Seminars and Colloquia by Series

Applications of Dynamical Belyi Maps

Series
Number Theory
Time
Wednesday, February 14, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Bella TobinAgnes Scott College

Unicritical polynomials, typically written in the form $z^d+c$, have been widely studied in arithmetic and complex dynamics and are characterized by their one finite critical point. The behavior of a map's critical points under iteration often determines the dynamics of the entire map. Rational maps where the critical points have a finite forward orbit are called post-critically finite (PCF), and these are of great interest in arithmetic dynamics. They are viewed as a dynamical analogue of abelian varieties with complex multiplication and often display interesting dynamical behavior. The family of (single-cycle normalized) dynamical Belyi polynomials have two fixed critical points, so they are PCF by construction, and these maps provide a new testing ground for conjectures and ideas related to post-critically finite polynomials. Using this family, we can begin to explore properties of polynomial maps with two critical points. In this talk we will discuss applications of this family in arithmetic dynamics; in particular, how this family can be used to determine more general reduction properties of PCF polynomials. 

The mysterious part of the fine curve graph

Series
Geometry Topology Student Seminar
Time
Wednesday, February 14, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Roberta ShapiroGeorgia Tech

The fine curve graph of a surface is a graph whose vertices are essential simple closed curves in the surface and whose edges connect disjoint curves. Following a rich history of hyperbolicity in various graphs based on surfaces, the fine curve was shown to be hyperbolic by Bowden–Hensel–Webb. Given how well-studied the curve graph and the case of “up to isotopy” is, we ask: what about the mysterious part of the fine curve graph not captured by isotopy classes? In this talk, we introduce the result that the subgraph of the fine curve graph spanned by curves in a single isotopy class is not hyperbolic; indeed, it contains a flat of EVERY dimension. Along the way, we will discuss how to not prove this theorem as we explore proofs of hyperbolicity of related complexes. This work is joint with Ryan Dickmann.

Logarithmic upper bounds in quantum transport for quasi-periodic Schroedinger operators.

Series
Math Physics Seminar
Time
Wednesday, February 14, 2024 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Matthew PowellSchool of Mathematics, Georgia Tech

Please Note: Available on zoom at: https://gatech.zoom.us/j/98258240051

We shall discuss the quantum dynamics associated with ergodic Schroedinger operators. Anderson localization (pure point spectrum with exponentially decaying eigenfunctions) has been obtained for a variety of ergodic operator families, but it is well known that Anderson localization is highly unstable and can also be destroyed by generic rank one perturbations. For quasiperiodic operators, it also sensitively depends on the arithmetic properties of the phase (a seemingly irrelevant parameter from the point of view of the physics of the problem) and doesn’t hold generically. These instabilities are also present for the physically relevant notion of dynamical localization.

In this talk we will introduce the notion of the transport exponent, explain its stability, and explain how logarithmic upper bounds may be obtained in the quasi-periodic setting for all relevant parameters. This is based on joint work with S. Jitomirskaya.

Combinatoric derivations in extremal graph theory and Sidorenko's conjecture (Daniel Brosch, University of Klagenfurt)

Series
Graph Theory Seminar
Time
Tuesday, February 13, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Daniel BroschUniversity of Klagenfurt

Sidorenko's conjecture can be formulated as "Let $H$ be a bipartite graph, and $\rho\in [0,1]$. Of all the graphs with edge density $\rho$, the graph(-limit) obtained by picking edges uniformly at random minimizes the homomorphism density of $H$." This conjecture, first formulated in 1991 by Sidorenko, has received considerable attention over the last decades, and yet remains open in the general case.
 
It was shown recently [Blekherman, Raymond, Singh, Thomas, 2020] that sums-of-squares in Razborov's flag algebra are not strong enough to prove even small, known cases of the conjecture. To circumvent this, we introduce a novel kind of derivation of flags. Due to their combinatoric nature, we can use them to systematically gain knowledge on global minimizers of problems in extremal graph theory. We combine them with the flag algebra method to find new proofs for various cases of Sidorenko's conjecture. 

Optimal localization for the Einstein constraints

Series
PDE Seminar
Time
Tuesday, February 13, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Philippe G. LeFlochSorbonne University and CNRS

I will discuss a nonlinear elliptic system of partial differential equations arising in Riemannian geometry and General Relativity. Specifically, I will present recent advances on the analysis of asymptotically Euclidean, initial data sets for Einstein’s field equations. In collaboration with Bruno Le Floch (Sorbonne University) I proved that solutions to the Einstein constraints can be glued together along possibly nested conical domains. The constructed solutions may have arbitrarily low decay at infinity, while enjoying (super-)harmonic estimates within possibly narrow cones at infinity. Importantly, our localized seed-to-solution method, as we call it, leads to a proof of a conjecture by Alessandro Carlotto and Richard Schoen on the localization problem at infinity, and generalize P. LeFloch and Nguyen’s theorem on the asymptotic localization problem. This lecture will be based on https://arxiv.org/abs/2312.17706

Transferable Neural Networks for Partial Differential Equations

Series
Applied and Computational Mathematics Seminar
Time
Monday, February 12, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005 and https://gatech.zoom.us/j/98355006347
Speaker
Lili JuUniversity of South Carolina

Transfer learning for partial differential equations (PDEs) is to develop a pre-trained neural network that can be used to solve a wide class of PDEs. Existing transfer learning approaches require much information about the target PDEs such as its formulation and/or data of its solution for pre-training. In this work, we propose to design transferable neural feature spaces for the shallow neural networks from purely function approximation perspectives without using PDE information. The construction of the feature space involves the re-parameterization of the hidden neurons and uses auxiliary functions to tune the resulting feature space. Theoretical analysis shows the high quality of the produced feature space, i.e., uniformly distributed neurons. We use the proposed feature space as the predetermined feature space of a random feature model, and use existing least squares solvers to obtain the weights of the output layer. Extensive numerical experiments verify the outstanding performance of our method, including significantly improved transferability, e.g., using the same feature space for various PDEs with different domains and boundary conditions, and the superior accuracy, e.g., several orders of magnitude smaller mean squared error than the state of the art methods.

New algebraic invariants of Legendrian links

Series
Geometry Topology Seminar
Time
Monday, February 12, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Lenhard NgDuke

For the past 25 years, a key player in contact topology has been the Floer-theoretic invariant called Legendrian contact homology. I'll discuss a package of new invariants for Legendrian knots and links that builds on Legendrian contact homology and is derived from rational symplectic field theory. This includes a Poisson bracket on Legendrian contact homology and a symplectic structure on augmentation varieties. Time permitting, I'll also describe an unexpected connection to cluster theory for a family of Legendrian links associated to positive braids. Parts of this are joint work in progress with Roger Casals, Honghao Gao, Linhui Shen, and Daping Weng.

Determinantal zeros and factorization of noncommutative polynomials

Series
Algebra Seminar
Time
Monday, February 12, 2024 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Jurij VolčičDrexel University

Please Note: There will be a pre-seminar (aimed toward grad students and postdocs) from 11:00 am to 11:30am in Skiles 005.

Hilbert's Nullstellensatz about zero sets of polynomials is one of the most fundamental correspondences between algebra and geometry. More recently, there has been an emerging interest in polynomial equations and inequalities in several matrix variables, prompted by developments in control systems, quantum information theory, operator algebras and optimization. The arising problems call for a suitable version of (real) algebraic geometry in noncommuting variables; with this in mind, the talk considers matricial sets where noncommutative polynomials attain singular values, and their algebraic counterparts.

Given a polynomial f in noncommuting variables, its free (singularity) locus is the set of all matrix tuples X such that f(X) is singular.  The talk focuses on the interplay between geometry of free loci (irreducible components, inclusions, eigenlevel sets, smooth points) and factorization in the free algebra. In particular, a Nullstellensatz for free loci is given, as well as a noncommutative variant of Bertini's irreducibility theorem and its consequences.

Legendrian knots and contact homology in R^3

Series
Geometry Topology Seminar Pre-talk
Time
Monday, February 12, 2024 - 00:45 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Lenhard NgDuke

This will be an introduction to Legendrian contact homology (LCH), a version of Floer homology that's important in contact topology, for the setting of Legendrian knots in R^3 with the standard contact structure. LCH is the homology of a differential graded algebra that can be defined combinatorially in terms of a diagram for the knot. We'll explore this combinatorial definition, with examples, and discuss some auxiliary invariants derived from LCH. No background about contact manifolds or Legendrian knots will be assumed.

ε-series by James Anderson, Sean Kafer, and Tantan Dai

Series
Combinatorics Seminar
Time
Friday, February 9, 2024 - 15:15 for 1 hour (actually 50 minutes)
Location
Skiles 308
Speaker
James Anderson, Sean Kafer, Tantan DaiGeorgia Tech

James Anderson: Odd coloring (resp, PCF coloring) is a stricter form of proper coloring in which every nonisolated vertex is required to have a color in its neighborhood with odd multiplicity (resp, with multiplicity 1). Using the discharging method, and a new tool which we call the Forb-Flex method, we improve the bounds on the odd and PCF chromatic number of planar graphs of girth 10 and 11, respectively.

Sean Kafer: Many classical combinatorial optimization problems (e.g. max weight matching, max weight matroid independent set, etc.) have formulations as linear programs (LPs) over 0/1 polytopes on which LP solvers could be applied.  However, there often exist bespoke algorithms for these problems which, by merit of being tailored to a specific domain, are both more efficient and conceptually nicer than running a generic LP solver on the associated LP.  We will discuss recent results which show that a number of such algorithms (e.g. the shortest augmenting path algorithm, the greedy algorithm, etc.) can be "executed" by the Simplex method for solving LPs, in the sense that the Simplex method can be made to generate the same sequence of solutions when applied to the appropriate corresponding LP.

Tantan Dai: There has been extensive research on Latin Squares. It is simple to construct a Latin Square with n rows and n columns. But how do we define a Latin Triangle? What are the rows? When does a Latin Triangle exist? How can we construct them? In this talk, I will discuss two types of Latin Triangles and the construction of a countable family of Latin Triangles.

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