+1 or −1 entries is very unlikely to have a factor of degree up to <math>n^{1/2-\epsilon}</math>.  Joint work with Sean O’Rourke.  The talk will also discuss joint work with UW-Madison
undergraduates Christian Borst, Evan Boyd, Claire Brekken, and Samantha Solberg, who were supported
by NSF grant DMS-1301690 and co-supervised by Melanie Matchett Wood.
== Thursday, September 29, [http://www.artsci.uc.edu/departments/math/fac_staff.html?eid=najnudjh&thecomp=uceprof Joseph Najnudel],  [http://www.artsci.uc.edu/departments/math.html University of Cincinnati]==
Title:  '''On the maximum of the characteristic polynomial of the Circular Beta Ensemble'''
In this talk, we present our result on the extremal values of (the logarithm of) the characteristic polynomial of a random unitary matrix whose spectrum is distributed according to the Circular Beta Ensemble. Using different techniques, it gives an improvement and a generalization of the previous recent results by Arguin, Belius, Bourgade on the one hand, and Paquette, Zeitouni on the other hand. They recently treated the CUE case, which corresponds to beta equal to 2.
== Thursday, October 6, No Seminar ==
== Thursday, October 13, No Seminar due to [http://sites.math.northwestern.edu/mwp/ Midwest Probability Colloquium] ==
For details, see [http://sites.math.northwestern.edu/mwp/ Midwest Probability Colloquium].
== Thursday, October 20, [http://www.math.harvard.edu/people/index.html Amol Aggarwal], [http://www.math.harvard.edu/ Harvard] ==
Title: Current Fluctuations of the Stationary ASEP and Six-Vertex Model
Abstract: We consider the following three models from statistical mechanics: the asymmetric simple exclusion process, the stochastic six-vertex model, and the ferroelectric symmetric six-vertex model. It had been predicted by the physics communities for some time that the limiting behavior for these models, run under certain classes of translation-invariant (stationary) boundary data, are governed by the large-time statistics of the stationary Kardar-Parisi-Zhang (KPZ) equation. The purpose of this talk is to explain these predictions in more detail and survey some of our recent work that verifies them.
== Thursday, October 27, [http://www.math.wisc.edu/~hung/ Hung Tran], [http://www.math.wisc.edu/ UW-Madison] ==
Title: '''Homogenization of non-convex Hamilton-Jacobi equations'''
Abstract: I will describe why it is hard to do homogenization for non-convex Hamilton-Jacobi equations and explain some recent results in this direction. I will also make a very brief connection to first passage percolation and address some challenging questions which appear in both directions. This is based on joint work with Qian and Yu.
== Thursday, November 3, Alejandro deAcosta, [http://math.case.edu/ Case-Western Reserve] ==
Title:  '''Large deviations for irreducible Markov chains with general state space'''
We study the large deviation principle for the empirical measure of general irreducible Markov chains in the tao topology for a broad class of initial distributions.  The roles of several rate functions, including the rate function based on the convergence parameter of the transform kernel and the Donsker-Varadhan rate function, are clarified.
== Thursday, November 10, [https://sites.google.com/a/wisc.edu/louisfan/home Louis Fan], [https://www.math.wisc.edu/ UW-Madison] ==
Title: '''Particle representations for (stochastic) reaction-diffusion equations'''
Reaction diffusion equations (RDE) is a popular tool to model complex spatial-temporal patterns including Turing patterns, traveling waves and periodic switching.
These models, however, ignore the stochasticity and individuality of many complex systems in nature. Recognizing this drawback, scientists are developing individual-based models for model selection purposes. The latter models are sometimes studied under the framework of interacting particle systems (IPS) by mathematicians, who prove scaling limit theorems to connect various IPS with RDE across scales.
In this talk, I will present some new limiting objects including SPDE on metric graphs and coupled SPDE. These SPDE reduce to RDE when the noise parameter tends to zero, therefore interpolates between IPS and RDE and identifies the source of stochasticity. Scaling limit theorems and novel duality formulas are obtained for these SPDE, which not only connect phenomena across scales, but also offer insights about the genealogies and time-asymptotic properties of certain population dynamics.  In particular, I will present rigorous results about the lineage dynamics for of a biased voter model introduced by Hallatschek and Nelson (2007).
== Thursday, November 24, No Seminar due to Thanksgiving ==
== Thursday, December 1, [http://math.columbia.edu/~hshen/ Hao Shen], [http://math.columbia.edu/~hshen/ Columbia] ==
Title:  '''On scaling limits of Open ASEP and Glauber dynamics of ferromagnetic models'''
We discuss two scaling limit results for discrete models converging to stochastic PDEs. The first is the asymmetric simple exclusion process in contact with sources and sinks at boundaries, called Open ASEP.  We prove that under weakly asymmetric scaling the height function converges to the KPZ equation with Neumann boundary conditions. The second is the Glauber dynamics of the Blume-Capel model (a generalization of Ising model), in two dimensions with Kac potential. We prove that the averaged spin field converges to the stochastic quantization equations. A common challenge in the proofs is how to identify the limiting process as the solution to the SPDE, and we will discuss how to overcome the difficulties in the two cases.(Based on joint works with Ivan Corwin and Hendrik Weber.)
== '''Colloquium''' Friday, December 2, [http://math.columbia.edu/~hshen/ Hao Shen], [http://math.columbia.edu/~hshen/ Columbia] ==
4pm, Van Vleck 9th floor
Title: '''Singular Stochastic Partial Differential Equations - How do they arise and what do they mean?'''
Abstract: Systems with random fluctuations are ubiquitous in the real world. Stochastic PDEs are default models for these random systems, just as PDEs are default models for deterministic systems. However, a large class of such stochastic PDEs were poorly understood until very recently: the presence of very singular random forcing as well as nonlinearities render it challenging to interpret what one even means by a ``solution". The recent breakthroughs by M. Hairer, M. Gubinelli and other researchers including the speaker not only established solution theories for these singular SPDEs, but also led to an explosion of new questions. These include scaling limits of random microscopic models, development of numerical schemes, ergodicity of random dynamical systems and a new approach to quantum field theory. In this talk we will discuss the main ideas of the recent solution theories of singular SPDEs, and how these SPDEs arise as limits of various important physical models.
== Thursday, January 28, [http://faculty.virginia.edu/petrov/ Leonid Petrov], [http://www.math.virginia.edu/ University of Virginia] ==
Title: '''The quantum integrable particle system on the line'''
I will discuss the higher spin six vertex model - an interacting  particle
system on the discrete 1d line in the Kardar--Parisi--Zhang universality
class. Observables of this system admit explicit contour integral expressions
which degenerate  to many known formulas of such type for other integrable
systems on the line in the KPZ class, including stochastic six vertex model,
ASEP, various <math>q</math>-TASEPs, and associated zero range processes. The structure
of the higher spin six vertex model (leading to contour integral formulas for
observables) is based on Cauchy summation identities for certain symmetric
rational functions, which in turn can be traced back to the sl2 Yang--Baxter
equation. This framework allows to also include space and spin inhomogeneities
into the picture, which leads to new particle systems with unusual phase
== Thursday, February 4, [http://homepages.math.uic.edu/~nenciu/Site/Contact.html Inina Nenciu], [http://www.math.uic.edu/ UIC], Joint Probability and Analysis Seminar ==
Title: '''On some concrete criteria for quantum and stochastic confinement'''
Abstract: In this talk we will present several recent results on criteria ensuring the confinement of a quantum or a stochastic particle to a bounded domain in <math>\mathbb{R}^n</math>. These criteria are given in terms of explicit growth and/or decay rates for the diffusion matrix and the drift potential close to the boundary of the domain. As an application of the general method, we will discuss several cases, including some where the background Riemannian manifold (induced by the diffusion matrix) is geodesically incomplete. These results are part of an ongoing joint project with G. Nenciu (IMAR, Bucharest, Romania).
== <span style="color:green">Friday, February 5</span>, [http://www.math.ku.dk/~d.cappelletti/index.html Daniele Cappelletti], [http://www.math.ku.dk/ Copenhagen University], speaks in the [http://www.math.wisc.edu/wiki/index.php/Applied/ACMS Applied Math Seminar], <span style="color:green">2:25pm in Room 901 </span>==
'''Note:''' Daniele Cappelletti is speaking in the [http://www.math.wisc.edu/wiki/index.php/Applied/ACMS Applied Math Seminar], but his research on stochastic reaction networks uses probability theory and is related to work of our own [http://www.math.wisc.edu/~anderson/ David Anderson].
Title: '''Deterministic and Stochastic Reaction Networks'''
Abstract:  Mathematical models of biochemical reaction networks are of great interest for the analysis of experimental data and theoretical biochemistry. Moreover, such models can be applied in a broader framework than that provided by biology. The classical deterministic model of a reaction network is a system of ordinary differential equations, and the standard stochastic model is a continuous-time Markov chain. A relationship between the dynamics of the two models can be found for compact time intervals, while the asymptotic behaviours of the two models may differ greatly. I will give an overview of these problems and show some recent development.
== Thursday, February 25, [http://www.princeton.edu/~rvan/ Ramon van Handel], [http://orfe.princeton.edu/ ORFE] and [http://www.pacm.princeton.edu/ PACM, Princeton] ==
Title: '''The norm of structured random matrices'''
Abstract: Understanding the spectral norm of random matrices is a problem
of basic interest in several areas of pure and applied mathematics. While
the spectral norm of classical random matrix models is well understood,
existing methods almost always fail to be sharp in the presence of
nontrivial structure. In this talk, I will discuss new bounds on the norm
of random matrices with independent entries that are sharp under mild
conditions. These bounds shed significant light on the nature of the
problem, and make it possible to easily address otherwise nontrivial
phenomena such as the phase transition of the spectral edge of random band
matrices. I will also discuss some conjectures whose resolution would
complete our understanding of the underlying probabilistic mechanisms.
== Thursday,  March 3, [http://www.math.wisc.edu/~janjigia/ Chris Janjigian], [http://www.math.wisc.edu/ UW-Madison] ==
Title: '''Large deviations for certain inhomogeneous corner growth models'''
The corner growth model is a classical model of growth in the plane and is connected to other familiar models such as directed last passage percolation and the TASEP through various geometric maps. In the case that the waiting times are i.i.d. with exponential or geometric marginals, the model is well understood: the shape function can be computed exactly, the fluctuations around the shape function are known to be given by the Tracy-Widom GUE distribution, and large deviation principles corresponding to this limit have been derived.
This talk considers the large deviation properties of a generalization of the classical model in which the rates of the exponential are drawn randomly in an appropriate way. We will discuss some exact computations of rate functions in the quenched and annealed versions of the model, along with some interesting properties of large deviations in this model. (Based on joint work with Elnur Emrah.)
== Thursday,  March 10, [http://www.math.wisc.edu/~jyin/jun-yin.html Jun Yin], [http://www.math.wisc.edu/ UW-Madison] ==
Title: '''Delocalization and Universality of band matrices.'''
Abstract: in this talk we introduce our new work on band matrices, whose eigenvectors and eigenvalues are  widely believed  to have the same asymptotic behaviors as those of Wigner matrices.
We proved that this conjecture is true as long as the bandwidth is wide enough.
== Thursday,  March 17, [http://www.math.wisc.edu/~roch/ Sebastien Roch], [http://www.math.wisc.edu/ UW-Madison] ==
Title: '''Recovering the Treelike Trend of Evolution Despite Extensive Lateral Genetic Transfer'''
Reconstructing the tree of life from molecular sequences is a fundamental problem in computational
biology. Modern data sets often contain large numbers of genes. That can complicate the reconstruction because different genes often undergo different evolutionary histories. This is the case in particular in the presence of lateral genetic transfer (LGT), where a gene is inherited from a distant species rather than an immediate ancestor. Such an event produces a gene tree which is distinct from (but related to) the species phylogeny. In this talk I will sketch recent results showing that, under a natural stochastic model of LGT, the species phylogeny can be reconstructed from gene trees despite surprisingly high rates of LGT.
== Thursday,  March 24, No Seminar, Spring Break ==
== Thursday,  March 31, [http://www.ssc.wisc.edu/~whs/ Bill Sandholm], [http://www.econ.wisc.edu/ Economics, UW-Madison] ==
Title: '''A Sample Path Large Deviation Principle for a Class of Population Processes'''
Abstract:  We establish a sample path large deviation principle for sequences of Markov chains arising in game theory and other applications. As the state spaces of these Markov chains are discrete grids in the simplex, our analysis must account for the fact that the processes run on a set with a boundary. A key step in the analysis establishes joint continuity properties of the state-dependent Cramer transform L(·,·), the running cost appearing in the large deviation principle rate function.
[http://www.ssc.wisc.edu/~whs/research/ldp.pdf paper preprint]
== Thursday,  April 7, No Seminar ==
== Thursday,  April 14, [https://www.math.wisc.edu/~jessica/ Jessica Lin], [https://www.math.wisc.edu/~jessica/ UW-Madison], Joint with [https://www.math.wisc.edu/wiki/index.php/PDE_Geometric_Analysis_seminar PDE Geometric Analysis seminar] ==
Title: '''Optimal Quantitative Error Estimates in Stochastic Homogenization for Elliptic Equations in Nondivergence Form'''
Abstract: I will present optimal quantitative error estimates in the
stochastic homogenization for uniformly elliptic equations in
nondivergence form. From the point of view of probability theory,
stochastic homogenization is equivalent to identifying a quenched
invariance principle for random walks in a balanced random
environment. Under strong independence assumptions on the environment,
the main argument relies on establishing an exponential version of the
Efron-Stein inequality. As an artifact of the optimal error estimates,
we obtain a regularity theory down to microscopic scale, which implies
estimates on the local integrability of the invariant measure
associated to the process. This talk is based on joint work with Scott
== Thursday,  April 21, [http://www.cims.nyu.edu/~bourgade/ Paul Bourgade], [https://www.cims.nyu.edu/ Courant Institute, NYU] ==
Title: '''Freezing and extremes of random unitary matrices'''
Abstract: A conjecture of Fyodorov, Hiary & Keating states that the maxima of the characteristic polynomial of random unitary matrices behave like the maxima of a specific class of Gaussian fields, the log-correlated Gaussian fields. We will outline the proof of the conjecture for the leading order of the maximum, and a freezing of the free energy related to the matrix model. This talk is based on a joint work with Louis-Pierre Arguin and David Belius.
== Thursday,  April 28, [http://www.ime.unicamp.br/~nancy/ Nancy Garcia], [http://www.ime.unicamp.br/conteudo/departamento-estatistica Statistics], [http://www.ime.unicamp.br/ IMECC], [http://www.unicamp.br/unicamp/ UNICAMP, Brazil] ==
Title: '''Rumor processes on <math>\mathbb{N}</math> and discrete renewal processe'''
Abstract: We study two rumor processes on the positive integers, the dynamics of which are related to an SI epidemic model with long range transmission. Start with one spreader at site <math>0</math> and ignorants situated at some other sites of <math>\mathbb{N}</math>. The spreaders transmit the information within a random distance on their right.  Depending on the initial distribution of the ignorants, we obtain  probability of survival, information on the distribution of the range of the rumor and limit theorems for the proportion of spreaders. The key step of our approach is to relate this model to the house-of-cards.
== Thursday,  May 5, [http://math.arizona.edu/~dianeholcomb/ Diane Holcomb], [http://math.arizona.edu/ University of Arizona]  ==
Title: '''Local limits of Dyson's Brownian Motion at multiple times'''
Abstract: Dyson's Brownian Motion may be thought of as a generalization of  Brownian Motion to the matrix setting. We  can study the eigenvalues of a Dyson's Brownian motion at multiple times. The resulting object has different "color" points corresponding to the eigenvalues at different times. Similar to a single time, the correlation functions of the process may be described in terms of determinantal formulas. We study the local behavior of the eigenvalues as we take the dimension of the associated matrix to infinity. The resulting limiting process in the bulk is again determinantal and is described with an "extended sine kernel." This work aims to give an alternate description of the limiting process in terms of the counting function. In this seminar I will go over the the description and methods for finding such a limit. This is work in progress and is joint with Elliot Paquette (Weizmann Institute).
== ==
== ==
[[Past Seminars]]
[[Past Seminars]]

Revision as of 15:23, 3 September 2017

Fall 2017

Thursdays in 901 Van Vleck Hall at 2:25 PM, unless otherwise noted. We usually end for questions at 3:15 PM.

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.

Thursday, September 7, 2017

Thursday, September 14, 2017

Thursday, September 21, 2017

Thursday, September 28, 2017

Thursday, October 5, 2017

Thursday, October 12, 2017

Thursday, October 19, 2017

Thursday, October 26, 2017

Thursday, November 2, 2017

Thursday, November 9, 2017

Thursday, November 16, 2017

Thursday, November 23, 2017

Thursday, November 30, 2017

Thursday, December 7, 2017

Thursday, December 14, 2017

Past Seminars