Seminars and Colloquia by Series

Optimal localization for the Einstein constraints

Series
PDE Seminar
Time
Tuesday, February 13, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Philippe G. LeFlochSorbonne University and CNRS

I will discuss a nonlinear elliptic system of partial differential equations arising in Riemannian geometry and General Relativity. Specifically, I will present recent advances on the analysis of asymptotically Euclidean, initial data sets for Einstein’s field equations. In collaboration with Bruno Le Floch (Sorbonne University) I proved that solutions to the Einstein constraints can be glued together along possibly nested conical domains. The constructed solutions may have arbitrarily low decay at infinity, while enjoying (super-)harmonic estimates within possibly narrow cones at infinity. Importantly, our localized seed-to-solution method, as we call it, leads to a proof of a conjecture by Alessandro Carlotto and Richard Schoen on the localization problem at infinity, and generalize P. LeFloch and Nguyen’s theorem on the asymptotic localization problem. This lecture will be based on https://arxiv.org/abs/2312.17706

New algebraic invariants of Legendrian links

Series
Geometry Topology Seminar
Time
Monday, February 12, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Lenhard NgDuke

For the past 25 years, a key player in contact topology has been the Floer-theoretic invariant called Legendrian contact homology. I'll discuss a package of new invariants for Legendrian knots and links that builds on Legendrian contact homology and is derived from rational symplectic field theory. This includes a Poisson bracket on Legendrian contact homology and a symplectic structure on augmentation varieties. Time permitting, I'll also describe an unexpected connection to cluster theory for a family of Legendrian links associated to positive braids. Parts of this are joint work in progress with Roger Casals, Honghao Gao, Linhui Shen, and Daping Weng.

Transferable Neural Networks for Partial Differential Equations

Series
Applied and Computational Mathematics Seminar
Time
Monday, February 12, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005 and https://gatech.zoom.us/j/98355006347
Speaker
Lili JuUniversity of South Carolina

Transfer learning for partial differential equations (PDEs) is to develop a pre-trained neural network that can be used to solve a wide class of PDEs. Existing transfer learning approaches require much information about the target PDEs such as its formulation and/or data of its solution for pre-training. In this work, we propose to design transferable neural feature spaces for the shallow neural networks from purely function approximation perspectives without using PDE information. The construction of the feature space involves the re-parameterization of the hidden neurons and uses auxiliary functions to tune the resulting feature space. Theoretical analysis shows the high quality of the produced feature space, i.e., uniformly distributed neurons. We use the proposed feature space as the predetermined feature space of a random feature model, and use existing least squares solvers to obtain the weights of the output layer. Extensive numerical experiments verify the outstanding performance of our method, including significantly improved transferability, e.g., using the same feature space for various PDEs with different domains and boundary conditions, and the superior accuracy, e.g., several orders of magnitude smaller mean squared error than the state of the art methods.

Determinantal zeros and factorization of noncommutative polynomials

Series
Algebra Seminar
Time
Monday, February 12, 2024 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Jurij VolčičDrexel University

Please Note: There will be a pre-seminar (aimed toward grad students and postdocs) from 11:00 am to 11:30am in Skiles 005.

Hilbert's Nullstellensatz about zero sets of polynomials is one of the most fundamental correspondences between algebra and geometry. More recently, there has been an emerging interest in polynomial equations and inequalities in several matrix variables, prompted by developments in control systems, quantum information theory, operator algebras and optimization. The arising problems call for a suitable version of (real) algebraic geometry in noncommuting variables; with this in mind, the talk considers matricial sets where noncommutative polynomials attain singular values, and their algebraic counterparts.

Given a polynomial f in noncommuting variables, its free (singularity) locus is the set of all matrix tuples X such that f(X) is singular.  The talk focuses on the interplay between geometry of free loci (irreducible components, inclusions, eigenlevel sets, smooth points) and factorization in the free algebra. In particular, a Nullstellensatz for free loci is given, as well as a noncommutative variant of Bertini's irreducibility theorem and its consequences.

Legendrian knots and contact homology in R^3

Series
Geometry Topology Seminar Pre-talk
Time
Monday, February 12, 2024 - 00:45 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Lenhard NgDuke

This will be an introduction to Legendrian contact homology (LCH), a version of Floer homology that's important in contact topology, for the setting of Legendrian knots in R^3 with the standard contact structure. LCH is the homology of a differential graded algebra that can be defined combinatorially in terms of a diagram for the knot. We'll explore this combinatorial definition, with examples, and discuss some auxiliary invariants derived from LCH. No background about contact manifolds or Legendrian knots will be assumed.

ε-series by James Anderson, Sean Kafer, and Tantan Dai

Series
Combinatorics Seminar
Time
Friday, February 9, 2024 - 15:15 for 1 hour (actually 50 minutes)
Location
Skiles 308
Speaker
James Anderson, Sean Kafer, Tantan DaiGeorgia Tech

James Anderson: Odd coloring (resp, PCF coloring) is a stricter form of proper coloring in which every nonisolated vertex is required to have a color in its neighborhood with odd multiplicity (resp, with multiplicity 1). Using the discharging method, and a new tool which we call the Forb-Flex method, we improve the bounds on the odd and PCF chromatic number of planar graphs of girth 10 and 11, respectively.

Sean Kafer: Many classical combinatorial optimization problems (e.g. max weight matching, max weight matroid independent set, etc.) have formulations as linear programs (LPs) over 0/1 polytopes on which LP solvers could be applied.  However, there often exist bespoke algorithms for these problems which, by merit of being tailored to a specific domain, are both more efficient and conceptually nicer than running a generic LP solver on the associated LP.  We will discuss recent results which show that a number of such algorithms (e.g. the shortest augmenting path algorithm, the greedy algorithm, etc.) can be "executed" by the Simplex method for solving LPs, in the sense that the Simplex method can be made to generate the same sequence of solutions when applied to the appropriate corresponding LP.

Tantan Dai: There has been extensive research on Latin Squares. It is simple to construct a Latin Square with n rows and n columns. But how do we define a Latin Triangle? What are the rows? When does a Latin Triangle exist? How can we construct them? In this talk, I will discuss two types of Latin Triangles and the construction of a countable family of Latin Triangles.

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An introduction to principal bundles and holonomy

Series
Geometry Topology Student Seminar
Time
Wednesday, February 7, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Dan IrvineGeorgia Tech

The concept of holonomy arises in many areas of mathematics, especially control theory. This concept is also related to the broader program of geometrization of forces in physics. In order to understand holonomy, we need to understand principal (fiber) bundles. In this talk I will explain U(1)-principal bundles by example. This explanation will be from the point-of-view of a geometer, but I will introduce the terminology of control theory. Finally, we will do a holonomy computation for a famous example of Aharonov and Bohm.

New lower bounds for sphere packings and independence sets via randomness

Series
Graph Theory Seminar
Time
Tuesday, February 6, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Marcus MichelenUniversity of Illinois, Chicago

We show new lower bounds for sphere packings in high dimensions and for independent sets in graphs with not too large co-degrees.  For dimension d, this achieves a sphere packing of density (1 + o(1)) d log d / 2^(d+1).  In general dimension, this provides the first asymptotically growing improvement for sphere packing lower bounds since Roger's bound of c*d/2^d in 1947.  The proof amounts to a random (very dense) discretization together with a new theorem on constructing independent sets on graphs with not too large co-degree.  Both steps will be discussed, and no knowledge of sphere packings will be assumed or required.  This is based on joint work with Marcelo Campos, Matthew Jenssen and Julian Sahasrabudhe.

Inviscid limit from Navier-Stokes to BV solutions of compressible Euler equations

Series
PDE Seminar
Time
Tuesday, February 6, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Geng ChenUniversity of Kansas

 In the realm of mathematical fluid dynamics, a formidable challenge lies in establishing inviscid limits from the Navier-Stokes equations to the Euler equations. The pursuit of solving this intricate problem, particularly concerning singular solutions, persists in both compressible and incompressible scenarios. In particular, compressible Euler equations are a typical system of hyperbolic conservation laws, whose solution forms shock waves in general.

 

In this talk, we will discuss the recent proof on the unique vanishing viscosity limit from Navier-Stokes equations to the BV solution of compressible Euler equations, for the general Cauchy Problem. Moreover, we extend our findings by establishing the well-posedness of such solutions within the broader class of inviscid limits of Navier-Stokes equations with locally bounded energy initial values.  This is a joint work with Kang and Vasseur, which can be found on arXiv:2401.09305.

 

The uniqueness and L2 stability of Euler equations, done by Chen-Krupa-Vasseur, will also be discussed in this talk.

Projective Rigidity of Circle Packings

Series
Geometry Topology Seminar
Time
Monday, February 5, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Mike WolfGeorgia Tech

We prove that the space of circle packings consistent with a given triangulation on a surface of genus at least two is projectively rigid, so that a packing on a complex projective surface is not deformable within that complex projective structure.  More broadly, we show that the space of circle packings is a (smooth)  submanifold within the space of complex projective structures on that surface.

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