Seminars and Colloquia by Series

Cohomology of moduli spaces of curves

Series
Algebra Seminar
Time
Monday, November 14, 2022 - 13:30 for 1 hour (actually 50 minutes)
Location
Clough 125 Classroom
Speaker
Sam Payne – The University of Texas, Austin –

Cohomology groups of moduli spaces of curves are fruitfully studied from several mathematical perspectives, including geometric group theory, stably homotopy theory, and quantum algebra.  Algebraic geometry endows these cohomology groups with additional structures (Hodge structures and Galois representations), and the Langlands program makes striking predictions about which such structures can appear.  In this talk, I will present recent results inspired by, and in some cases surpassing, such predictions.  These include the vanishing of odd cohomology on moduli spaces of stable curves in degrees less than 11, generators and relations for H^11, and new constructions of unstable cohomology on M_g.  


Based on joint work with Jonas Bergström and Carel Faber; with Sam Canning and Hannah Larson; with Melody Chan and Søren Galatius; and with Thomas Willwacher. 

Statistical and non-statistical dynamics in doubly intermittent maps

Series
CDSNS Colloquium
Time
Friday, November 11, 2022 - 15:30 for 1 hour (actually 50 minutes)
Location
In-person in Skiles 005
Speaker
Stefano Luzzatto – Abdus Salam International Centre for Theoretical Physics (ICTP) –

Please Note: In-person. Streaming available via zoom: Zoom link: https://us06web.zoom.us/j/83392531099?pwd=UHh2MDFMcGErbzFtMHBZTmNZQXM0dz09

 

We introduce a large family of one-dimensional full branch maps which generalize the classical “intermittency maps” by admitting two neutral fixed points and possibly also critical points and/or singularities. We study the statistical properties of these maps for various parameter values, including the existence (and non-existence) of physical measures, and their properties such as decay of correlations and limit theorems. In particular we describe a new mechanism for relatively persistent non-statistical chaotic dynamics. This is joint work with Douglas Coates and Muhammad Mubarak.

Self-Adjusting Data Structures

Series
ACO Student Seminar
Time
Friday, November 11, 2022 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Robert Tarjan – Princeton University –

Please Note: Robert Tarjan is the James S. McDonnell Distinguished University Professor of Computer Science at Princeton University. He has held academic positions at Cornell, Berkeley, Stanford, and NYU, and industrial research positions at Bell Labs, NEC, HP, Microsoft, and Intertrust Technologies. He has invented or co-invented many of the most efficient known data structures and graph algorithms. He was awarded the first Nevanlinna Prize from the International Mathematical Union in 1982 for “for outstanding contributions to mathematical aspects of information science,” the Turing Award in 1986 with John Hopcroft for “fundamental achievements in the design and analysis of algorithms and data structures,” and the Paris Kanellakis Award in Theory and Practice in 1999 with Daniel Sleator for the invention of splay trees. He is a member of the U.S. National Academy of Sciences, the U. S. National Academy of Engineering, the American Academy of Arts and Sciences, and the American Philosophical Society.

Data structures are everywhere in computer software.  Classical data structures are specially designed to make each individual operation fast.  A more flexible approach is to design the structure so that it adapts to its use.  This idea has produced data structures that perform well in practice and have surprisingly good performance guarantees.  In this talk I’ll review some recent work on such data structures, specifically on self-adjusting search trees and self-adjusting heaps.

Predicting The Weather, 4d-Var, Hybrid Tangent Linear Models, and JEDI

Series
Time
Friday, November 11, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
Online
Speaker
Christian Sampson – The Joint Center for Satellite Data Assimilation (JCSDA)

https://gatech.zoom.us/j/95197085752?pwd=WmtJUVdvM1l6aUJBbHNJWTVKcVdmdz09

Weather modeling in conjunction with Data Assimilation (DA) has proven to provide effective weather forecasts that can both help you plan your day to save your life. We often refer to the combination of weather models and DA as Numerical Weather Prediction (NWP). One of the most widely employed DA methods in NWP is a variational method called 4d-Var. In this method, a cost function involving the model background error and a series of observations over time is minimized to find the best initial condition from which to run your model so that model forecast is consistent with observations. 4d-Var has been shown to provide the most reliable weather forecasts to date, but is not without its pitfalls. In particular, 4d-Var depends heavily on a tangent linear model (TLM) and an adjoint to the tangent linear model. While conceptually simple, coding these two elements is extremely time intensive and difficult. A small change in the larger weather model can induce months of work on its TLM and adjoint delaying the benefits of improvements on the model side. In this talk I will introduce the 4d-var method in general and present work on a Hybrid Tangent Linear Model (HTLM) developed in [Payne 2021] which is aimed at improving TLMs as well as allowing the use of incomplete TLMs when model physics changes. I will also touch on the Joint Effort for Data Integration (JEDI) project which now includes an HTLM and how you can use JEDI for DA.

A Keller-Lieb-Thirring Inequality for Dirac operators.

Series
Math Physics Seminar
Time
Thursday, November 10, 2022 - 16:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Hanne Van Den Bosch – University of Chile –

The classical Keller-Lieb-Thirring inequality bounds the ground state energy of a Schrödinger operator by a Lebesgue norm of the potential. This problem can be rewritten as a minimization problem for the Rayleigh quotient over both the eigenfunction and the potential. It is then straightforward to see that the best potential is a power of the eigenfunction, and the optimal eigenfunction satisfies a nonlinear Schrödinger equation. 

This talk concerns the analogous question for the smallest eigenvalue in the gap of a massive Dirac operator. This eigenvalue is not characterized by a minimization problem. By using a suitable Birman-Schwinger operator, we show that for sufficiently small potentials in Lebesgue spaces, an optimal potential  and eigenfunction exists. Moreover, the corresponding eigenfunction solves a nonlinear Dirac equation.

This is joint work with Jean Dolbeaults, David Gontier and Fabio Pizzichillo

Join Zoom Meeting:  https://gatech.zoom.us/j/91396672718

Breaking the curse of dimensionality for boundary value PDE in high dimensions

Series
Stochastics Seminar
Time
Thursday, November 10, 2022 - 15:30 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Ionel Popescu – University of Bucharest and Simion Stoilow Institute of Mathematics –

Zoom link to the seminar: https://gatech.zoom.us/j/91330848866

I will show how to construct a numerical scheme for solutions to linear Dirichlet-Poisson boundary problems which does not suffer of the curse of dimensionality. In fact we show that as the dimension increases, the complexity of this  scheme increases only (low degree) polynomially with the dimension. The key is a subtle use of walk on spheres combined with a concentration inequality. As a byproduct we show that this result has a simple consequence in terms of neural networks for the approximation of the solution. This is joint work with Iulian Cimpean, Arghir Zarnescu, Lucian Beznea and Oana Lupascu.

Coprime matchings and lonely runners

Series
Colloquia
Time
Thursday, November 10, 2022 - 11:00 for
Location
Skiles 006
Speaker
Tom Bohman – Carnegie Mellon University –

Suppose n runners are running on a circular track of circumference 1, with all runners starting at the same time and place. Each runner proceeds at their own constant speed. We say that a runner is lonely at some point in time if the distance around the track to the nearest other runner is at least 1/n. For example, if there two runners then there will come a moment when they are at anitpodal points on the track, and at this moment both runners are lonely. The lonely runner conjecture asserts that for every runner there is a point in time when that runner is lonely. This conjecture is over 50 years old and remains widely open.

A coprime matching of two sets of integers is a matching that pairs every element of one set with a coprime element of the other set. We present a recent partial result on the lonely runner conjecture. Coprime matchings of intervals of integers play an central role in the proof of this result.

Joint work with Fei Peng

An introduction to Nonlinear Algebra

Series
Research Horizons Seminar
Time
Wednesday, November 9, 2022 - 12:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Papri Dey – Georgia Institute of Technology –
 Nonlinear algebra is a newly evolving field which borrows ideas from the various core areas of mathematics.
     In this talk, the theoretical and computational aspects of nonlinear algebra emerging from algebraic geometry, tropical geometry, tensor algebra, and semidefinite programming will be briefly discussed and demonstrated with examples.
     This talk is mainly based on the book "Invitation to Nonlinear Algebra" by Mateusz Michalek and Bernd Sturmfels.

 

Uniform linear inviscid damping near monotonic shear flows in the whole space

Series
PDE Seminar
Time
Tuesday, November 8, 2022 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Hao Jia – University of Minnesota –

In recent years tremendous progress was made in understanding the ``inviscid damping" phenomenon near shear flows and vortices, which are steady states for the 2d incompressible Euler equation, especially at the linearized level. However, in real fluids viscosity plays an important role. It is natural to ask if incorporating the small but crucial viscosity term (and thus considering the Navier Stokes equation in a high Reynolds number regime instead of Euler equations) could change the dynamics in any dramatic way. It turns out that for the perturbative regime near a spectrally stable monotonic shear flows in an infinite periodic channel (to avoid boundary layers and long wave instabilities), we can prove uniform-in-viscosity inviscid damping. The proof introduces techniques that provide a unified treatment of the classical Orr-Sommerfeld equation in a way analogous to Rayleigh equations. 

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