Seminars and Colloquia by Series

Boothby Wang Fibrations, K-Contact Structures and Regularity

Series
Geometry Topology Student Seminar
Time
Wednesday, October 24, 2018 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Surena Hozoori – Georgia Institute of Technology –
Boothby Wang fibrations are historically important examples of contact manifolds and it turns out that we can equip these contact manifolds with extra structures, namely K-contact structures. Based on the study of the relation of these examples and the regularity properties of the corresponding Reeb vector fields, works of Boothby, Wang, Thomas and Rukimbira gives a classification of K-contact structures.

On the fifth Busemann-Petty problem

Series
Analysis Seminar
Time
Wednesday, October 24, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Dmirty Ryabogin – Kent State University –
In 1956, Busemann and Petty posed a series of questions about symmetric convex bodies, of which only the first one has been solved.Their fifth problem asks the following.Let K be an origin symmetric convex body in the n-dimensional Euclidean space and let H_x be a hyperplane passing through the origin orthogonal to a unit direction x. Consider a hyperplane G parallel to H_x and supporting to K and let C(K,x)=vol(K\cap H_x)dist (0, G). (proportional to the volume of the cone spanned by the secion and the support point). If there exists a constant C such that for all directions x we have C(K,x)=C, does it follow that K is an ellipsoid?We give an affirmative answer to this problem for bodies sufficiently close to the Euclidean ball in the Banach Mazur distance.This is a joint work with Maria Alfonseca, Fedor Nazarov and Vlad Yaskin.

Introduction to geometric tomography

Series
High Dimensional Seminar
Time
Wednesday, October 24, 2018 - 12:55 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Dmitry Ryabogin – Kent State University –

We will discuss several open problems concerning unique determination of convex bodies in the n-dimensional Euclidean space given some information about their projections or sectionson all sub-spaces of dimension n-1. We will also present some related results.

Knot invariants and algebraic structures based on knots

Series
Research Horizons Seminar
Time
Wednesday, October 24, 2018 - 12:20 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Thang Le – Georgia Tech
A knot is a simple closed curve in the 3-space. Knots appeared as one of the first objects of study in topology. At first knot theory was rather isolated in mathematics. Lately due to newly discovered invariants and newly established connections to other branches of mathematics, knot theory has become an attractive and fertile area where many interesting, intriguing ideas collide. In this talk we discuss a new class of knot invariants coming out of the Jones polynomial and an algebra of surfaces based on knots (skein algebra) which has connections to many important objects including hyperbolic structures of surfaces and quantum groups. The talk is elementary.

Genuine Equivariant Operads

Series
Geometry Topology Seminar
Time
Monday, October 22, 2018 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Luis Alexandre Pereira – Georgia Tech
A fundamental result in equivariant homotopy theory due to Elmendorf states that the homotopy theory of G-spaces, with w.e.s measured on all fixed points, is equivalent to the homotopy theory of G-coefficient systems in spaces, with w.e.s measured at each level of the system. Furthermore, Elmendorf’s result is rather robust: analogue results can be shown to hold for, among others, the categories of (topological) categories and operads. However, it has been known for some time that in the G-operad case such a result does not capture the ”correct” notion of weak equivalence, a fact made particularly clear in work of Blumberg and Hill discussing a whole lattice of ”commutative operads with only some norms” that are not distinguished at all by the notion of w.e. suggested above. In this talk I will talk about part of a joint project which aims at providing a more diagrammatic understanding of Blumberg and Hill’s work using a notion of G-trees, which are a generalization of the trees of Cisinski-Moerdijk-Weiss. More specifically, I will describe a new algebraic structure, which we dub a ”genuine equivariant operad”, which naturally arises from the study of G-trees and which allows us to state the ”correct” analogue of Elmendorf’s theorem for G-operads.

The Fractional Laplacian: Approximation and Applications

Series
Applied and Computational Mathematics Seminar
Time
Monday, October 22, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Professor Hans-Werner van Wyk – Auburn University
The fractional Laplacian is a non-local spatial operator describing anomalous diffusion processes, which have been observed abundantly in nature. Despite its many similarities with the classical Laplacian in unbounded domains, its definition on bounded regions is more problematic. So is its numerical discretization. Difficulties arise as a result of the integral kernel's singularity at the origin as well as its unbounded support. In this talk, we discuss a novel finite difference method to discretize the fractional Laplacian in hypersingular integral form. By introducing a splitting parameter, we first formulate the fractional Laplacian as the weighted integral of a function with a weaker singularity, and then approximate it by a weighted trapezoidal rule. Our method generalizes the standard finite difference approximation of the classical Laplacian and exhibits the same quadratic convergence rate, for any fractional power in (0, 2), under sufficient regularity conditions. We present theoretical error bounds and demonstrate our method by applying it to the fractional Poisson equation. The accompanying numerical examples verify our results, as well as give additional insight into the convergence behavior of our method.

Invariant Manifolds Associated to Non-resonant Spectral Subspaces

Series
Dynamical Systems Working Seminar
Time
Friday, October 19, 2018 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 156
Speaker
Jiaqi Yang – GT Math
We show that, if the linearization of a map at a fixed point leaves invariant a spectral subspace, and some non-resonance conditions are satisfied. Then the map leaves invariant a smooth (as smooth as the map) manifold, which is unique among C^L invariant manifolds. Here, L only depends on the spectrum of the linearization. This is based on a work of Prof. Rafael de la Llave.

Nearly orthogonal vectors

Series
Combinatorics Seminar
Time
Friday, October 19, 2018 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Boris Bukh – Carnegie Mellon University
How can d+k vectors in R^d be arranged so that they are as close to orthogonal as possible? We show intimate connection of this problem to the problem of equiangular lines, and to the problem of bounding the first moment of isotropic measures. Using these connections, we pin down the answer precisely for several values of k and establish asymptotics for all k. Joint work with Chris Cox.

Hyperfields, Ordered Blueprints, and Moduli Spaces of Matroids

Series
Algebra Seminar
Time
Friday, October 19, 2018 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Matt Baker – Georgia Tech –
I will begin with a gentle introduction to hyperrings and hyperfields (originally introduced by Krasner for number-theoretic reasons), and then discuss a far-reaching generalization, Oliver Lorscheid’s theory of ordered blueprints. Two key examples of hyperfields are the hyperfield of signs S and the tropical hyperfield T. An ordering on a field K is the same thing as a homomorphism to S, and a (real) valuation on K is the same thing as a homomorphism to T. In particular, the T-points of an ordered blue scheme over K are closely related to Berkovich’s theory of analytic spaces.I will discuss a common generalization, in this language, of Descartes' Rule of Signs (which involves polynomials over S) and the theory of Newton Polygons (which involves polynomials over T). I will then introduce matroids over hyperfields (as well as certain more general kinds of ordered blueprints). Matroids over S are classically called oriented matroids, and matroids over T are also known as valuated matroids. I will explain how the theory of ordered blueprints and ordered blue schemes allow us to construct a "moduli space of matroids”, which is the analogue in the theory of ordered blue schemes of the usual Grassmannian variety in algebraic geometry. This is joint work with Nathan Bowler and Oliver Lorscheid.

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