Week of October 13, 2019

Mon Oct 14, 2019
4:00pm - TBA - Logic Set Theory
Kevin Duanmu - (UC Berkeley)
Mixing times and Hitting times for general Markov processes using Nonstandard Analysis

Nonstandard analysis, a powerful machinery derived from mathematical logic, has had many applications in probability theory as well as stochastic processes. Nonstandard analysis allows construction of a single object---a hyperfinite probability space---which satisfies all the first order logical properties of a finite probability space, but which can be simultaneously viewed as a measure-theoretical probability space via the Loeb construction. As a consequence, the hyperfinite/measure duality has proven to be particularly in porting discrete results into their continuous settings. 

In this talk, for every general-state-space discrete-time Markov process satisfying appropriate conditions, we construct a hyperfinite Markov process which has all the basic order logical properties of a finite Markov process to represent it.  We show that the mixing time and the hitting time agree with each other up to some multiplicative constants for discrete-time general-state-space reversible Markov processes satisfying certain condition. Finally, we show that our result is applicable to a large class of Gibbs samplers and Metropolis-Hasting algorithms.

Tue Oct 15, 2019
11:00am to 12:00pm - RH 340N - Combinatorics and Probability
Kyle Luh - (Harvard University)
Eigenvalue gaps of sparse random matrices

We will discuss some recent work on quantifying the gaps between eigenvalues of sparse random matrices.  Before these results, it was not even known if the eigenvalues were distinct for this class of random matrices.  We will also touch upon how these results relate to random graphs, the graph isomorphism problem and nodal domains.  This is joint work with Van Vu and Patrick Lopatto.

2:00pm to 3:00pm - 510R - Working Group in Information Theory
Kathryn Dover - (UCI)
 Submodularity of Entropy, Han's Inequality, and Shearer’s Lemma

Working Group in Information Theory is a self-educational project in the department. Techniques based on information theory have become essential in high-dimensional probability, theoretical computer science and statistical learning theory. On the other hand, information theory is not taught systematically. The goal of this group is to close this gap.

This week, we will discuss Section 1.4 and 1.5 of the lecture notes of Wu and Polyanski: 
http://people.lids.mit.edu/yp/homepage/papers.html

 

4:00pm - RH306 - Differential Geometry
Paula Burkhardt-Guim - (UC Berkeley)
Pointwise lower scalar curvature bounds for C^0 metrics via regularizing Ricci flow

We propose a class of local definitions of weak lower scalar curvature bounds that is well defined for C^0 metrics. We show the following: that our definitions are stable under greater-than-second-order perturbation of the metric, that there exists a reasonable notion of a Ricci flow starting from C^0 initial data which is smooth for positive times, and that the weak lower scalar curvature bounds are preserved under evolution by the Ricci flow from C^0 initial data.

Thu Oct 17, 2019
2:00pm to 3:00pm - RH 340 P - Mathematical Physics
Lior Alon - (Technion)
On a universal limit conjecture for the nodal count statistics of quantum graphs

 

 
 Understanding the statistical properties of Laplacian eigenfunctions in general and their nodal sets in particular, have an important role in the field of spectral geometry, and interest both mathematicians and physicists. A quantum graph is a system of a metric graph with a self-adjoint Schrodinger operator. It was proven for quantum graphs that the number of points on
 which each eigenfunction vanish (also known as the nodal count) is
 bounded away from the spectral position of the eigenvalue by the first Betti number of the graph. A remarkable result by Berkolaiko and Weyand showed that the nodal surplus is equal to a magnetic stability index of the corresponding eigenvalue. A similar result for discrete graphs holds as well proved first by Berkoliako and later by Colin deVerdiere.
 
 Both from the nodal count point of view and the magnetic point of view, it is interesting to consider the distribution of these indices over the spectrum. In our work, we show that such a density exists and defines a nodal count distribution. Moreover, this distribution is symmetric, which allows deducing the topology of a graph from its nodal count. Although for general graphs we can not a priori calculate the nodal count distribution, we proved that a certain family of graphs will have a binomial distribution. As a corollary, given any sequence of graphs from that family with an increasing number of cycles, the sequence of nodal count distributions, properly normalized, will converge to a normal distribution. 
A numerical study indicates that this property might be universal and led us to state the following conjecture. For every sequence of graphs with an increasing number of cycles, the corresponding sequence of properly normalized nodal count distributions will converge to a normal distribution. 
 In my talk, I will present our latest results extending the number
 of families of graphs for which we can prove the conjecture.
 
This talk is based on joint works with Ram Band (Technion) and Gregory Berkolaiko (Texas A&M)

3:00pm to 4:00pm - RH 440R - Number Theory
Alexandra Florea - (Columbia University)
Moments of cubic L-functions over function fields
I will focus on the mean value of $L$-functions associated to cubic characters over $\mathbb{F}_q[t]$ when $q \equiv 1 \pmod 3$. I will explain how to obtain an asymptotic formula which relies on obtaining cancellation in averages of cubic Gauss sums over functions fields. I will also talk about the corresponding non-Kummer case when $q \equiv 2 \pmod 3$ and I will explain why this setting is somewhat easier to handle than the Kummer case, which allows us to prove some better results. This is joint work with Chantal David and Matilde Lalin.

 

Fri Oct 18, 2019
1:00pm - RH 440R - Cryptography
Alice Silverberg - (UCI)
Introduction to NTRU encryption

This talk will give an introduction to NTRU encryption.

3:00pm to 4:00pm - RH 440R - Nonlinear PDEs
Antonio De Rosa - (Courant Institute, NYU)
Elliptic integrands in geometric variational problems

I will present the recent tools I have developed to prove existence and regularity properties of the critical points of anisotropic functionals. In particular, I will provide the anisotropic extension of Allard's celebrated rectifiability theorem and its applications to the anisotropic Plateau problem. Three corollaries are the solutions to the formulations of the Plateau problem introduced by Reifenberg, by Harrison-Pugh and by Almgren-David. Furthermore, I will present the anisotropic counterpart of Allard's compactness theorem for integral varifolds. To conclude, I will focus on the anisotropic isoperimetric problem: I will provide the anisotropic counterpart of Alexandrov's characterization of volume-constrained critical points among finite perimeter sets. Moreover I will derive stability inequalities associated to this rigidity theorem. 

Some of the presented results are joint works with De Lellis, De Philippis, Ghiraldin, Gioffré, Kolasinski and Santilli.

4:00pm - PSCB 140 - Graduate Seminar
Jen McIntosh - (NSA)
The #1 job to take after graduation

Dr. Jen McIntosh is a UCI alum who started at the National Security Agency (NSA) over 15 years ago as an Applied Research Mathematician. Her journey with NSA has been typically atypical, in that most mathematicians have a world of choice and opportunities to explore as interests and mission needs evolve - whether math-y or not.

Currently she is in the Senior Technical Development Program (STDP), a mid- to late-career program designed to foster expertise in areas of strategic importance to the Agency. Her current passion is bridging math, psychology, business, and other areas that make up decision science - enhancing decision-making with information and data.

She'll talk a little about her journey, the diversity of mathematical fields she's practiced (from common sense to cryptanalysis), and she'll dedicate most of the time for questions about life at NSA and the wealth of career opportunities for mathematicians at any phase of their careers.