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.

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 $\lambda_1, \dots, \lambda_k$ of $R$, let

\[\lambda_1 \cdot A + \cdots + \lambda_k \cdot A = \{\lambda_1 a_1 + \cdots + \lambda_k a_k : a_1, \dots, a_k \in A\}.\]

Such sums of dilates have attracted considerable attention in recent years, with the basic problem asking for an estimate on the minimum size of $|\lambda_1 \cdot A + \cdots + \lambda_k \cdot A|$ 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 $t^{k-1}$ contains an independent set of size $c n \frac{(\log t)^{1/(k-1)}}{t}$ 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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