Week of January 19, 2020

Tue Jan 21, 2020
2:00pm to 2:50pm - RH 510R - Working Group in Information Theory
Sonky Ung - (UC Irvine)
Geometric interpretation of mutual information

This week, we will discuss Section 3.2 of the lecture notes of Wu and Polyanski:

http://people.lids.mit.edu/yp/homepage/papers.html

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.

3:00pm to 3:50pm - 306 RH - Special Colloquium
Ken Ascher - (Princeton)
Compactifying moduli spaces in algebraic geometry

Algebraic geometry is concerned with algebraic varieties, which can be understood as solution sets of polynomial equations. At the heart of research is the classification of algebraic varieties, and a geometric solution is provided in the form of a moduli space. Roughly speaking, a moduli space is itself an algebro-geometric object whose points represent equivalence classes of algebraic varieties of a fixed type. This talk begins with the moduli space of curves, which parametrizes equivalence classes of complex algebraic curves (i.e. Riemann surfaces) of a fixed genus. This moduli space, like most moduli spaces appearing in algebraic geometry, is not a compact space. A celebrated result of Deligne and Mumford provides a geometric way to compactify this space. The goal of this talk is to discuss recent progress towards compactifying moduli spaces of higher dimensional complex algebraic varieties (e.g. complex algebraic surfaces).

4:00pm to 5:00pm - RH 306 - Differential Geometry
Luca Spolaor - (UCSD)
Epsilon-regularity for minimal surfaces near quadratic cones

Every area-minimizing hypercone having only an isolated singularity fits into a foliation by smooth, area-minimizing hypersurfaces asymptotic to the cone itself. In this talk I will present the following epsilon-regularity result: every minimal surfaces lying sufficiently close to a minimizing quadratic cone (for example, the Simons' cone), is a perturbation of either the cone itself, or some leaf of its associated foliation. This result also implies the Bernstein-type result of Simon-Solomon, which characterizes area-minimizing hypersurfaces asymptotic to a quadratic cone as either the cone itself, or some leaf of the foliation, and it also allows to study convergence to singular minimal hyper surfaces. This is a joint result with N. Edelen

4:00pm to 5:00pm - RH 306 - Analysis
Luca Spolaor - (UCSD)
Epsilon-regularity for minimal surfaces near quadratic cones

Joint with Differential Geometry seminar.

 

Every area-minimizing hypercone having only an isolated singularity fits into a foliation by smooth, area-minimizing hypersurfaces asymptotic to the cone itself. In this talk I will present the following epsilon-regularity result: every minimal surfaces lying sufficiently close to a minimizing quadratic cone (for example, the Simons' cone), is a perturbation of either the cone itself, or some leaf of its associated foliation. This result also implies the Bernstein-type result of Simon-Solomon, which characterizes area-minimizing hypersurfaces asymptotic to a quadratic cone as either the cone itself, or some leaf of the foliation, and it also allows to study convergence to singular minimal hyper surfaces. This is a joint result with N. Edelen

Thu Jan 23, 2020
3:00pm to 4:00pm - RH 306 - Algebra
Siamak Yassemi - (University of Tehran and IPM)
Castelnuovo-Mumford Regularity of Edge Ideals of Graphs
Let $\mathbb{K}$ be a field and $S = \mathbb{K}[x_1,\ldots,x_n]$ be the

polynomial ring in $n$ variables over $\mathbb{K}$. Suppose that $M$ is a graded $S$-module with minimal free resolution

$$0 \longrightarrow \cdots \longrightarrow \bigoplus_{j}S(-j)^{\beta_{1,j}(M)} \longrightarrow \bigoplus_{j}S(-j)^{\beta_{0,j}(M)} \longrightarrow M \longrightarrow 0.$$

 

The Castelnuovo--Mumford regularity (or simply, regularity) of $M$, denote by ${\rm reg}(M)$, is defined as follows:

$${\rm reg}(M)={\rm max}\{j-i|\ \beta_{i,j}(M)\neq0\}.$$

 

We survey a number of recent studies of the Castelnuovo-Mumford regularity of the ideals related to a graph and their (symbolic) powers. Our focus is on the bounds and exact values for the regularity in terms of combinatorial data from associated graphs. This research program has produced many exciting results and, at the same time, opened many further interesting questions and conjectures. 

Fri Jan 24, 2020
3:00pm to 3:50pm - RH 306 - Special Colloquium
Alexandra Florea - (Columbia University)
Moments in families of L-functions

The moments of the Riemann zeta function were introduced by Hardy and Littlewood more than 100 years ago, in an attempt to prove the Lindelöf hypothesis, which provides a strong upper bound on the size of the Riemann zeta function on the critical line. Since then, moments became central objects of study in number theory. I will give an overview of the problem of computing moments in different families of L-functions, and I will discuss some of the applications. For example, I will explain how one can extract information about the values of L-functions at special points by computing moments of the L-functions in question.

4:00pm to 4:50pm - RH 306 - Inverse Problems
Yimin Zhong - (UCI)
A hybrid inverse problem of fluorescence ultrasound modulated optical tomography in the diffusive regime

We investigate a hybrid inverse problem in fluorescence ultrasound modulated optical tomography (fUMOT) in the diffusive regime. We prove that the boundary measurement of the photon currents allows unique and stable reconstructions of the absorption coefficient of the fluorophores at the excitation frequency and the quantum efficiency coefficient simultaneously, provided that some background medium parameters are known. Reconstruction algorithms are proposed and numerically implemented as well.