Regularization of probability measures through free addition and convolution semigroups II

Speaker: 

Gregory Zitelli

Institution: 

UCI

Time: 

Thursday, December 6, 2018 - 12:00pm to 1:00pm

Host: 

Location: 

340P

This is a continuation of the talk from last week about the regularizing properties of free additive convolution. We introduce and discuss a variety of tools from noncommutative probability, notably the operations of free, boolean, and monotone additive and multiplicative convolutions, the Belinschi–Nica semigroup, the free divisibility indicator, and complex subordination. The goal of the talk is to provide a broad overview of free probability, including a variety of curious equations and identities, from a complex analytic viewpoint. The talk is based on the works of Belinschi, Bercovici, Nica, Arizmendi, and Hasebe.

A maximum entropy approach to approximating the number of graphs with given degree sequence

Speaker: 

Adrien Peltzer

Institution: 

UCI

Time: 

Thursday, November 15, 2018 - 12:00pm to 1:00pm

Location: 

340P

Let G(D) be the set of all graphs with degree sequence d. The Erdos-Gallai conditions give the necessary and sufficient conditions for the existence of a graph with degree sequence d. If G(D) is nonempty, how do we approximate the size of all such graphs? I will discuss a maximum entropy approach to this problem. It involves considering G(D) as the set of integer points of a certain polytope in R^(n choose 2) and constructing a probability distribution that is constant on this set of points. Using a concentration result, we can use this distribution to approximate the size of G(D).

 

Beating very small probabilities -- upper bounding the number of Hadamard matrices

Speaker: 

Asaf Ferber

Institution: 

MIT

Time: 

Thursday, October 25, 2018 - 12:00pm

Host: 

Location: 

340P

A Hadamard matrix of order n is an n\times n, \pm 1 matrix for which every two rows are orthogonal. It is not hard to check that if n is not divisible by 4, then a Hadamard matrix of order n does not exists. On the other hand, it is widely open to prove the existence of Hadamard matrices of order 4n for all n. 

 

Years of investigation show that Hadamard matrices are very hard to find. In particular, as an easy exercise, one can show that the probability for a random matrix to be Hadamard is at most 2^{-n^2/2+o(n^2)}. 

 

In this talk I'll convince ourselves that Hadamard matrices are even harder to find! that is, the probability for a random matrix to be Hadamard is at most 2^{(1+\varepsilon)n^2/} for some small (but fixed) epsilon>0. Note that the gain is of the form 2^{-\theta(n^2)}, which, at least according to my knowledge, does not follow from any standard large deviation inequality. 

 

I'll give a purely combinatorial proof that does not appear on the paper (or elsewhere..) I have with Jain and Zhao on this topic.  This proof was basically our key for giving improved anticoncentration inequalities for classical results by Halasz regarding random sums of vectors (I won't discuss the anticoncentration bounds though...).

Subscribe to RSS - Stub