Seminars and Colloquia by Series

Decoupling and applications: a journey from continuous to discrete

Series
Job Candidate Talk
Time
Thursday, February 6, 2020 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Ciprian DemeterIndiana University

Decoupling is a Fourier analytic tool that  has repeatedly proved its extraordinary potential for a broad range of applications to number theory (counting solutions to Diophantine systems, estimates for the growth of the Riemann zeta), PDEs (Strichartz estimates, local smoothing for the wave equation, convergence of solutions to the initial data), geometric measure theory (the Falconer distance conjecture)  and harmonic analysis (the Restriction Conjecture). The abstract theorems are formulated and proved in a continuous framework, for arbitrary functions with spectrum supported near curved manifolds. At this level of generality, the proofs involve no number theory, but rely instead on  wave packet analysis and incidence geometry related to the Kakeya phenomenon.   The special case when the spectrum is localized near lattice points leads to unexpected  solutions of conjectures once thought to pertain to the realm of number theory. 

Lattice polytopes in representation theory and geometry

Series
Job Candidate Talk
Time
Tuesday, February 4, 2020 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Ricky LiuMath, North Carolina State University

Lattice polytopes play an important role in combinatorics due to their intricate geometric structure as well as their enumerative properties. In this talk, we will discuss several instances in which lattice point enumeration of lattice polytopes relates to problems in algebraic combinatorics, particularly the representation theory of GL(n) and related groups. We will also see how certain types of algebraic constructions have polytopal counterparts. This talk is based on joint work with Karola Mészáros and Avery St. Dizier.

Geometric statistics for shape analysis of bioimaging data

Series
Job Candidate Talk
Time
Thursday, January 23, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Nina MiolaneStanford University

The advances in bioimaging techniques have enabled us to access the 3D shapes of a variety of structures: organs, cells, proteins. Since biological shapes are related to physiological functions, statistical analyses in biomedical research are poised to incorporate more shape data. This leads to the question: how do we define quantitative descriptions of shape variability from images?

Mathematically, landmarks’ shapes, curve shapes, or surface shapes can be seen as the remainder after we have filtered out the corresponding object’s position and orientation. As such, shape data belong to quotient spaces, which are non-Euclidean spaces.

In this talk, I introduce “Geometric statistics”, a statistical theory for data belonging to non-Euclidean spaces. In the context of shape data analysis, I use geometric statistics to prove mathematically and experimentally that the “template shape estimation” algorithm, used for more than 15 years in biomedical imaging and signal processing, has an asymptotic bias. As an alternative, I present variational autoencoders (VAEs) and discuss the accuracy-speed trade-off of these procedures. I show how to use VAEs to estimate biomolecular shapes from cryo-electron microscopy (cryo-EM) images. This study opens the door to unsupervised fast (cryo-EM) biological shape estimation and analysis.

Matroids, log-concavity, and expanders

Series
Job Candidate Talk
Time
Thursday, January 23, 2020 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Cynthia VinzantNorth Carolina State University

Abstract:  Matroids are combinatorial objects that model various types of independence. They appear several fields mathematics, including graph theory, combinatorial optimization, and algebraic geometry. In this talk, I will introduce the theory of matroids along with the closely related class of polynomials called strongly log-concave polynomials. Strong log-concavity is a functional property of a real multivariate polynomial that translates to useful conditions on its coefficients. Discrete probability distributions defined by these coefficients inherit several of these nice properties.  I will discuss the beautiful real and combinatorial geometry underlying these polynomials and describe applications to random walks on the faces of simplicial complexes. Consequences include proofs of Mason's conjecture that the sequence of numbers of independent sets of a matroid is ultra log-concave and the Mihail-Vazirani conjecture that the basis exchange graph of a matroid has expansion at least one. This is based on joint work with Nima Anari, Kuikui Liu, and Shayan Oveis Gharan.

Extremal Problems in Discrete Geometry

Series
Job Candidate Talk
Time
Tuesday, January 14, 2020 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Zilin JiangMIT

What is the smallest total width of a collection of strips that cover a disk in the plane? How many lines through the origin pairwise separated by the same angle can be placed in 3-dimensional space? What about higher-dimensions?

These extremal problems in Discrete Geometry look deceitfully simple, yet some of them remain unsolved for an extended period or have been partly solved only recently following great efforts. In this talk, I will discuss two longstanding problems: Fejes Tóth’s zone conjecture and a problem on equiangular lines with a fixed angle.

No specific background will be needed to enjoy the talk.

Elliptic integrands in geometric variational problems

Series
Job Candidate Talk
Time
Thursday, January 9, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Antonio De RosaNYU

Elliptic integrands are used to model anisotropic energies in variational problems. These energies are employed in a variety of applications, such as crystal structures, capillarity problems and gravitational fields, to account for preferred inhomogeneous and directionally dependent configurations. After a brief introduction to variational problems involving elliptic integrands, I will present an overview of the techniques I have developed to prove existence, regularity and uniqueness properties of the critical points of anisotropic energies. In particular, I will present the anisotropic extension of Allard's rectifiability theorem and its applications to the Plateau problem. Furthermore, I will describe the anisotropic counterpart of Alexandrov's characterization of volume-constrained critical points. Finally, I will mention some of my ongoing and future research projects.

Analysis and computation of nonlocal models

Series
Job Candidate Talk
Time
Tuesday, January 7, 2020 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Xiaochuan TianUniversity of Texas at Austin
Nonlocal models are experiencing a firm upswing recently as more realistic alternatives to the conventional local models for studying various phenomena from physics and biology to materials and social sciences. In this talk, I will describe our recent effort in taming the computational challenges for nonlocal models. I will first highlight a family of numerical schemes -- the asymptotically compatible schemes -- for nonlocal models that are robust with the modeling parameter approaching an asymptotic limit. Second, I will discuss nonlocal-to-local coupling techniques so as to improve the computational efficiency of using nonlocal models. This also motivates the development of new mathematical results -- for instance, a new trace theorem that extends the classical results. 
 
Although new nonlocal models have been gaining popularity in various applications, they often appear as phenomenological models, such as the peridynamics model in fracture mechanics. Here I will illustrate how to characterize the origin of nonlocality through homogenization of wave propagation in periodic media. 

Random matrix theory and supersymmetry techniques

Series
Job Candidate Talk
Time
Monday, January 6, 2020 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Tatyana ShcherbynaPrinceton University
Starting from the works of Erdos, Yau, Schlein with coauthors, significant progress in understanding universal behavior of many random graph and random matrix models were achieved. However for random matrices with a spatial structure, our understanding is still very limited.  In this talk I am going to overview applications of another approach to the study of the local eigenvalue statistics in random matrix theory based on so-called supersymmetry techniques (SUSY). The SUSY approach is based on the representation of the determinant as an integral over the Grassmann (anticommuting) variables. Combining this representation with the representation of an inverse determinant as an integral over the Gaussian complex field, SUSY allows to obtain an integral representation for the main spectral characteristics of random matrices such as limiting density, correlation functions, the resolvent's elements, etc. This method is widely (and successfully) used in the physics literature and is potentially very powerful but the rigorous control of the integral representations, which can be obtained by this method, is quite difficult, and it requires powerful analytic and statistical mechanics tools. In this talk we will discuss some recent progress in application of SUSY  to the analysis of local spectral characteristics of the prominent ensemble of random band matrices, i.e. random matrices whose entries become negligible if their distance from the main diagonal exceeds a certain parameter called the band width. 
 

Geometry and analysis of degenerating Calabi-Yau manifolds

Series
Job Candidate Talk
Time
Thursday, December 5, 2019 - 12:15 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Ruobing ZhangSUNY Stony Brook

This talk concerns a naturally occurring family of Calabi-Yau manifolds that degenerates in the sense of metric geometry, algebraic geometry and nonlinear PDE. A primary tool in analyzing their behavior is the recently developed regularity theory. We will give a precise description of arising singularities and explain possible generalizations. 

Pages