Seminars and Colloquia by Series

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.

Elliptic surfaces from the perspective of Kirby Calculus

Series
Geometry Topology Seminar Pre-talk
Time
Monday, September 18, 2023 - 13:45 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Charles SteinNYU

Elliptic surfaces are some of the most well-behaved families of smooth, simply-connected four-manifolds from the geometric and analytic perspective. Many of their smooth invariants are easily computable and they carry a fibration structure which makes it possible to modify them by various surgical operations. However, elliptic surfaces have large Euler characteristics which means even their simplest handle-decompositions appear to be quite complicated. In this seminar, we will learn how to draw several different handle diagrams of elliptic surfaces which show explicitly many of their nice properties. This will allow us to see many useful properties of elliptic surfaces combinatorially, and gives insight into the constructions of their exotic smooth structures. 

Global Optimization of Analytic Functions over Compact Domains

Series
Algebra Seminar
Time
Monday, September 18, 2023 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Georgy ScholtenSorbonne Université

Please Note: There will be a pre-seminar (aimed toward grad students and postdocs) from 11:00 am-11:30 am in Skiles 006.

In this talk, we introduce a new method for minimizing analytic Morse functions over compact domains through the use of polynomial approximations. This is, in essence, an effective application of the Stone-Weierstrass Theorem, as we seek to extend a local method to a global setting, through the construction of polynomial approximants satisfying an arbitrary set precision in L-infty norm. The critical points of the polynomial approximant are computed exactly, using methods from computer algebra. Our Main Theorem states probabilistic conditions for capturing all local minima of the objective function $f$ over the compact domain. We present a probabilistic method, iterative on the degree, to construct the lowest degree possible least-squares polynomial approximants of $f$ which attains a desired precision over the domain. We then compute the critical points of the approximant and initialize local minimization methods on the objective function $f$ at these points, in order to recover the totality of the local minima of $f$ over the domain.

An efficient way to discretize a sphere

Series
Combinatorics Seminar
Time
Friday, September 15, 2023 - 15:15 for 1 hour (actually 50 minutes)
Location
Skiles 308
Speaker
Galyna LivshytsGeorgia Tech

We discuss small-ball probability estimates of the smallest singular value of a rather general ensemble of random matrices which we call “inhomogeneous”. One of the novel ingredients of our family of universality results is an efficient discretization procedure, applicable under unusually mild assumption. Most of the talk will focus on explaining the ideas behind the proof of the first ingredient. Partially based on the joint work with Tikhomirov and Vershynin, and an ongoing joint work with Fernandez and Tatarko. We will also mention a related work on the cube minimal dispersion, joint with Litvak.

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.

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