|
4:00pm to 5:00pm - RH 306 - Applied and Computational Mathematics Halyun Jeong - (UCLA) Linear Convergence of Kaczmarz Methods With Sparse Constraints In this talk, I will discuss recent developments in hybrid methods that combine the Kaczmarz method (KZ) and the iterative thresholding (IHT) method to solve linear systems with sparse constraints. The Kaczmarz method (KZ) and its variants, which are types of stochastic gradient descent (SGD) methods, have been extensively studied due to their simplicity and efficiency in solving linear equation systems. The iterative thresholding (IHT) method has gained popularity in various research fields, including compressed sensing or sparse linear regression. Recently, a hybrid method called Kaczmarz-based IHT (KZIHT) has been proposed, combining the benefits of both approaches, but its theoretical guarantees are missing. In this paper, we provide the first theoretical convergence guarantees for KZIHT by showing that it converges linearly to the solution of a system with sparsity constraints up to optimal statistical bias when the reshuffling data sampling scheme is used. We also propose the Kaczmarz with periodic thresholding (KZPT) method, which generalizes KZIHT by applying the thresholding operation for every certain number of KZ iterations and by employing two different types of step sizes. I will discuss several numerical experiments to support our theory; This is joint work with Deanna Needell. |
|
4:00pm to 5:30pm - RH 440 R - Logic Set Theory Jan Grebik - (UCLA) Lossless expansion and measure hyperfiniteness Abstract: The notions of measure hyperfiniteness and measure reducibility of countable Borel equivalence relations are variants of the usual notions of hyperfiniteness and Borel reducibility. Conley and Miller proved that every basis for the countable Borel equivalence relations strictly above E_0 under measure reducibility is uncountable and asked whether there is a "measure successor of E_0"—i.e. a countable Borel equivalence relation E such that E is not measure reducible to E_0 and any F which is measure reducible to E is either equivalent to E or measure reducible to E_0. In an ongoing work with Patrick Lutz, we have isolated a combinatorial condition on a Borel group action (a strong form of expansion that we call "lossless expansion" after a similar property which is studied in computer science and finite combinatorics) which implies that the associated orbit equivalence relation is a measure successor of E_0. We have also found several examples of group actions which are plausible candidates for satisfying this condition. In this talk, I will explain the context for Conley and Miller's question, the condition that we have isolated and discuss some of the candidate examples we have identified. All of this is joint work with Patrick Lutz. |
|
3:00pm to 4:00pm - RH 306 - Number Theory Roger Van Peski - (KTH) New universal limits for cokernels of random matrix products Since 1980s work of Cohen-Lenstra and Friedman-Washington, many (pseudo-)random groups in number theory, combinatorics and topology have been conjectured---and sometimes proven---to match certain universal distributions, which appear as large-N limits of cokernels of N x N random matrices over $\mathbb{Z}$ or $\mathbb{Z}_p$.
In this talk I discuss a new such distribution, the cokernel of a product of k independent matrices. For each fixed k, it converges to a universal distribution, generalizing in a natural way the k=1 case of the Cohen-Lenstra distribution. As time permits I will discuss the case when the number of products k goes to infinity along with N. Then the groups do not converge, but the fluctuations of their ranks and other statistics still approach limit distributions related to a new interacting particle system, the 'reflecting Poisson sea'.
Based on https://arxiv.org/abs/2209.14957v2 (with Hoi Nguyen) and https://arxiv.org/abs/2312.11702, https://arxiv.org/abs/2310.12275. |
|
2:00pm to 3:00pm - 510R Rowland Hall - Combinatorics and Probability Jan Grebik - (UCLA) Measurable tilings Let $(X,\mu)$ be a standard probability space and $G\curvearrowright (X,\mu)$ be a measure-preserving action of a group $G$ on $X$. The general problem that we consider is to understand the structure of measurable tilings $F\oplus A=X$ of $X$ by a measurable tile $A\subseteq X$ shifted by a finite set $F\subseteq G$, thus the shifts $f\cdot A$, $f\in F$ partition $X$ up to null sets. The motivation comes from the theory of (paradoxical) equidecompositions and tilings in $\mathbb{R}^n$. After a summary of recent results that concern the spheres and tori, I will focus on the intersection of these cases, that is, the case of the circle. Using the structure theorem of Greenfeld and Tao for tilings of $\mathbb{Z}^d$, we show that measurable tilings of the circle can be reduced to tilings of finite cyclic groups. This is a joint work with Conley and Pikhurko, and Greenfeld, Rozhon and Tao. |
|
9:00am to 9:50am - Zoom - Inverse Problems Roel Snieder - (Colorado School of Mines) Variations and healing of the seismic velocity |
|
4:00pm to 4:50pm - RH 306 - Colloquium Konstantin Khanin - (Toronto University) On KPZ universality and stochastic flows We will start by introducing the phenomenon of the KPZ (Kardar-Parisi-Zhang) universality. KPZ problem was a very active research area in the last 20 years. The area of KPZ is essentially interdisciplinary. It is related to such fields as probability theory, statistical mechanics, mathematical physics, PDE, SPDE, random dynamics, random matrices, and random geometry, to name a few. In most general form the problem can be formulated in the following way. Consider random geometry on the two-dimensional plane. The main aim is to understand the asymptotic statistical properties of the length of the geodesic connecting two points, which are far away from each other, in the limit as distance between the endpoints tends to zero. One also wants to study the geometry of random geodesics, in particular how much they deviate from a straight line. It turn out that the limiting statistics for both the length and the deviation is universal, that is it does not depend on the details of the random geometry. Moreover, many limiting probability distributions can be found explicitly. In the second part of the talk we will proceed with discussion of the geometrical approach to the problem of the KPZ universality which provides an even broader point of view on the problem of universal statistical behavior. No previous knowledge of the subject will be assumed. |