The delocalization conjecture for random band matrices

Speaker: 

Jun Yin

Institution: 

UCLA

Time: 

Wednesday, February 26, 2025 - 3:00pm to 4:00pm

Location: 

510R Rowland Hall

In this talk, we present a new proof of the delocalization conjecture for random band matrices. The conjecture asserts that for an $N\times N$ random matrix with bandwidth $W$, the eigenvectors in the bulk of the spectrum are delocalized provided that $W \gg N^{1/2}$. Moreover, in this regime, the eigenvalue distribution aligns with that of Gaussian random ensembles (i.e., GOE or GUE). Our proof employs a novel loop hierarchy method and leverages the sum-zero property, a concept that was instrumental in the previous work on high-dimensional random matrices. 

 

This work is a joint collaboration with H.T. Yau.

Locally Sampleable Uniform Symmetric Distributions

Speaker: 

Kewen Wu

Speaker Link: 

https://shlw.github.io

Institution: 

UC Berkeley

Time: 

Wednesday, January 29, 2025 - 3:00pm to 4:00pm

Host: 

Marcelo Sales

Location: 

Rowland Hall 510R

We characterize the power of constant-depth Boolean circuits in generating uniform symmetric distributions. Let f:{0,1}m→{0,1}n be a Boolean function where each output bit of f depends only on O(1) input bits. Assume the output distribution of f on uniform input bits is close to a uniform distribution D with a symmetric support. We show that D is essentially one of the following six possibilities: (1) point distribution on 0n, (2) point distribution on 1n, (3) uniform over {0n,1n}, (4) uniform over strings with even Hamming weights, (5) uniform over strings with odd Hamming weights, and (6) uniform over all strings. This confirms a conjecture of Filmus, Leigh, Riazanov, and Sokolov (RANDOM 2023).
Joint work with Daniel M. Kane and Anthony Ostuni.

Adapted optimal transport for stochastic processes

Speaker: 

Daniel Bartl

Institution: 

University of Vienna

Time: 

Wednesday, March 12, 2025 - 3:00pm to 4:00pm

Location: 

Rowland Hall 510R

I will discuss adapted transport theory and the adapted Wasserstein distance, which extend classical transport theory from probability measures to stochastic processes by incorporating the temporal flow of information. This adaptation addresses key limitations of classical transport when dealing with time-dependent data. 

I will highlight how, unlike other topologies for stochastic processes, the adapted Wasserstein distance ensures continuity for fundamental probabilistic operations, including the Doob decomposition, optimal stopping, and stochastic control. Additionally, I will explore how adapted transport preserves many desirable properties of classical transport theory, making it a powerful tool for analyzing stochastic systems.

Bias and Free Infinite Divisibility

Speaker: 

Todd Kemp

Institution: 

UCSD

Time: 

Wednesday, January 22, 2025 - 3:00pm to 4:00pm

Location: 

510R Rowland Hall

Size bias and other distributional transforms play an important role in sampling problems.  They are also very useful tools in sharp Normal (and other distributional) approximation, giving slick Stein's method quantitative proofs of Central Limit Theorems.  Recently, Goldstein and Schmook discovered a connection between size bias and infinitely divisible distributions, yielding a new kind of Levy--Khintchine formula for positively supported distributions.

 

In this talk, I will discuss joint work with Goldstein exploring the free probability analogues of bias transforms, with applications to freely infinitely divisible distributions.  In most cases the classical results are transferable, but with a significant change in perspective required.  Among other things, this approach gives a probabilistic meaning to the (free) Levy--Khintchine measure, and a new result simply characterizing those distributions that are positively freely infinitely divisible.

A new approach to critical phenomena in long-range models

Speaker: 

Tom Hutchcroft

Institution: 

Caltech

Time: 

Wednesday, February 12, 2025 - 3:00pm to 4:00pm

Location: 

510R Rowland Hall

Many statistical mechanics models on the lattice (including percolation, self-avoiding walk, the Ising model and so on) have natural "long-range" versions in which vertices interact not only with their neighbours, but with all other vertices in a way that decays with the distance. When this decay is described by a power-law, it can lead to new kinds of critical phenomena that are not present in the short-range models. In this talk I will describe a new approach to the study of these models, leading to some striking new results with surprisingly easy proofs

Mean field results for queueing models with infection

Speaker: 

Thomas Mountford

Institution: 

EPFL

Time: 

Wednesday, February 5, 2025 - 3:00pm to 4:00pm

Location: 

Rowland Hall 510R

This is joint work in progress with Zhe Wang. We consider the model introduced by Baccelli, Foss and Schneer where we consider a closed system of n queues where clients can infect other clients in a contact process like manner. We show in the super critical phase convergence to the "invariant measure".

Can we spot a fake?

Speaker: 

Roman Vershynin

Institution: 

UC Irvine

Time: 

Wednesday, November 20, 2024 - 3:00pm to 4:00pm

Location: 

Rowland Hall 510R

The ongoing AI revolution is transforming human-computer interaction. But not all interaction is good: generative AI has made it easy to create fake data: images, news, and soon videos. But what is a fake? Can we define it mathematically? Which fakes can we detect and which cannot? I will suggest a simple probabilistic framework, where these questions can lead to concrete open (and fun!) problems in mathematics. This talk is based on a joint work with Shahar Mendelson and Grigoris Paouris: https://arxiv.org/abs/2410.18880

Sums of dilates

Speaker: 

Jeck Lim

Speaker Link: 

https://www.its.caltech.edu/~jlim/

Institution: 

Caltech

Time: 

Wednesday, October 30, 2024 - 3:00pm to 4:00pm

Host: 

Marcelo Sales

Location: 

510R Rowland Hall

For any subset A of a commutative ring R (or, more generally, an R-module M) and any elements λ1,…,λk of R, let

λ1⋅A+⋯+λk⋅A={λ1a1+⋯+λkak:a1,…,ak∈A}.

Such sums of dilates have attracted considerable attention in recent years, with the basic problem asking for an estimate on the minimum size of |λ1⋅A+⋯+λk⋅A| given |A|. In this talk, I will discuss various generalizations and settings of this problem, and share recent progress. This is based on joint work with David Conlon.

Independence number of hypergraphs under degree conditions

Speaker: 

Marcelo Sales

Institution: 

UCI

Time: 

Wednesday, October 23, 2024 - 3:00pm to 4:00pm

Host: 

Marcelo Sales

Location: 

510R Rowland Hall

A well-known result of Ajtai et al. from 1982 states that every k-graph H on n vertices, with girth at least five, and average degree tk−1 contains an independent set of size cn(logt)1/(k−1)t for some c>0. In this talk, we explore a related problem where we relax the girth condition, allowing certain cycles of length 2, 3, and 4. We will also present lower bounds on the size of independent sets in hypergraphs under specific degree conditions. This is joint work with Vojtěch Rödl and Yi Zhao. 

Corners in Quasirandom Groups via Sparse Mixing

Speaker: 

Anthony Ostuni

Speaker Link: 

https://aostuni.github.io/

Institution: 

UCSD

Time: 

Wednesday, December 4, 2024 - 3:00pm to 4:00pm

Host: 

Marcelo Sales

Location: 

510R Rowland Hall

We improve the best known upper bounds on the density of corner-free sets over quasirandom groups from inverse poly-logarithmic to quasi-polynomial. We make similarly substantial improvements to the best known lower bounds on the communication complexity of a large class of permutation functions in the 3-player Number-on-Forehead model. Underpinning both results is a general combinatorial theorem that extends the recent work of Kelley, Lovett, and Meka (STOC'24), itself a development of ideas from the breakthrough result of Kelley and Meka on three-term arithmetic progressions (FOCS'23).

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