Seminars and Colloquia by Series

Inverse Problems, Imaging and Tensor Decomposition

Series
Job Candidate Talk
Time
Tuesday, March 3, 2020 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Joe KileelProgram in Applied and Computational Mathematics, Princeton University

Perspectives from numerical optimization and computational algebra are
brought to bear on a scientific application and a data science
application.  In the first part of the talk, I will discuss
cryo-electron microscopy (cryo-EM), an imaging technique to determine
the 3-D shape of macromolecules from many noisy 2-D projections,
recognized by the 2017 Chemistry Nobel Prize.  Mathematically, cryo-EM
presents a particularly rich inverse problem, with unknown
orientations, extreme noise, big data and conformational
heterogeneity. In particular, this motivates a general framework for
statistical estimation under compact group actions, connecting
information theory and group invariant theory.  In the second part of
the talk, I will discuss tensor rank decomposition, a higher-order
variant of PCA broadly applicable in data science.  A fast algorithm
is introduced and analyzed, combining ideas of Sylvester and the power
method.

Large stochastic systems of interacting particles

Series
Job Candidate Talk
Time
Thursday, February 20, 2020 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Pierre-Emmanuel JabinUniversity of Maryland, College Park

I will present some recent results, obtained with D. Bresch and Z. Wang, on large stochastic many-particle or multi-agent systems. Because such systems are conceptually simple but exhibit a wide range of emerging macroscopic behaviors, they are now employed in a large variety of applications from Physics (plasmas, galaxy formation...) to the Biosciences, Economy, Social Sciences...

The number of agents or particles is typically quite large, with 10^20-10^25 particles in many Physics settings for example and just as many equations. Analytical or numerical studies of such systems are potentially very complex  leading to the key question as to whether it is possible to reduce this complexity, notably thanks to the notion of propagation of chaos (agents remaining almost uncorrelated).

To derive this propagation of chaos, we have introduced a novel analytical method, which led to the resolution of two long-standing conjectures:
_The quantitative derivation of the 2-dimensional incompressible Navier-Stokes system from the point vortices dynamics;
_The derivation of the mean-field limit for attractive singular interactions such as in the Keller-Segel model for chemotaxis and some Coulomb gases.

Descriptive combinatorics and the probabilistic method

Series
Job Candidate Talk
Time
Tuesday, February 18, 2020 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Anton BernshteynCarnegie Mellon University (CMU)

Descriptive combinatorics studies the interaction between classical combinatorial concepts, such as graph colorings and matchings, and notions from measure theory and topology. Results in this area enable one to apply combinatorial techniques to problems in other (seemingly unrelated) branches of mathematics, such as the study of dynamical systems. In this talk I will give an introduction to descriptive combinatorics and discuss some recent progress concerning a particular family of combinatorial tools---the probabilistic method---and its applications in the descriptive setting.

Quasiperiodic Schrodinger operators: nonperturbative analysis of small denominators, universal self-similarity, and critical phenomena.

Series
Job Candidate Talk
Time
Tuesday, February 11, 2020 - 11:00 for 1 hour (actually 50 minutes)
Location
TBA
Speaker
Svetlana JitomirskayaUCI

We will give a brief introduction to the spectral theory of ergodic operators. Then we discuss several remarkable spectral phenomena present in the class of quasiperiodic operators, as well as the nonperturbative approach to small denominator problems that has been behind much of the related progress.  In particular, we will talk about the almost Mathieu (aka Harper's) operator - a model heavily studied in physics literature and linked to several Nobel prizes (in addition to one Fields medal). We will describe several results on this model that resolve some long-standing conjectures.

Characterizing Smoothness of Quotients

Series
Job Candidate Talk
Time
Monday, February 10, 2020 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Matthew SatrianoUniversity of Waterloo

Given an action of a finite group $G$ on a complex vector space $V$, the Chevalley-Shephard-Todd Theorem gives a beautiful characterization for when the quotient variety $V/G$ is smooth. In his 1986 ICM address, Popov asked whether this criterion could be extended to the case of Lie groups. I will discuss my contributions to this problem and some intriguing questions in combinatorics that this raises. This is based on joint work with Dan Edidin.

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.

Arithmetic, Geometry, and the Hodge and Tate Conjectures for self-products of some K3 surfaces

Series
Job Candidate Talk
Time
Monday, January 27, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jaclyn LangUniversité Paris 13

Although studying numbers seems to have little to do with shapes, geometry has become an indispensable tool in number theory during the last 70 years. Deligne's proof of the Weil Conjectures, Wiles's proof of Fermat's Last Theorem, and Faltings's proof of the Mordell Conjecture all require machinery from Grothendieck's algebraic geometry. It is less frequent to find instances where tools from number theory have been used to deduce theorems in geometry. In this talk, we will introduce one tool from each of these subjects -- Galois representations in number theory and cohomology in geometry -- and explain how arithmetic can be used as a tool to prove some important conjectures in geometry. More precisely, we will discuss ongoing joint work with Laure Flapan in which we prove the Hodge and Tate Conjectures for self-products of 16 K3 surfaces using arithmetic techniques.

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.