Seminars and Colloquia by Series

The profinite topology on a group

Series
Geometry Topology Seminar Pre-talk
Time
Monday, February 6, 2023 - 12:30 for 1 hour (actually 50 minutes)
Location
Speaker
Tam Cheetham-WestRice University

The finite index subgroups of a finitely presented group generate a topology on the group. We will discuss using examples how this relates to the organization of a group's finite quotients, and introduce the ideas of profinite rigidity and flexibility. 

Central Curve in Semidefinite Programming

Series
Algebra Seminar
Time
Monday, February 6, 2023 - 10:20 for 1.5 hours (actually 80 minutes)
Location
Skiles 005
Speaker
Isabelle ShankarPortland State University

The Zariski closure of the central path (which interior point algorithms track in convex optimization problems such as linear and semidefinite programs) is an algebraic curve, called the central curve. Its degree has been studied in relation to the complexity of these interior point algorithms.  We show that the degree of the central curve for generic semidefinite programs is equal to the maximum likelihood degree of linear concentration models.  This is joint work with Serkan Hoşten and Angélica Torres.

 

Sampling with Riemannian Hamiltonian Monte Carlo in a Constrained Space

Series
ACO Student Seminar
Time
Friday, February 3, 2023 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Yunbum KookGeorgia Tech CS

We demonstrate for the first time that ill-conditioned, non-smooth, constrained distributions in very high dimensions, upwards of 100,000, can be sampled efficiently in practice. Our algorithm incorporates constraints into the Riemannian version of Hamiltonian Monte Carlo and maintains sparsity. This allows us to achieve a mixing rate independent of condition numbers. On benchmark data sets from systems biology and linear programming, our algorithm outperforms existing packages by orders of magnitude. In particular, we achieve a 1,000-fold speed-up for sampling from the largest published human metabolic network (RECON3D). Our package has been incorporated into the COBRA toolbox. This is joint work with Yin Tat Lee, Ruoqi Shen, and Santosh Vempala.

On Extremal Polynomials: 4. Estimates of Chebyshev Numbers and Weakly Equilibrium Cantor-type Sets

Series
Mathematical Physics and Analysis Working Seminar
Time
Friday, February 3, 2023 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Burak HatinogluGeorgia Institute of Technology

We will continue to discuss lower and upper estimates of Widom factors. We will also introduce Cantor-type sets, constructed as the intersection of the level domains for simple sequences of polynomials. Using these Cantor-type sets we will prove some results on growth of Widom factors.

Nonsingular Poisson suspensions

Series
CDSNS Colloquium
Time
Friday, February 3, 2023 - 11:00 for 1 hour (actually 50 minutes)
Location
Online
Speaker
Oleksandr DanilenkoInstitute for Low Temperature Physics and Engineering

 https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT09

Let T be an invertible measure preserving transformation of a standard infinite measure space (X,m). Then a Poisson suspension (X*,m*,T*) of the dynamical system (X,m,T) is a well studied object in ergodic theory (especially for the last 20 years). It has physical applications as a model for the ideal gas consisting of countably many non-interacting particles. A natural problem is to develop a nonsingular counterpart of the theory of Poisson suspensions. The following will be enlightened in the talk:

--- description of the m-nonsingular (i.e. preserving the equivalence class of m) transformations T such that T* is m*-nonsingular
---algebraic and topological properties of the group of all m*-nonsingular Poisson suspensions
--- an interplay between dynamical properties of T and T*
--- an example of a "phase transition" in the ergodic properties of T* depending on the scaling of m
--- applications to Kazhdan property (T), stationary (nonsingular) group actions and the Furstenberg entropy.

(joint work with Z. Kosloff and E. Roy)

 

Sets of non-Lyapunov behaviour for transfer matrices of Schroedinger operators

Series
Math Physics Seminar
Time
Thursday, February 2, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
ONLINE and Skiles room 005
Speaker
Sasha SodinQueen Mary University of London

We shall discuss the asymptotics of singular values of the transfer matrices of ergodic Schroedinger and block-Schroedinger  operators. At a fixed value of the spectral parameter, the logarithmic asymptotics is almost surely given by the Lyapunov exponents; however, this is not, in general, true simultaneously for all the values of the parameter.  We shall try to explain the importance of these sets in various problems of spectral theory, and then review some of the earlier works on the subject and present some new results. Based on joint work with I. Goldsheid.

This talk will be online.  Meeting ID: 919 5236 6315.  Pleas note the unusual time!

Continuous combinatorics and natural quasirandomness

Series
Job Candidate Talk
Time
Wednesday, February 1, 2023 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Leonardo CoreglianoInstitute for Advanced Study

The theory of graph quasirandomness studies graphs that "look like" samples of the Erdős--Rényi
random graph $G_{n,p}$. The upshot of the theory is that several ways of comparing a sequence with
the random graph turn out to be equivalent. For example, two equivalent characterizations of
quasirandom graph sequences is as those that are uniquely colorable or uniquely orderable, that is,
all colorings (orderings, respectively) of the graphs "look approximately the same". Since then,
generalizations of the theory of quasirandomness have been obtained in an ad hoc way for several
different combinatorial objects, such as digraphs, tournaments, hypergraphs, permutations, etc.

The theory of graph quasirandomness was one of the main motivations for the development of the
theory of limits of graph sequences, graphons. Similarly to quasirandomness, generalizations of
graphons were obtained in an ad hoc way for several combinatorial objects. However, differently from
quasirandomness, for the theory of limits of combinatorial objects (continuous combinatorics), the
theories of flag algebras and theons developed limits of arbitrary combinatorial objects in a
uniform and general framework.

In this talk, I will present the theory of natural quasirandomness, which provides a uniform and
general treatment of quasirandomness in the same setting as continuous combinatorics. The talk will
focus on the first main result of natural quasirandomness: the equivalence of unique colorability
and unique orderability for arbitrary combinatorial objects. Although the theory heavily uses the
language and techniques of continuous combinatorics from both flag algebras and theons, no
familiarity with the topic is required as I will also briefly cover all definitions and theorems
necessary.

This talk is based on joint work with Alexander A. Razborov.

Optimal control of stochastic delay differential equations

Series
PDE Seminar
Time
Tuesday, January 31, 2023 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Filippo de FeoPolitecnico di Milano

In this talk we will discuss an optimal control problem for stochastic differential delay equations. We will only consider the case with delays in the state. We will show how to rewrite the problem in a suitable infinite-dimensional Hilbert space. Then using the dynamic programming approach we will characterize the value function of the problem as the unique viscosity solution of an infinite dimensional Hamilton-Jacobi-Bellman equation.  We will discuss partial C^{1}-regularity of the value function. This regularity result is particularly interesting since it permits to construct a candidate optimal feedback map which may allow to find an optimal feedback control. Finally we will discuss some ideas about the case in which delays also appear in the controls.

This is a joint work with S. Federico and A. Święch.

Higher Complex Structures and Hitchin Components

Series
Geometry Topology Seminar
Time
Monday, January 30, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Alex NolteRice/Georgia Tech

A source of richness in Teichmüller theory is that Teichmüller spaces have descriptions both in terms of group representations and in terms of hyperbolic structures and complex structures. A program in higher-rank Teichmüller theory is to understand to what extent there are analogous geometric interpretations of Hitchin components. In this talk, we will give a natural description of the SL(3,R) Hitchin component in terms of higher complex structures as first described by Fock and Thomas. Along the way, we will describe higher complex structures in terms of jets and discuss intrinsic structural features of Fock-Thomas spaces.

Towards a theory of complexity of sampling, inspired by optimization

Series
Job Candidate Talk
Time
Monday, January 30, 2023 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006 and https://gatech.zoom.us/j/91578357941?pwd=QS9malIvMVJqaWhpT0xqdWtxMCs1QT09
Speaker
Sinho ChewiMIT

Sampling is a fundamental and widespread algorithmic primitive that lies at the heart of Bayesian inference and scientific computing, among other disciplines. Recent years have seen a flood of works aimed at laying down the theoretical underpinnings of sampling, in analogy to the fruitful and widely used theory of convex optimization. In this talk, I will discuss some of my work in this area, focusing on new convergence guarantees obtained via a proximal algorithm for sampling, as well as a new framework for studying the complexity of non-log-concave sampling.

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