Department of


Seminar Calendar
for Probability Seminar events the year of Wednesday, November 25, 2020.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
     October 2020          November 2020          December 2020    
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Thursday, January 30, 2020

2:00 pm in 347 Altgeld Hall,Thursday, January 30, 2020

The Semicircle Law for Wigner Matrices

Kesav Krishnan (UIUC Math)

Abstract: I will introduce Wigner Matrices and their universal properties. I will then state the semi-circle law and sketch out three district proofs, in analogy to the proof of the usual central limit theorem. Talk 1 will sketch out the proof via the Stieltjes transform and via the energy entropy balance.

Thursday, February 6, 2020

2:00 pm in 347 Altgeld Hall,Thursday, February 6, 2020

The Semicircle Law for Wigner Matrices Part 2

Kesav Krishnan (UIUC Math)

Abstract: I will introduce Wigner Matrices and their universal properties. I will then state the semi-circle law and sketch out three district proofs, in analogy to the proof of the usual central limit theorem. talk two will discuss the proof based on the method of moments and its relation to enumerative combinatorics.

Thursday, February 13, 2020

2:00 pm in 347 Altgeld Hall,Thursday, February 13, 2020

Distribution of eigenvalues of random matrices (part I)

Peixue Wu (UIUC Math)

Abstract: Last time we proved a famous semicircular law for the limit distribution of the empirical measure of the eigenvalues of Wigner's matrix (i.i.d. under the symmetry restriction). When we go over the proof in detail, we find two essential ingredients to the proof: 1. Stochastic independence of the entries. 2. Most matrix entries are centered and have the same variance. Using the similar idea (methods of moments) we will show that semicircular law holds for a much larger class of random matrices. We will also talk about the joint distribution for the eigenvalues of the Gaussian Orthogonal (Unitary) Ensembles (GOE or GUE).

Thursday, February 20, 2020

2:00 pm in 347 Altgeld Hall,Thursday, February 20, 2020

Distribution of eigenvalues of random matrices (part II)

Peixue Wu (UIUC Math)

Abstract: Last time we proved the classical Wigner's semicircular law for Wigner matrix. This time I will state a dynamical version of the semicircular law, which implies the classical Wigner's semicircular law. Our main tool will be stochastic analysis.

4:00 pm in 243 Altgeld Hall,Thursday, February 20, 2020

Variable-order time-fractional partial differential equations: modeling and analysis

Hong Wang (University of South Carolina)

Abstract: Fractional partial differential equations (FPDEs) provide more accurate descriptions of anomalously diffusive transport of solute in heterogeneous porous media than integer-order PDEs do, because they generate solutions with power law (instead of exponentially) decaying tails that were observed in field tests. However, solutions to time-fractional PDEs (tFPDEs) have nonphysical singularity at the initial time t=0, which does not seem physically relevant to anomalously diffusive transport they model and makes many error estimates to their numerical approximations in the literature that were proved under the full regularity assumption of the true solutions in appropriate. The reason lies in the incompatibility between the nonlocality of the power law decaying tail of the solutions and the locality of the initial condition. But there is no consensus on how to correct the nonphysical behavior of tFPDEs. We argue that the order of a physically correct tFPDE model should vary smoothly near the initial time to account for the impact of the locality of the initial condition. Moreover, variable-order tFPDEs themselves also occur in a variety of applications. However, rigorous analysis on variable-order tFPDEs is meager. We outline the proof of the wellposedness and smoothing properties of tFPDEs. More precisely, we prove that their solutions have the similar regularities to their integer-order analogues if the order has an integer limit at the initial time or have the same singularity near the initial time as their constant-order analogues otherwise.

Tuesday, February 25, 2020

2:00 pm in 345 Altgeld Hall,Tuesday, February 25, 2020

Multiple SLE from a loop measure perspective

Vivian Healey (U Chicago Math)

Abstract: I will discuss the role of Brownian loop measure in the study of Schramm-Loewner evolution. This powerful perspective allows us to apply intuition from discrete models (in particular, the λ-SAW model) to the study of SLE while simultaneously reducing many SLE computations to problems of stochastic calculus. I will discuss recent work on multiple radial SLE that employs this method, including the construction of global multiple radial SLE and its links to locally independent SLE and Dyson Brownian motion. (Joint work with Gregory F. Lawler.)

Thursday, February 27, 2020

2:00 pm in 347 Altgeld Hall,Thursday, February 27, 2020

Tracy Widom Distribution and Spherical Spin Glass (Part I)

Qiang Wu (UIUC Math)

Abstract: We studied the global behavior of eigenvalues of random matrices in previous talks. This time we are going to zoom into the bulk to study some local behavior of eigenvalues. In particular, the edge scaling limit of largest eigenvalue is given by the Tracy-widom (TW) distribution, which as a universal object also appears in some other areas, like growth process, spin system and many other interacting particle systems. Taking GUE as our example, we will try to derive the TW distribution represented as a Fredholm determinant with Airy Kernel. Time permits, we will briefly go through the integral representation of TW, and some universality results even extended to the underlying integrable system for general beta ensembles.

Tuesday, March 3, 2020

2:00 pm in 345 Altgeld Hall,Tuesday, March 3, 2020

Heat kernel of fractional Laplacian with Hardy drift via desingularizing weights

Damir Kinzebulatov   [email] (Universite Laval)

Abstract: We establish sharp two-sided bounds on the heat kernel of the fractional Laplacian, perturbed by a drift having critical-order singularity, using the method of desingularizing weights. This is joint work with Yu.A.Semenov (Toronto) and K.Szczypkowski (Wroclaw).

Thursday, March 12, 2020

2:00 pm in 347 Altgeld Hall,Thursday, March 12, 2020

Tracy-Widom distribution and spherical spin glass (Part II)

Qiang Wu (UIUC Math)

Abstract: I will talk about the connection between spherical spin glass(SSK) and random matrices, in particular, the fluctuation of free energy in SSK on low temperatures regime is given by GOE Tracy-Widom distribution.

Tuesday, April 7, 2020

2:00 pm in 345 Altgeld Hall,Tuesday, April 7, 2020

**Rescheduled due to COVID-19 campus-shutdown**

Michael Perlmutter   [email] (Michigan State University)

Abstract: To Be Announced

Tuesday, September 22, 2020

2:00 pm in Zoom Meeting (email for info),Tuesday, September 22, 2020

Empirical measures, geodesic lengths, and a variational formula in first-passage percolation 

Erik Bates (University of Wisconsin-Madison)

Abstract: We consider the standard first-passage percolation model on Z^d, in which each edge is assigned an i.i.d. nonnegative weight, and the passage time between any two points is the smallest total weight of a nearest-neighbor path between them.  Our primary interest is in the empirical measures of edge-weights observed along geodesics from 0 to ne_1.  For various dense families of edge-weight distributions, we prove that these measures converge weakly to a deterministic limit as n tends to infinity.  The key tool is a new variational formula for the time constant.  In this talk, I will derive this formula and discuss its implications for the convergence of both empirical measures and lengths of geodesics.

Wednesday, September 23, 2020

3:00 pm in Zoom Meeting (email for info),Wednesday, September 23, 2020

Empirical distributions (and another variational formula) in first-passage percolation on trees

Erik Bates (University of Wisconsin-Madison)

Abstract: This talk will address the same topic as Tuesday's probability seminar, but in a simplified setting: we replace the d-dimensional integer lattice with the infinite d-ary tree. In this "mean-field" case, we are able to completely resolve the question of convergence for empirical distributions along FPP geodesics. The proof, which will be presented in full, leads to two interesting threads of discussion. The first concerns the difference between lattice percolation and tree percolation; here I invite discussion of an open problem. The second thread is a corollary of our methods, namely a fact about convergence of measures which I have not seen before. The talk should be accessible to all, and I will not assume that any part of Tuesday's seminar is remembered.

Tuesday, October 6, 2020

2:00 pm in Zoom Meeting (email for info),Tuesday, October 6, 2020

Degenerate diffusions and optimal convex bodies

Yair Shenfeld (MIT)

Abstract: It has been known since antiquity that the ball is the unique shape which minimizes surface area among all shapes of equal volume. Remarkably, if in addition to the volume, the average width of the shape is fixed, the optimal solutions become non-unique and non-smooth. This is just a very special instance of optimization problems arising in convex geometry whose optimal solutions are bizarre and still conjectural. In this talk, I will explain how the study of these optimal shapes is intertwined with the spectral structure of a certain diffusion operator on the sphere, and how we (with Ramon van Handel) solved many of these problems. No prior knowledge of convex geometry is assumed.

Tuesday, October 13, 2020

2:00 pm in Zoom Meeting (email for info),Tuesday, October 13, 2020

Stochastically modeled reaction network

Jinsu Kim (University of California Irvine)

Abstract: A reaction network is a graphical configuration that describes an interaction between species (molecules). If the abundances of the network system are small, then the randomness inherent in the molecular interactions is important to the system dynamics, and the abundances are modeled stochastically as a jump by jump fashion continuous-time Markov chain. One of the challenging issues facing researchers who study biological systems is the often extraordinarily complicated structure of their interaction networks. Thus, how to characterize network structures that induce characteristic behaviors of the system dynamics is one of the major open questions in this literature. In this talk, I will provide a class of reaction networks whose associated stochastic process is stable. I will also provide how this stability can be used in system biology and control theory.

Tuesday, October 20, 2020

2:00 pm in Zoom Meeting (email for info),Tuesday, October 20, 2020

Multiple points of Gaussian random fields

Cheuk Yin Lee (École polytechnique fédérale de Lausanne)

Abstract: This talk is concerned with multiple points (or self-intersections) of multivariate Gaussian random fields. Typically, the existence of multiple points depends on the dimensions of the domain and state space, and the existence problem is more difficult to solve when a random field is in its critical dimension. Under the framework of Dalang, Mueller and Xiao (2017), we prove that for a class of Gaussian random fields, multiple points do not exist in critical dimension. Our approach is based on a covering argument of Talagrand (1998). The result can be applied to fractional Brownian sheets and systems of linear SPDEs such as stochastic heat equations and wave equations driven by space-time white noise or colored noise. Joint work with Robert Dalang, Carl Mueller and Yimin Xiao.

Monday, October 26, 2020

2:00 pm in Zoom Meeting (email for info),Monday, October 26, 2020

An introduction to random walks in random environments

Jonathon Peterson (Purdue University)

Abstract: I will give an introduction to the model of random walks in random environments, paying particular attention to the one-dimensional case. I will give sketches of proofs for some of the basic results, including criteria for recurrence/transience, limiting velocity, and a central limit theorem.

Tuesday, October 27, 2020

2:00 pm in Zoom Meeting (email for info),Tuesday, October 27, 2020

Quantitative quenched CLTs for one-dimensional random walks in random environments

Jonathon Peterson (Purdue University)

Abstract: The Berry-Esseen estimates give quantitative error estimates on the CLT for sums of i.i.d. random variables, and the polynomial decay rate for the error depends on moment bounds of the i.i.d. random variables with the optimal $1/\sqrt{n}$ rate of convergence obtained under a third moment assumption. In this talk we will prove quantitative error bounds for quenched CLTs of both the position and hitting times of one dimensional random walks in random environments (RWRE). For the quantitative CLTs for the hitting times we prove that our rates our optimal. This talk is based on joint works with Sungwon Ahn.

Tuesday, November 3, 2020

2:00 pm in Zoom Meeting (email for info),Tuesday, November 3, 2020

The Scattering Transform: a Wavelet-based Neural Network for Statistical Feature Extraction

Michael Perlmutter (UCLA)

Abstract: The scattering transform is a mathematical framework for understanding convolutional neural networks (CNNs) introduced by S. Mallat. Similar to more traditional CNNs, the scattering transform is a deep, feed-forward network featuring an alternating cascade of convolutions and nonlinearities. However, it differs by using predesigned, wavelet filters rather than filters that are learned from training data. This leads to a network that provably has desirable invariance and stability properties. In addition to preforming well on standard machine learning tasks such as image recognition. The scattering transform can also be used to extract statistical information about stochastic processes with stationary increments. The expected scattering moments are a novel form of nonparametric statistics which can be used to distinguish random textures with identical power spectrums. I will present Mallat's original construction as well as several novel variations designed for structured data such as graphs. In particular, I will discuss the use of one of the more recent construction to distinguish between different types of randomly generated graphs.

Tuesday, November 10, 2020

2:00 pm in Zoom Meeting (email for info),Tuesday, November 10, 2020

Entropy dissipation for degenerate diffusion process

Qi Feng (USC)

Abstract: A drift-diffusion process with non-degenerate diffusion coefficient matrix posses good properties (under certain conditions): convergence to equilibrium, entropy dissipation rate, etc. The degenerate drift-diffusion process possess degenerate/rectangular diffusion coefficient matrix, which makes it difficult to govern the convergence property and entropy dissipation rate by drift-diffusion coefficients on its own because of lacking control for the system. In general, the degenerate drift-diffusion is intrinsically equipped with sub-Riemannian struc- ture defined by the diffusion coefficients. We propose a new methodology to systematically study general drift-diffusion process through sub-Riemannian geometry and Wasserstein geometry. We generalize the Bakry-Emery calculus and Gamma z (Baudoin-Garofalo) calculus to define a new notion of sub-Riemannian Ricci curvature tensor. With the new Ricci curvature tensor, we are able to establish generalized curvature dimension bounds on sub-Riemannnian manifolds which goes beyond step two condition. As application, for the first time, we establish analytical bounds for logarithmic Sobolev inequalities for the weighted measure in a compact region on displacement group(SE(2)). Our result also provides entropy dissipation rate for Langevin dynamics with gradient drift and variable temperature matrix. The talk is based on joint works with Wuchen Li.

Tuesday, December 1, 2020

2:00 pm in Zoom Meeting (email for info),Tuesday, December 1, 2020

Joint distribution of Busemann functions in corner growth models

Wai-Tong (Louis) Fan (Indiana University)

Abstract: The 1+1 dimensional corner growth model with exponential weights is a centrally important exactly solvable model in the Kardar-Parisi-Zhang class of statistical mechanical models. While significant progress has been made on the fluctuations of the growing random shape, understanding of the optimal paths, or geodesics, is less developed. The Busemann function is a useful analytical tool for studying geodesics. We present the joint distribution of the Busemann functions, simultaneously in all directions of growth, in terms of mappings that represent FIFO (first-in-first-out) queues. As applications of this description we derive a marked point process representation for the Busemann function across a single lattice edge and point out its implication on structure of semi-infinite geodesics. This is joint work with Timo Seppäläinen.