Seminars and Colloquia by Series

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 Jabin – University 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.

Small Ball Probability for the Smallest Singular Value of a Complex Random Matrix

Series
High Dimensional Seminar
Time
Wednesday, February 19, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Michail Sarantis – GeorgiaTech

Let $N_n$ be an $n\times n$ matrix whose entries are i.i.d. copies of a random variable $\zeta=\xi+i\xi'$, where $\xi,\xi'$ are i.i.d., mean zero, variance one, subgaussian random variables. We will present a result of Luh, according to which the probability that $N_n$ has a real eigenvalue is exponentially small in $n$. An interesting part of the proof is a small ball probability estimate for the smallest singular value of a complex perturbation $M_n=M+N_n$ of the original matrix.

Lifting Covers to Braided Embeddings

Series
Geometry Topology Student Seminar
Time
Wednesday, February 19, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Sudipta Kolay – Georgia Tech

An embedding of a manifold into a trivial disc bundle over another manifold is called braided if projection onto the first factor gives a branched cover. This notion generalizes closed braids in the solid torus, and gives an explicit way to construct many embeddings in higher dimensions. In this talk, we will discuss when a covering map of surfaces lift to a braided embedding.

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 Bernshteyn – Carnegie 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.

Dual spaces and Noetherian operators

Series
Student Algebraic Geometry Seminar
Time
Monday, February 17, 2020 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Marc Härkönen – Georgia Tech –

Noetherian operators are a set of differential operators that encode the scheme structure of a primary ideal. We propose a framework for studying primary ideals numerically by using a combination of witness sets and Noetherian operators. We will also present a method for computing Noetherian operators using numerical data. 

The first step in the theory of Noetherian operators are the Macaulay dual spaces. Indeed, for an ideal that is primary over a maximal ideal corresponding to a rational point, the generators of the dual space are a valid set of Noetherian operators. We will start by presenting basic ideas, results and algorithms in the classical dual space theory, and then revisit some of these ideas in the context of Noetherian operators.

Oriented Matroids and Combinatorial Neural Codes

Series
Algebra Seminar
Time
Monday, February 17, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Zvi Rosen – Florida Atlantic University

A combinatorial neural code is convex if it arises as the intersection pattern of convex open subsets of Euclidean space. We relate the emerging theory of convex neural codes to the established theory of oriented matroids, both categorically and with respect to feasibility and complexity. By way of this connection, we prove that all convex codes are related to some representable oriented matroid, and we show that deciding whether a neural code is convex is NP-hard.

Spaces of trees and fatgraphs for string topology and moduli spaces

Series
Geometry Topology Seminar
Time
Monday, February 17, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Kate Poirier – CUNY - City College of Technology –

Spaces of fatgraphs have long been used to study a variety of topics in math and physics. In this talk, we introduce two spaces of fatgraphs arising in string topology—one which parameterizes operations on chains of the free loop space of a manifold and one which parametrizes operations on Hochschild cochains of a “V-infinity” algebra. We present a conjecture relating these two spaces to one another and to the moduli space of Riemann surfaces. We also introduce polyhedra called “assocoipahedra” which generalize Stasheff’s associahedra to algebras with a compatible co-inner product. Assocoipahedra are used to prove that the dioperad governing V-infinity algebras satisfies certain algebraic properties. 

Structure-Preserving Numerical Method for Stochastic Nonlinear Schrodinger Equation

Series
Applied and Computational Mathematics Seminar
Time
Monday, February 17, 2020 - 13:50 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Cui, Jianbo – Georgia Tech math –

It's know that when discretizing stochastic ordinary equation with non-globally Lipschitz coefficient, the traditional numerical method, like
Euler method, may be divergent and not converge in strong or weak sense. For stochastic partial different equation with non-globally Lipschitz
coefficient, there exists fewer result on the strong and weak convergence results of numerical methods. In this talk, we will discuss several numerical schemes approximating stochastic Schrodinger Equation.  Under certain condition, we show that the exponential integrability preserving schemes are strongly and weakly convergent with positive orders.

Algebraic definitions for string topology

Series
Geometry Topology Seminar Pre-talk
Time
Monday, February 17, 2020 - 12:45 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Kate Poirier – CUNY - City College of Technology –

String topology studies various algebraic structures given by intersecting loops in a manifold, as well as those on the Hochschild chains or homology of an algebra. In this preparatory talk, we survey a collection of such structures and their relationships with one another.

A rigorous proof of Batchelor's law for passive scalar turbulence

Series
CDSNS Colloquium
Time
Monday, February 17, 2020 - 11:15 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Alex Blumenthal – University of Maryland and Georgia Tech –

Batchelor's law describes the power law spectrum of the turbulent regime of passive scalars (e.g., temperature or a dilute concentration of some tracer chemical) advected by an incompressible fluid (e.g., the Navier-Stokes equations at fixed Reynolds number), in the limit of vanishingly low molecular diffusivity. Predicted in 1959, it has been confirmed empirically in a variety of experiments, e.g. salinity concentrations among ocean currents. On the other hand, as with many turbulent regimes in physics, a true predictive theory from first principles has been missing (even a non-rigorous one), and there has been some controversy regarding the extent to which Batchelor's law is universal. 

 

In this talk, I will present a program of research, joint with Jacob Bedrossian (UMD) and Sam Punshon-Smith (Brown), which has rigorously proved Batchelor's law for passive scalars advected by the Navier-Stokes equations on the periodic box subjected to Sobolev regular, white-in-time body forces. The proof is a synthesis of techniques from dynamical systems and smooth ergodic theory, stochastics/probability, and fluid mechanics. To our knowledge, this work constitutes the first mathematically rigorous proof of a turbulent power law spectrum. It also establishes a template for predictive theories of passive scalar turbulence in more general settings, providing a strong argument for the universality of Batchelor's law. 

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