Seminar time: Thursdays 10:20-11:15am.
Location: Cockins Hall 212
January 9 (first week of regular classes)
January 16: Parisa Fatheddin (OSU)
Title: Large Deviation Principle for Stochastic Navier-Stokes and Schrodinger equations
Abstract: The asymptotic behavior of stochastic Navier-Stokes and stochastic Schrodinger equations will be presented. Theorems such as large and moderate deviations as well as law of iterated logarithm and central limit theorem will be discussed. For large and moderate deviations results employ two methods: the classical Azencott method that relies on time discretization and the more modern technique weak convergence approach that was introduced in 2008. These are from joint work with Zhaoyang Qiu and Hannelore Lisei.
January 23: Jimmy He (OSU)
Title: Random growth models with half space geometry
Abstract: Random growth models in 1+1 dimension capture the behavior of interfaces evolving in the presence of noise. These models are expected to exhibit universal behavior, but we are still far from proving such results even in relatively simple models. A key development which has led to recent progress is the discovery of exact formulas for certain models with rich algebraic structure, leading to asymptotic results. I will survey some of these results, with a focus on models where a single boundary wall is present, as well as applications to rates of convergence for a Markov chain.
February 6: Wlodzimierz Bryc (University of Cincinnati)
Title: Open TASEP in steady state as the top marginal of a two-layer ensemble
Abstract: I will discuss the stationary measure for the Totally Asymmetric Simple Exclusion Process (TASEP) on a segment with open boundaries, represented as the top marginal of a two-layer measure. This two-layer representation facilitates the analysis of particle density (or height function) fluctuations for parameters near the so-called triple point, provides a concise proof of the large deviations principle, and yields asymptotic fluctuations around a random limit for parameters on the coexistence line.
The talk is based on papers with Yizao Wang, Joseph Najnudel and Pavel Zatitskii.
February 20: Yizao Wang (University of Cincinnati)
Title: Scaling limits of stationary measures of open ASEP
Abstract: In a series of recent developments, the scaling limits of stationary measures of open ASEP have been computed with various choices of parameters, and for some choices the limits are stationary measures of the open KPZ equation and the conjectured open KPZ fixed point. In this talk, we shall review the methodology underlying these developments based on the Askey-Wilson (signed) measures. The talk is based on several joint works with Wlodek Bryc, Wesolowski Jacek, Alexey Kuznetsov, and Zongrui Yang.
February 27:
March 6:
March 13: spring break
March 20:
March 27:
April 3: Benjamin Schweinhart (George Mason)
Title: Further Directions in Homological Percolation
Abstract: For well-behaved random subsets of Euclidean space the percolation threshold coincides with the emergence of periodic paths on the torus. Homological percolation generalizes the latter event in terms of higher-dimensional topological features. That is, a random subset of the d-torus exhibits i-dimensional homological percolation if it contains a “giant” i-dimensional cycle spanning the torus.
In previous work with Matt Kahle and Paul Duncan, we studied homological percolation in higher-dimensional analogues of bond percolation on the square lattice and site percolation on the triangular lattice. In particular, we proved that p=1/2 is a sharp threshold for i-dimensional homological percolation on a 2i-dimensional torus in both models. I will review these results and discuss more recent extensions to Voronoi percolation (joint with Morgan Shuman) and the plaquette random cluster model (joint with Paul Duncan). The latter suggests the use of homological percolation as a stopping condition for simulations of Potts lattice gauge theory, generalizing the invaded cluster algorithm for the Potts model (joint with Anthony Pizzimenti and Paul Duncan).
April 10:
April 17: Brian Hall (University of Notre Dame)
Title: Heat flow on polynomials with connections to random matrix theory
Abstract: I will the heat flow on polynomials in two cases: (1) the backward heat flow on polynomials with real roots, and (2) the general heat flow on polynomials with complex roots. In both cases, we will find that the problem is described by a PDE and has close connections to random matrix theory. The talk will be self-contained and have lots of pictures and animations. Based on joint work with Ching-Wei Ho, Jonas Jalowy, and Zakhar Kabluchko.
April 24: Xuan Wu (University of Illinois Urbana-Champaign)
Title: Applications of optimal transport to non-intersecting paths
Abstract: Non-intersecting random paths naturally arise in probability theory and mathematical physics, appearing in settings such as level curves of random surfaces and the evolution of eigenvalues of random matrices. Understanding the regularity of these paths is crucial for analyzing their local structures and asymptotic behaviors. Previously, path regularity has been achieved through Gibbs resampling techniques. In this talk, we introduce a novel approach that uses optimal transport to establish path regularity. This method sharpens several previous estimates and yields new results that were previously inaccessible.
May 1: