Seminars and Colloquia by Series

Conformal mappings and integrability of surface dynamics

Series
Math Physics Seminar
Time
Thursday, November 2, 2023 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 005 and online at https://gatech.zoom.us/j/99225468139
Speaker
Pavel LushnikovDepartment of Mathematics and Statistics, University of New Mexico

A fully nonlinear surface dynamics of the time dependent potential flow of ideal incompressible fluid with a free surface is considered in two dimensional geometry. Arbitrary large surface waves can be efficiently characterized through a time-dependent conformal mapping of a fluid domain into the lower complex half-plane. We reformulate the exact Eulerian dynamics through a non-canonical nonlocal Hamiltonian system for the pair of new conformal variables. We also consider a generalized hydrodynamics for two components of superfluid Helium which has the same non-canonical Hamiltonian structure. In both cases the fluid dynamics is fully characterized by the complex singularities in the upper complex half-plane of the conformal map and the complex velocity. Analytical continuation through the branch cuts generically results in the Riemann surface with infinite number of sheets including Stokes wave, An infinite family of solutions with moving poles are found on the Riemann surface. Residues of poles are the constants of motion. These constants commute with each other in the sense of underlying non-canonical Hamiltonian dynamics which provides an argument in support of the conjecture of complete Hamiltonian integrability of surface dynamics. If we consider initial conditions with short branch cuts then fluid dynamics is reduced to the complex Hopf equation for the complex velocity coupled with the complex transport equation for the conformal mapping. These equations are fully integrable by characteristics producing the infinite family of solutions, including the pairs of moving square root branch points. The solutions are compared with the simulations of the full Eulerian dynamics giving excellent agreement.

Estimation and Inference in Tensor Mixed-Membership Blockmodels

Series
Stochastics Seminar
Time
Thursday, November 2, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Joshua AgterbergUniversity of Pennsylvania

Higher-order multiway data is ubiquitous in machine learning and statistics and often exhibits community-like structures, where each component (node) along each different mode has a community membership associated with it. In this talk we propose the tensor mixed-membership blockmodel, a generalization of the tensor blockmodel positing that memberships need not be discrete, but instead are convex combinations of latent communities. We first study the problem of estimating community memberships, and we show that a tensor generalization of a matrix algorithm can consistently estimate communities at a rate that improves relative to the matrix setting, provided one takes the tensor structure into account. Next, we study the problem of testing whether two nodes have the same community memberships, and we show that a tensor analogue of a matrix test statistic can yield consistent testing with a tighter local power guarantee relative to the matrix setting. If time permits we will also examine the performance of our estimation procedure on flight data. This talk is based on two recent works with Anru Zhang.

Exploiting low-dimensional data structures in deep learning

Series
School of Mathematics Colloquium
Time
Thursday, November 2, 2023 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Wenjing LiaoGeorgia Tech

In the past decade, deep learning has made astonishing breakthroughs in various real-world applications. It is a common belief that deep neural networks are good at learning various geometric structures hidden in data sets, such as rich local regularities, global symmetries, or repetitive patterns. One of the central interests in deep learning theory is to understand why deep neural networks are successful, and how they utilize low-dimensional data structures. In this talk, I will present some statistical learning theory of deep neural networks where data exhibit low-dimensional structures, such as lying on a low-dimensional manifold. The learning tasks include regression, classification, feature representation and operator learning. When data are sampled on a low-dimensional manifold, the sample complexity crucially depends on the intrinsic dimension of the manifold instead of the ambient dimension of the data. These results demonstrate that deep neural networks are adaptive to low-dimensional geometric structures of data sets.

Vanishing of Brauer classes on K3 surfaces under reduction

Series
Number Theory
Time
Wednesday, November 1, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Salim TayouHarvard University

Given a Brauer class on a K3 surface over a number field, we prove that there exists infinitely many primes where the reduction of the Brauer class vanishes, under some mild assumptions. This answers a question of Frei--Hassett--Várilly-Alvarado. The proof uses Arakelov intersection theory on GSpin Shimura varieties. If time permits, I will explain some applications to rationality questions. The results in this talk are joint work with Davesh Maulik.

Higher dimensional fractal uncertainty

Series
Analysis Seminar
Time
Wednesday, November 1, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Alex CohenMIT

The fractal uncertainty principle (FUP) roughly says that a function and its Fourier transform cannot both be concentrated on a fractal set. These were introduced to harmonic analysis in order to prove new results in quantum chaos: if eigenfunctions on hyperbolic manifolds concentrated in unexpected ways, that would contradict the FUP. Bourgain and Dyatlov proved FUP over the real numbers, and in this talk I will discuss an extension to higher dimensions. The bulk of the work is constructing certain plurisubharmonic functions on C^n. 

Classifying Legendrian Positive Torus Knots

Series
Geometry Topology Student Seminar
Time
Wednesday, November 1, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Tom RodewaldGeorgia Tech

Legendrian knots are an important kind of knot in contact topology. One of their invariants,  the Thurston-Bennequin number, has an upper bound for any given knot type, called max-tb. Using convex surface theory, we will compute the max-tb of positive torus knots and show that two max-tb positive torus knots are Legendrian isotopic. If time permits, we will show that any non max-tb positive torus knot is obtained from the unique max-tb positive torus knot by a sequence of stabilizations. 

The Erdős-Szekeres problem

Series
Graph Theory Seminar
Time
Tuesday, October 31, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Cosmin PohoataEmory University

For every natural number n, if we start with sufficiently many points in R^d in general position there will always exist n points in convex position. The problem of determining quantitative bounds for this statement is known as the Erdős-Szekeres problem, and is one of the oldest problems in Ramsey theory. We will discuss some of its history, along with the recent developments in the plane and in higher dimensions.

Long time behavior in cosmological Einstein-Belinski-Zakharov spacetimes

Series
PDE Seminar
Time
Tuesday, October 31, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Online: https://gatech.zoom.us/j/95574359880?pwd=cGpCa3J1MFRkY0RUeU1xVFJRV0x3dz09
Speaker
Claudio MuñozUniversidad de Chile

In this talk I will present some recent results in collaboration with Jessica Trespalacios where we consider Einstein-Belinski-Zakharov spacetimes and prove local and global existence, long time behavior of possibly large solutions and some applications to gravisolitons of Kasner type.

The Burau representation and shapes of polyhedra by Ethan Dlugie

Series
Geometry Topology Seminar
Time
Monday, October 30, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Speaker

The Burau representation is a kind of homological representation of braid groups that has been around for around a century. It remains mysterious in many ways and is of particular interest because of its relation to quantum invariants of knots and links such as the Jones polynomial. In recent work, I came across a relationship between this representation and a moduli space of Euclidean cone metrics on spheres (think e.g. convex polyhedra) first examined by Thurston. After introducing the relevant definitions, I'll explain a bit about this connection and how I used the geometric structure on this moduli space to exactly identify the kernel of the Burau representation after evaluating its formal parameter at complex roots of unity. There will be many pictures!

Lie algebra representations, flag manifolds, and combinatorics. An old story with new twists

Series
Algebra Seminar
Time
Monday, October 30, 2023 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Cristian LenartSUNY Albany

Please Note: There will be a pre-seminar from 11am to 11:30am (aimed toward grad students and postdocs) in Skiles 006.

The connections between representations of complex semisimple Lie algebras and the geometry of the corresponding flag manifolds have a long history. Moreover, combinatorics plays an important role in the related computations. My talk is devoted to new aspects of this story. On the Lie algebra side, I consider certain modules for quantum affine algebras. I discuss their relationship with Macdonald polynomials, which generalize the irreducible characters of simple Lie algebras. On the geometric side, I consider the quantum K-theory of flag manifolds, which is a K-theoretic generalization of quantum cohomology. A new combinatorial model, known as the quantum alcove model, is also presented. The talk is based on joint work with S. Naito, D. Sagaki, A. Schilling, and M. Shimozono.

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