Seminars and Colloquia by Series

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.

Structure-Preserving Methods for Nonlinear Hyperbolic Waves

Series
Applied and Computational Mathematics Seminar
Time
Monday, February 5, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005 and https://gatech.zoom.us/j/98355006347
Speaker
Philippe G. LeFlochSorbonne University and CNRS

Many numerical methods have been developed in the past years for computing weak solutions (with shock waves) to nonlinear hyperbolic conservation laws. My research, specifically, concerns the design of well-balanced numerical algorithms that preserve certain key structure of these equations in various applications, including for problems involving moving phase boundaries and other scale-dependent interfaces. In particular, in this lecture, I will focus on the evolution of a compressible fluid in spherical symmetry on a Schwarzschild curved background, for which I have designed a class of well-balanced numerical algorithms up to third-order of accuracy. Both the relativistic Burgers-Schwarzschild model and the relativistic Euler-Schwarzschild model were considered, and the proposed numerical algorithm took advantage of the explicit or implicit forms available for the stationary solutions of these models. The schemes follow the finite volume methodology and preserve the stationary solutions and, most importantly, allow us to investigate the global asymptotic behavior of such flows and determine the asymptotic behavior of the mass density and velocity field of the fluid. Blog: philippelefloch.org

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.

Two short talks

Series
Algebra Seminar
Time
Monday, February 5, 2024 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
May Cai and Matt BakerGeorgia Tech

This special algebra seminar will feature short talks by our very own May Cai and Matt Baker, who will speak on the following topics: 

May Cai: The completion problem asks one to take a partial observation of some underlying object, and try to recover the original observation. Concretely, we have some object of interest, and a point in the image of that object under a projection map, and want to understand the fiber of this point under this map. In particular, for log-linear models, which are the restrictions of toric varieties to the probability simplex, under certain mild conditions, when this fiber is finite it turns out to have exactly either one or two entries. This is joint work with Cecilie Olesen Recke and Thomas Yahl.

Matt Baker: The determinant of a skew-symmetric matrix has a canonical square root given by the Pfaffian. Similarly, the resultant of two reciprocal polynomials of even degree has a canonical square root given by their reciprocant. Computing the reciprocant of two cyclotomic polynomials yields a short and elegant proof of the Law of Quadratic Reciprocity.

The Riesz Transform and Rectifiability of Measures

Series
Analysis Working Seminar
Time
Wednesday, January 31, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Ben Jaye

 This will be an expository talk which aims to introduce some problems in harmonic analysis and geometric measure theory concerning the geometry of a measure for which an associated integral operator is well behaved.  As an example, we shall prove a result of Mattila and Preiss concerning the relationship between the rectifiability of a measure and the existence of the Riesz transform in the sense of principle value.

Braid Groups are Linear

Series
Geometry Topology Student Seminar
Time
Wednesday, January 31, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jacob GuyneeGeorgia Tech

The Johnson filtration is a filtration of the mapping class group induced by the action of the mapping class group on the lower central series of the fundamental group of a surface.  A theorem of Johnson tells us that the first term of this filtration, called the Torelli group, is finitely generated for surfaces of genus at least 3.  We will explain work of Ershov-He and Church-Ershov-Putman, which uses Johnson's result to show that the kth term of the Johnson filtration is finitely generated for surfaces of genus g at least 2k - 1.  Time permitting, we will also discuss some extensions of these ideas.  In particular, we will explain how to show that the terms of the Johnson filtration are finitely presented assuming the Torelli group is finitely presented.

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