Seminars and Colloquia by Series

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.

Algebraization theorems in p-adic geometry

Series
Job Candidate Talk
Time
Tuesday, January 30, 2024 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Abhishek OswalMichigan State University

In recent years, algebraization theorems arising from model theory, in particular o-minimality, have been a crucial ingredient in several breakthroughs in arithmetic geometry and Hodge theory. In this talk, I'll present some of my recent work on p-adic versions of these model theoretic algebraization criteria, with a focus on two different applications of this circle of ideas. The first being an algebraization theorem in the context of Shimura varieties, which are vaguely speaking moduli spaces of Hodge structures. The second being in the context of non-abelian Hodge theory, in the setting of moduli spaces of flat connections and local systems.

Zoom: https://gatech.zoom.us/j/95425627723

Heegaard Floer Homology and Closed Exotic 4-Manifolds

Series
Geometry Topology Seminar
Time
Monday, January 29, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Adam LevineDuke

We discuss new methods for using the Heegaard Floer homology of hypersurfaces to distinguish between smooth closed 4-manifolds that are homeomorphic but non-diffeomorphic. Specifically, for a 4-manifold X with b_1(X)=1, the minimum rank of the reduced Heegaard Floer homology of any embedded 3-manifold X representing a generator of H_1(X) gives a diffeomorphism invariant of X. We use this invariant to distinguish certain infinite families of exotic 4-manifolds that cannot be distinguished by previously known techniques. Using related ideas, we also provide the first known examples of (non-simply-connected) exotic 4-manifolds with negative definite intersection form. This is joint work with Tye Lidman and Lisa Piccirillo.

Polynomials with Lorentzian Signature over Cones, and Perron-Frobenius Theorem

Series
Algebra Seminar
Time
Monday, January 29, 2024 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Papri DeyGeorgia Tech

Please Note: There is no pre-seminar this time.

 The classical theorems of Perron and Frobenius, which explore spectral properties of nonnegative matrices, have been extensively examined and generalized from various perspectives, including a cone-theoretic (geometric) viewpoint. Concurrently, in the past decade, there has been a notable effort to fuse the techniques of algebraic geometry and combinatorics in an exploration of Lorentzian polynomials by Brändén and Huh, also known as completely log-concave polynomials (CLC) by Anari et.al. or strongly log-concave polynomials by Gurvits.

 

In this talk, I will discuss my ongoing joint work with Greg Blekherman regarding the class of polynomials with Lorentzian signature (PLS) defined over closed convex cones. This class encompasses various special polynomials, including Lorentzian polynomials over the nonnegative orthant and hyperbolic polynomials over hyperbolicity cones. We establish a compelling connection between PLS over a self-dual cone K and the generalized Perron Frobenius theorem over K. This connection enables us to provide an alternative necessary and sufficient condition to characterize the Lorentzian polynomials.

Exotic 4-Manifolds

Series
Geometry Topology Seminar Pre-talk
Time
Monday, January 29, 2024 - 12:45 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Adam LevineDuke

A central theme in 4-dimensional topology is the search for exotic 4-manifolds, i.e. families of smooth manifolds that are homeomorphic not diffeomorphic. We will survey some basic results in this area.

Chromatic quasisymmetric functions

Series
Combinatorics Seminar
Time
Friday, January 26, 2024 - 15:15 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Sarah MasonWake Forest University

 Every graph is associated to a symmetric function constructed from proper colorings of the graph.  The Stanley-Stembridge conjecture posits that the expansion of the chromatic symmetric function into the elementary symmetric functions has positive coefficients for a certain class of graphs.  We explore a potential new approach to the Stanley-Stembridge Conjecture using combinatorial objects called "special rim hooks" and connect this to the "chromatic quasisymmetric functions" introduced by Shareshian and Wachs as a generalization of chromatic symmetric functions.  This is joint work with Meagan Hodge.

Measure classification problems in smooth dynamics

Series
Job Candidate Talk
Time
Thursday, January 25, 2024 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006; Zoom streaming available
Speaker
Asaf KatzU Michigan

Please Note: Zoom link: https://gatech.zoom.us/j/98245747313?pwd=RmFtcmlWYjBncXJTOU00NFMvSVNsZz09 Meeting ID: 982 4574 7313 Passcode: SoM

Abstract: Classifying the invariant measures for a given dynamical system represents a fundamental challenge.

In the field of homogeneous dynamics, several important theorems give us an essentially complete picture. Moving away from homogeneous dynamics — results are more difficult to come byA recent development in Teichmuller dynamics — the celebrated magic wand theorem of Eskin–Mirzakhani, proved by their factorization technique gives one such example.
 
I will explain an implementation of the factorization technique by Eskin–Mirzakhani in smooth dynamics, aiming to classify u-Gibbs states for non-integrable Anosov actionsMoreover, I will try to explain some applications of the theorem, including a result of Avila–Crovosier–Eskin–Potrie–Wilkinson–Zhang towards Gogolev’s conjecture on actions over the 3D torus.

Positive curvature implies existence of isoperimetric sets?

Series
Analysis Seminar
Time
Wednesday, January 24, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Federico GlaudoPrinceton University

Over the past decade, a rich theory of existence for the isoperimetric problem in spaces of nonnegative curvature has been established by multiple authors.
We will briefly review this theory, with a special focus on the reasons why one may expect the isoperimetric problem to have a solution in any nonnegatively curved space: it is true for large enough volumes, it is true if the ambient is 2-dimensional, and it is true under appropriate assumptions on the ambient space at infinity.

The main topic of the talk will be the presentation of a counterexample to this "intuition": a 3-dimensional manifold of positive sectional curvature without isoperimetric sets for small volumes.
This is a joint work with G. Antonelli.

Three perspectives on B_3

Series
Geometry Topology Student Seminar
Time
Wednesday, January 24, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Akash NarayananGeorgia Tech

Braid groups are relatively simple to describe, but they have deep and intricate connections to many different areas of math. We will discuss three specific instances where the braid group on 3 strands arises in geometry and knot theory. In exploring connections between these perspectives, we will take a detour into the world of elliptic curves and their moduli space. As a result, we will see that these three perspectives are actually the same. Time permitting, we will explore generalizations of this to the braid group on n strands for n > 3.

Topology, geometry and adaptivity in soft and living matter

Series
Job Candidate Talk
Time
Tuesday, January 23, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Vishal PatilStanford University

Title: Topology, geometry and adaptivity in soft and living matter

Abstract:

Topology and adaptivity play fundamental roles in controlling the dynamics of biological and physical systems, from chromosomal DNA and biofilms to cilia carpets and worm collectives. How topological rules govern the self-adaptive dynamics of living matter remains poorly understood. Here we investigate the interplay between topology, geometry and reconfigurability in knotted and tangled matter. We first identify topological counting rules which predict the relative mechanical stability of human-designed knots, by developing a mapping between elastic knots and long-range ferromagnetic spin systems. Building upon this framework, we then examine the adaptive topological dynamics exhibited by California blackworms, which form living tangled structures in minutes but can rapidly untangle in milliseconds. Using blackworm locomotion datasets, we construct stochastic trajectory equations that explain how the dynamics of individual active filaments controls their emergent topological state. To further understand how tangled matter, along with more general biological networks, adapt to their surroundings, we introduce a theory of adaptive elastic networks which can learn mechanical information. By identifying how topology and adaptivity produce stable yet responsive structures, these results have applications in understanding broad classes of adaptive, self-optimizing biological systems.

 

Zoom: https://gatech.zoom.us/j/93619173236?pwd=ZGNRZUZ2emNJbG5pRzgzMnlFL1dzQT09

 

 

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