All talks are in Kelley Auditorium in the Lecture Halls building. See this map for directions to the venue from both hotels.
Registration is in the lobby. Elevator access is through Shannon Hall.
Times are in Central Daylight Time (UTC-5).
Tuesday May 20th
9:00-9:30 Registration and Refreshments
9:30-9:45 Opening Remarks
9:45-10:45 Keynote by SRS Varahdan
10:45-11:00 Coffee break
11:00-12:00 Keynote by Elton Hsu
12:00-1:30 Lunch
1:30-2:30 Keynote by Brian Rider
2:30-3:00 Iddo Ben-Ari
3:00-3:15 Coffee Break
3:15-4:15 Keynote by Mike Cranston
4:15-4:45 Marcel Hudiani
4:45-5:00 Coffee Break
5:00-5:30 Maximillian Newman
5:30-6:00 Janos Englander
Wednesday May 21th
9:00-9:30 Registration and Refreshments
9:30-10:30 Keynote by Krzysztof Burdzy
10:30-10:45 Coffee Break
10:45-11:15 Yehuda Pinchover
11:15-11:45 Hongyi Chen
11:45-12:00 Group Photo
12:00-1:30 Lunch
1:30-2:30 Keynote by Rami Atar
2:30-3:00 Shannon Starr
3:00-3:15 Coffee Break
3:15-4:15 Keynote by Ron Peled
4:15-4:45 Douglas Rizzolo
6:00-7:30 Banquet at Gokul kosher Indian restaurant followed by drinks at International Tap House — Delmar
Thursday May 22th
9:00-9:30 Refreshments
9:30-10:30 Keynote by Jay Rosen
10:30-11:00 Jonathon Peterson
11:00-11:15 Coffee Break
11:15-11:45 Martin Kolb (online)
11:45-12:15 Zhen-Qing Chen (online)
12:15 -12:30 Closing Remarks
Free boundary problems and particle systems
Particle systems described macroscopically via free boundary problems (FBP) include the N-branching Brownian motion, of branching Brownian particles on the line with removal of the leftmost particle upon each branching, and the Atlas model, of Brownian particles on the line where the leftmost particle is equipped with a positive drift. I will describe a weak FBP formulation with which one can establish the hydrodynamic limit for generalizations of these models, and a related "free obstacle" problem aimed at N-branching random walks where removals occur at the most densely populated site. The talk is partly based on joint works with Amarjit Budhiraja and with Leonid Mytnik and Gershon Wolansky.
Characterization of Quasistationary Distributions
We provide a complete description of Quasistationary Distributions (QSDs) for Markov chains with a unique absorbing state and an irreducible set of non-absorbing states. As is well-known, every QSD has an associated absorption parameter describing the exponential tail of the absorption time under the law of the process with the QSD as the initial distribution. The analysis associated with the existence and representation of QSDs corresponding to a given parameter is according to whether the moment generating function of the absorption time starting from any non-absorbing state evaluated at the parameter is finite or infinite, the finite or infinite moment generating function regimes, respectively. For parameters in the finite regime, it is shown that when they exist, all QSDs are in the convex cone of a Martin entry boundary of sub-probability measures associated with the parameter. The infinite regime corresponds to at most one parameter value and at most one QSD. In this regime, when a QSD exists, it is unique and can be represented by a renewal-type formula. Multiple applications to the findings are presented, including revisiting some of the main classical results in the area.
Simple nonlinear PDEs inspired by billiards
How many times can n billiard balls collide in the open d-dimensional space? I will provide some estimates. I will explain how the above question leads to a "pinned billiard balls'' model. On a large scale, the model seems to have a hydrodynamic limit. The parameters of the conjectured limit should satisfy simple nonlinear PDEs. While the existence and properties of the conjectured hydrodynamic limit are open questions, I will provide a quite complete analysis of the conjectured PDEs.
Global Geometry and the Parabolic Anderson Model on Compact Manifolds
We introduce a family of intrinsic Gaussian noises on compact manifolds that we call “colored noise” on manifolds. With this noise, we study the parabolic Anderson model (PAM) on compact manifolds. Under some curvature conditions, we show the well-posedness of the PAM and provide some preliminary (but sharp) bounds on the second moment of the solution. It is interesting to see that global geometry plays a role in obtaining the well-posedness of the equation.
Dirichlet problem for integro-differential operators
Consider an integro-differential operator L in the Euclidean space that has both diffusive and non-local parts. We will show under some mild conditions, there is a Feller process having strong Feller property associated with it. We then discuss the solvability of the Dirichlet problem for L and its probabilistic representation.
Sampling Using Zeta Functions
Zeta functions arise in a variety of contexts. The integers, ideals in Dedekind domains, polynomials over finite fields provide a few examples. In this paper, we will discuss using zeta functions to sample these objects. This leads to very elegant analysis of the statistical properties of the sampled objects. One example is the famous Erdos-Kac Central Limit Theorem. More detailed analysis can be achieved using ideas of mod-Poisson convergence. The talk will summarize joint work with Elton Hsu, Mariia Khodiakova, Thomas Mountford and Adrien Peltzer.
Tree Builder Random Walk
We investigate a self-interacting random walk, in a dynamically evolving environment, which is a random tree built by the walker itself, as it walks around. That is, the walker and the graph on which it moves mutually influence each other. At discrete time units, right before stepping, the walker adds a random number (possibly zero) leaves to its current position. We assume that the number of leaves at each time unit are independent, but we do not assume that they are identically distributed, resulting thus in a time in-homogeneous setting. Properties of the walk (transience/recurrence, getting stuck) as well as the structure of the generated random trees (limiting degree, distribution, maximal degree etc.) are discussed. A coupling with the well-known preferential attachment model of Barabási and Albert turns out to be useful in the appropriate regime. This is joint work with R. Ribeiro (Denver/Rio de Janeiro), G. Iacobelli (Rio de Janeiro) and G. Pete (Budapest).
The Parisi Formula via Stochastic Analysis
The Parisi formula is a fundamental result in spin glass theory. It gives a variational characterization of the asymptotic limit of the expected free energy. The upper bound is a consequence of an interpolation identity due to F. Guerra and the lower bound is a celebrated result of M. Talagrand. In this talk I will present a new approach to (an enhanced version of) Guerra's identity using stochastic analysis, more specifically Brownian motion and Ito’s calculus. This approach is suggested by the form of the Parisi formula in which the solution of a Hamilton-Jacobi equation is involved. It helps in many ways to illuminate the original method of Guerra (basically a discretized version of Girsanov’s transform) and suggests a possible similar approach to the significantly deeper lower bound, which has been intensively studied since Talagrand’s work. Among the techniques from stochastic analysis we will use include path space integration by parts for the Wiener measure, Girsanov’s transform (i.e., exponential martingales), and probabilistic representation of solutions to (linear) partial differential equations. The key observation is that the nonlinear Hamilton-Jacobi partial differentiation equation figuring in Parisi’s variation formula becomes linear after differentiating with respect to Guerra’s interpolation parameter, thus bringing the full strength of stochastic analysis based on Ito’s calculus into play. It is hoped that this approach will shed some lights on the much more difficult lower bound in the Parisi formula.
Singular Diffusion Limit of a Tagged Particle in Zero Range Processes with Sinai-type Random Environment
We derive a singular diffusion limit for the position of a tagged particle in zero range interacting particle processes on a one dimensional torus with a Sinai-type random environment via two steps. In the first step, a regularization is introduced by averaging the random environment over an epsilon neighborhood. With respect to such an environment, we find the macroscopic dynamics of the tagged particle in terms of the hydrodynamic density depending on epsilon as the diffusively scaled limit of its microscopic dynamics. In the second step, we take the epsilon regularization to zero and obtain a convergence in distribution to a diffusion process making use of the para-controlled limit of the hydrodynamic density which solves a singular SPDE. Joint work with Sunder Sethuraman and Claudio Landim.
Brownian motion with partial resetting conditioned to stay positive
In the talk we condition one-dimensional Brownian motion with partial resetting to stay positive using the most natural limit procedure. The limiting process is a three-dimensional Bessel process with partial resetting but with a reduced resetting rate. In some sense this topic connects some of the early work with much more recent work of Ross.
Quenched limit laws for coalescing random walks
In genetics, certain statistical quantities are governed by a tree structure, called a coalescent, whose law is determined by a scaling limit of random walks on random graphs. Classical coalescent theory describes a limiting law for these coalescents by, implicitly, averaging over these graphs. Recent work has examined what random limiting laws one may describe when one instead conditions on these random graphs. For a diploid Moran model with self-fertilization, we establish a phase transition between deterministic and stochastic quenched limits as one varies the rate of self-fertilization and explore some properties of the limiting random measures. The phase transition is tied to a certain propagation of chaos.
Euclidean random permutations
We discuss the cycle structure of Euclidean random permutations: random permutations of points in R^d which are biased towards the identity in the underlying geometry. We identify sub-critical, critical and super-critical regimes and the limiting distributions of the cycle lengths in all cases. We also highlight analogies with models of random band matrices. Based on joint works with Dor Elboim and Alexey Gladkich.
Scaling limit of the "true" self-avoiding random walk
The “true” self-avoiding random walk (TSAW) is a model for random motion that is very strongly self-repelling. Over 25 years ago, Toth showed that the one-dimensional version of this walk is strongly super-diffusive in the sense that the position of the walk after n steps should converge in distribution when scaled by n^(2/3). Toth was able to characterize what the limiting distribution of the walk would be if it exists, but he was not able to prove that the limiting distribution does in fact exist. Later, motivated by Toth’s earlier results, Toth and Werner constructed a continuous time limit called the true self-repelling motion which they argued should be the functional scaling limit for the TSAW, but once again they did not prove this functional limiting distribution. In this talk I will show how one can use joint Ray-Knight theorems to prove that the TSAW does indeed converge in distribution to the true self-repelling motion. This is based on joint work with Elena Kosygina.
The Landis conjecture via Liouville comparison principle and criticality theory
Landis' conjecture is concerned with growth bounds for nontrivial solutions of the Schrödinger equation on R^d whose potential is bounded by one. We study this problem for real valued potentials on noncompact manifolds and on discrete infinite countable graphs. Joint works with Ujjal Das and Matthias Keller.
A matrix model for conditioned Stochastic Airy
There are three basic flavors of local limit theorems in random matrix theory, connected to the spectral bulk and the so-called soft and hard edges. There also abound a collection of more exotic limits which arise in models that posses degenerate (or “non-regular”) points in their equilibrium measure. What is more, there is typically a natural double scaling about these non-regular points, producing limit laws that transition between the more familiar basic flavors. Here I will describe a general beta matrix model for which, conjecturally, the appropriate double scaling limit is the Stochastic Airy Operator, conditioned on having no eigenvalues below a fixed level. The conjectural bit relates to a question about the Martin boundary for a certain diffusion process, which I hope will amuse our guest of honor.
Local limits of permutations avoiding a pattern of length three
We discuss the problem of finding local limits of permutations avoiding a pattern of length 3. This problem addressed by Ross Pinsky in 2020 paper where he identified the limits in all cases except for the case of 321-avoiding permutations. In this talk we will discuss a unified approach based on local limits of Galton-Watson trees that allows us to identify the limits in all cases. Joint work with Jungeun Park (University of Delaware).
Exact moduli of continuity for the local times of rebirthed Markov processes
Let Y be a transient symmetric Borel right process with state space S. A fully rebirthed version of Y is an extension of Y so that instead of terminating at the end of its lifetime, it is immediately reborn with a probability measure μ on S. This rebirthed version of Y is a recurrent Borel right process with state space S which is not symmetric. Using a new approach, we apply properties of Gaussian processes to obtain exact local and uniform moduli of continuity for the local times of many rebirthed Markov processes.
About Pinsky's combinatorial approach to Ulam's problem
Pinsky showed a combinatorial mapping relating two interesting quantities in probability: the variance of the number of length-k subsequences of random permutations which are increasing; and the number of pairs of random walks with a prescribed number of collisions. We apply large deviations to Pinsky's formula to rederive an old result of Lifshitz and Pittel for the variance of the total number of increasing subsequences of a random permutation, of any length.
The story of the Polaron
The Polaron is a quantum mechanical object and behavior of its ground state can be expressed in terms of Brownian motion. The question is the following. x(t) is three dimensional Brownian motion. What matters are the increments {x(t) - x(s)}. A tilted measure Q^{α,T} is defined by