Seminars and Colloquia by Series

An Interactive Introduction to Surface Bundles

Series
Geometry Topology Student Seminar
Time
Wednesday, September 13, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jaden WangGeorgia Tech

Surface bundles lie in the intersection of many areas of math: algebraic topology, 2–4 dimensional topology, geometric group theory, algebraic geometry, and even number theory! However, we still know relatively little about surface bundles, especially compared to vector bundles. In this interactive talk, I will present the general (and beautiful) fiber bundle theory, including characteristic classes, as a starting point, and you the audience will get to specialize the general theory to surface bundles, with rewards! The talk aims to be accessible to anyone who had exposure to algebraic topology. This is also part one of three talks about surface bundles I will give this semester.

Spectral stability for periodic waves in some Hamiltonian systems

Series
PDE Seminar
Time
Tuesday, September 12, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Atanas StefanovUniversity of Alabama at Birmingham

A lot of recent work in the theory of partial differential equations has focused on the existence and stability properties of special solutions for Hamiltonian PDE’s.  

We review some recent works (joint with Hakkaev and Stanislavova), for spatially periodic traveling waves and their stability properties. We concentrate on three examples, namely the Benney system, the Zakharov system and the KdV-NLS model. We consider several standard explicit solutions, given in terms of Jacobi elliptic functions. We provide explicit and complete description of their stability properties. Our analysis is based on the careful examination of the spectral properties of the linearized operators, combined with recent advances in the Hamiltonian instability index formalism.

Convexity and rigidity of hypersurfaces in Cartan-Hadamard manifolds

Series
Geometry Topology Seminar
Time
Monday, September 11, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Mohammad GhomiGeorgia Tech

We show that in Cartan-Hadamard manifolds M^n, n≥ 3, closed infinitesimally convex hypersurfaces S bound convex flat regions, if curvature of M^n vanishes on tangent planes of S. This encompasses Chern-Lashof characterization of convex hypersurfaces in Euclidean space, and some results of Greene-Wu-Gromov on rigidity of Cartan-Hadamard manifolds. It follows that closed simply connected surfaces in M^3 with minimal total absolute curvature bound Euclidean convex bodies, as stated by M. Gromov in 1985. The proofs employ the Gauss-Codazzi equations, a generalization of Schur comparison theorem to CAT(0) spaces, and other techniques from Alexandrov geometry outlined by A. Petrunin, including Reshetnyak’s majorization theorem, and Kirszbraun’s extension theorem.

The Principal Minor Map

Series
Algebra Seminar
Time
Monday, September 11, 2023 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Abeer Al AhmadiehGeorgia Tech

The principal minor map takes an n x n square matrix and maps it to the 2^n-length vector of its principal minors. In this talk, I will describe both the fiber and the image of this map. In 1986, Loewy proposed a sufficient condition for the fiber to be a single point up to diagonal equivalence. I will provide a necessary and sufficient condition for the fiber to be a single point. Additionally, I will describe the image of the space of complex matrices using a characterization of determinantal representations of multiaffine polynomials, based on the factorization of their Rayleigh differences. Using these techniques I will give equations and inequalities characterizing the images of the spaces of real and complex symmetric and Hermitian matrices. This is based on joint research with Cynthia Vinzant.

Introductions to convex sets in CAT(0) space

Series
Geometry Topology Seminar Pre-talk
Time
Monday, September 11, 2023 - 12:45 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Mohammad GhomiGeorgia Tech

A CAT(0) space is a geodesic metric space where triangles are thinner than comparison triangles in a Euclidean plane. Prime examples of CAT(0) spaces are Cartan-Hadamard manifolds: complete simply connected Riemannian spaces with nonpositive curvature, which include Euclidean and Hyperbolic space as special cases. The triangle condition ensures that every pair of points in a CAT(0) space can be connected by a unique geodesic. A subset of a CAT(0) space is convex if it contains the geodesic connecting every pair of its points. We will give a quick survey of classical results in differential geometry on characterization of convex sets, such the theorems of Hadamard and  of Chern-Lashof, and also cover other background from the theory of CAT(0) spaces and Alexandrov geometry, including the rigidity theorem of Greene-Wu-Gromov, which will lead to the new results in the second talk.
 

Chip-firing, served three ways

Series
Algebra Student Seminar
Time
Friday, September 8, 2023 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Daniel HwangGeorgia Tech
Chip-firing asks a simple question: Given a group of people and
an initial integer distribution of dollars among the people including people
in debt, can we redistribute the money so that no one ends up in debt? This
simple question with its origins in combinatorics can be reformulated using
concepts from linear algebra, graph theory, and even divisors in Riemann
surfaces. In this expository presentation, we will cover the original chip-
firing problem, along with three different approaches to solving this problem:
utilizing the Laplacian, Dhar’s algorithm, and a graph-theoretic version of

the Riemann-Roch theorem by Baker and Norine.

Spectral clustering in the geometric block model

Series
Stochastics Seminar
Time
Thursday, September 7, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Shuangping LiStanford

Gaussian mixture block models are distributions over graphs that strive to model modern networks: to generate a graph from such a model, we associate each vertex with a latent feature vector sampled from a mixture of Gaussians, and we add edge if and only if the feature vectors are sufficiently similar. The different components of the Gaussian mixture represent the fact that there may be different types of nodes with different distributions over features---for example, in a social network each component represents the different attributes of a distinct community. Natural algorithmic tasks associated with these networks are embedding (recovering the latent feature vectors) and clustering (grouping nodes by their mixture component).

In this talk, we focus on clustering and embedding graphs sampled from high-dimensional Gaussian mixture block models, where the dimension of the latent feature vectors goes to infinity as the size of the network goes to infinity. This high-dimensional setting is most appropriate in the context of modern networks, in which we think of the latent feature space as being high-dimensional. We analyze the performance of canonical spectral clustering and embedding algorithms for such graphs in the case of 2-component spherical Gaussian mixtures and begin to sketch out the information-computation landscape for clustering and embedding in these models.

This is based on joint work with Tselil Schramm.

On displacement concavity of the relative entropy

Series
Analysis Seminar
Time
Wednesday, September 6, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Liran RotemTechnion

It is known for many years that various inequalities in convex geometry have information-theoretic analogues. The most well known example is the Entropy power inequality which corresponds to the Brunn-Minkowski inequality, but the theory of optimal transport allows to prove even better analogues. 

At the same time, in recent years there is a lot of interest in the role of symmetry in Brunn-Minkowski type inequalities. There are many open conjectures in this direction, but also a few proven theorems such as the Gaussian Dimensional Brunn-Minkowski inequality. In this talk we will discuss the natural question — do the known information-theoretic inequalities similarly improve in the presence of symmetry?  I will present some cases where the answer is positive together with some open problems. 

Based on joint work with Gautam Aishwarya. 

Spheres can knot in 4 dimensions

Series
Geometry Topology Student Seminar
Time
Wednesday, September 6, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Sean EliGeorgia Tech

You are probably familiar with the concept of a knot in 3 space: a tangled string that can't be pushed and pulled into an untangled one. We briefly show how to prove mathematical knots are in fact knotted, and discuss some conditions which guarantee unknotting. We then give explicit examples of knotted 2-spheres in 4 space, and discuss 2-sphere version of the familiar theorems. A large part of the talk is practice with visualizing objects in 4 dimensional space. We will also prove some elementary facts to give a sense of what working with these objects feels like. Time permitting we will describe know knotted 2-spheres were used to give evidence for the smooth 4D Poincare conjecture, one of the guiding problems in the field.

Selection of standing waves at small energy for NLS with a trapping potential in 1 D

Series
PDE Seminar
Time
Tuesday, September 5, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Online: https://gatech.zoom.us/j/95574359880?pwd=cGpCa3J1MFRkY0RUeU1xVFJRV0x3dz09
Speaker
Scipio CuccagnaUniversita` di Trieste

Due to linear superposition, solutions of a Linear Schrodinger Equation with a trapping potential,  produce a discrete  quasiperiodic part. When  a nonlinear perturbation is turned on,  it is known in principle, and proved in various situations,  that at small energies there is a phenomenon of standing wave selection where, up to radiation,  quasiperiodicity breaks down and there is convergence to a periodic wave.  We will discuss  this phenomenon in 1 D, where cubic nonlinearities are long range perturbations of the linear equations. Our aim is to show that a very effective framework to see these phenomena is provided by   a combination of the dispersion theory of  Kowalczyk, Martel and Munoz  along with  Maeda's  notion of Refined Profile.

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