Seminars and Colloquia by Series

Contact structures, open books, and convex surfaces

Series
Geometry Topology Seminar Pre-talk
Time
Monday, September 25, 2023 - 12:45 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Joseph BreenUniversity of Iowa

This talk will include background information on contact structures and open book decompositions of 3-manifolds and the relationship between them. I will state the necessary definitions and include a number of concrete examples. I will also review some convex surface theory, which is the tool used in the main talk to investigate the contact structure – open book relationship.

Phase-shifted, exponentially small nanopterons in a model of KdV coupled to an oscillatory field

Series
CDSNS Colloquium
Time
Friday, September 22, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 249
Speaker
Tim FaverKennesaw State University

We develop nanopteron solutions for a coupled system of singularly perturbed ordinary differential equations.  To leading order, one equation governs the traveling wave profile for the Korteweg-de Vries (KdV) equation, while the other models a simple harmonic oscillator whose small mass is the problem’s natural small parameter.  A nanopteron solution consists of the superposition of an exponentially localized term and a small-amplitude periodic term.  We construct two families of nanopterons.  In the first, the periodic amplitude is fixed to be exponentially small but nonzero, and an auxiliary phase shift is introduced in the periodic term to meet a hidden solvability condition lurking within the problem.  In the second, the phase shift is fixed as a (more or less) arbitrary value, and now the periodic amplitude is selected to satisfy the solvability condition.  These constructions adapt different techniques due to Beale and Lombardi for related systems and is intended as the first step in a broader program uniting the flexible framework of Beale’s methods with the precision of Lombardi’s for applications to various problems in lattice dynamical systems.  As a more immediate application, we use the results for the model problem to solve a system of coupled KdV-KdV equations that models the propagation of certain surface water waves.

Electromagnetism and Falling Cats

Series
Geometry Topology Working Seminar
Time
Friday, September 22, 2023 - 14:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 006
Speaker
Daniel IrvineGeorgia Institute of Technology

In this talk I will develop a parallel between the classical field theory of electromagnetism and geometric mechanics of animal locomotion. I will illustrate this parallel using some informative examples from the two disciplines. In the realm of electromagnetism, we will investigate the magnetic monopole, as classically as possible. In the realm of animal locomotion, we will investigate the aphorism that a cat dropped (from a safe height) upside-down always lands on her feet. It turns out that both of these phenomena are caused by the presence of non-trivial topology.

No prior knowledge of classical field theory will be assumed, and this talk may continue into a second session at a later date.

k-Blocks and forbidden induced subgraphs

Series
Colloquia
Time
Thursday, September 21, 2023 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Maria ChudnovskyPrinceton University

Atlanta Combinatorics Colloquium Hosted by Georgia Tech

A k-block in a graph is a set of k vertices every two of which are joined by k vertex disjoint paths. By a result of Weissauer, graphs with no k-blocks admit tree-decompositions with especially useful structure. While several constructions show that it is probably very difficult to characterize induced subgraph obstructions to bounded tree width, a lot can be said about graphs with no k-blocks. On the other hand, forbidding induced subgraphs places significant restrictions on the structure of a k-block in a graph. We will discuss this phenomenon and its consequences on the study of tree-decompositions in classes of graphs defined by forbidden induced subgraphs.

Curie-Weiss Model under $\ell^{p}$ constraint

Series
Stochastics Seminar
Time
Thursday, September 21, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Daesung KimGeorgia Tech

We consider the Ising Curie-Weiss model on the complete graph constrained under a given $\ell_{p}$ norm. For $p=\infty$, it reduces to the classical Ising Curie-Weiss model. We prove that for all $p\ge 2$, there exists a critical inverse temperature $\beta_{c}(p)$ such that for $\beta<\beta_{c}(p)$, the magnetization is concentrated at zero and satisfies an appropriate Gaussian CLT. On the other hand, for $\beta>\beta_{c}(p)$, the magnetization is not concentrated at zero similar to the classical case. We further generalize the model for general symmetric spin distributions and prove similar phase transition. In this talk, we discuss a brief overview of classical Curie-Weiss model, a generalized Hubbard-Stratonovich transforms, and how we apply the transform to Curie-Weiss model under $\ell^p$ constraint. This is based on joint work with Partha Dey.

Magic functions for the Smyth-Siegel trace problem

Series
Number Theory
Time
Wednesday, September 20, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Naser SardariPenn State

We study the Schur-Siegel-Smyth trace problem. We introduce a new linear programming problem that inclues Smyths' constraints, and we give an exact solution to it. This improves the best known lower bound on the Siegel trace problem which is based on Smyths' method. In a special case, we recover Siegel's original upper bound.  Our method unifies Siegel's and Smyth's work under the same framework. This is joint work with Bryce Orloski.

Flag Hardy space theory—an answer to a question by E.M. Stein.

Series
Analysis Seminar
Time
Wednesday, September 20, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ji LiMacquarie University


In 1999, Washington University in Saint Louis hosted a conference on Harmonic Analysis to celebrate the 70th birthday of G. Weiss. In his talk in flag singular integral operators, E. M. Stein asked “What is the Hardy space theory in the flag setting?” In our recent paper, we characterise completely a flag Hardy space on the Heisenberg group. It is a proper subspace of the classical one-parameter Hardy space of Folland and Stein that was studied by Christ and Geller. Our space is useful in several applications, including the endpoint boundedness for certain singular integrals associated with the Sub-Laplacian on Heisenberg groups, and representations of flag BMO functions.

An introduction to Morse theory and Morse homology

Series
Geometry Topology Student Seminar
Time
Wednesday, September 20, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Akash NarayananGeorgia Institute of Technology

Morse theory analyzes the topology of a smooth manifold by studying the behavior of its real-valued functions. From this, one obtains a well-behaved homology theory which provides further information about the manifold and places constraints on the smooth functions it admits. This idea has proven to be useful in approaching topological problems, playing an essential role in Smale's solution to the generalized Poincare conjecture in dimensions greater than 4. Morse theory has been adapted to study complex manifolds, and even algebraic varieties over more general fields, but the underlying principles remain the same. In this talk, we will define the basic notions of Morse theory and describe some of the fundamental results. Then we will define Morse homology and discuss some important corollaries and applications. 

Exploiting low-dimensional structures in machine learning and PDE simulations

Series
PDE Seminar
Time
Tuesday, September 19, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Wenjing LiaoGeorgia Tech

Many data in real-world applications are in a high-dimensional space but exhibit low-dimensional structures. In mathematics, these data can be modeled as random samples on a low-dimensional manifold. I will talk about machine learning tasks like regression and classification, as well as PDE simulations. We consider deep learning as a tool to solve these problems. 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. Our results demonstrate that deep neural networks can utilize low-dimensional geometric structures of data in machine learning and PDE simulations.

Corks Equivalent to Fintushel-Stern Knot-Surgery

Series
Geometry Topology Seminar
Time
Monday, September 18, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Charles SteinNYU

Fintushel and Stern’s knot surgery constructions has been a central source of exotic 4-manifolds since its introduction in 1997. In the simply connected setting, it is known that there are also embedded corks in knot-surgered manifolds whose twists undo the knot surgery. This has been known abstractly since the construction was first given, but the explicit corks and embeddings have remained elusive. We will give an algorithmic process for transforming a generic Kirby diagram of a simply-connected knot surgered 4-manifold into one which contains an explicit cork whose twist undoes the surgery: answering the question. Along the way we will discuss $S^2\times S^2$-stable diffeomorphisms of knot-surgered 4-manifolds, and their relationship to the existence of corks.

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