Seminars and Colloquia by Series

The Heilbronn triangle problem

Series
Additional Talks and Lectures
Time
Friday, September 20, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Cosmin PohoataEmory University

The Heilbronn triangle problem is a classical problem in discrete geometry with several old and new connections to various topics in extremal and additive combinatorics, graph theory, incidence geometry, harmonic analysis, and projection theory. In this talk, we will give an overview of some of these connections, and discuss some recent developments. Based on joint work with Alex Cohen and Dmitrii Zakharov.

Pseudo-Maximum Likelihood Theory for High-Dimension Rank-One Inference

Series
Stochastics Seminar
Time
Thursday, September 19, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Justin KoUniversity of Waterloo

We consider the task of estimating a rank-one matrix from noisy observations. Models that fall in this framework include community detection and spiked Wigner models. In this talk, I will discuss pseudo-maximum likelihood theory for such inference problems. We provide a variational formula for the asymptotic maximum pseudo-likelihood and characterize the asymptotic performance of pseudo maximum likelihood estimators. We will also discuss the implications of these findings to least squares estimators. Our approach uses the recent connections between statistical inference and statistical physics, and in particular the connection between the maximum likelihood and the ground state of a modified spin glass.

Based on joint work with Curtis Grant and Aukosh Jagannath.

Homology cobordism and Heegaard Floer homology

Series
School of Mathematics Colloquium
Time
Thursday, September 19, 2024 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Jen HomGeorgia Tech

Under the operation of connected sum, the set of three-manifolds form a monoid. Modulo an equivalence relation called homology cobordism, this monoid (of homology spheres) becomes a group. What is the structure of this group? What families of three-manifolds generate (or don’t generate) this group? We give some answers to these questions using Heegaard Floer homology. This is joint work with (various subsets of) I. Dai, K. Hendricks, M. Stoffregen, L. Truong, and I. Zemke.

Galois groups of reciprocal polynomials and the van der Waerden-Bhargava theorem

Series
Number Theory
Time
Wednesday, September 18, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Evan O'DorneyCarnegie Mellon University

Given a random polynomial f of degree n with integer coefficients each drawn uniformly and independently from an interval [-H, H], what is the probability that the Galois group of the roots of f is NOT the full symmetric group Sₙ? In 1936, van der Waerden conjectured that the answer should be of order 1/H, with the dominant contribution coming from f with a rational root. This conjecture was finally resolved by Bhargava in 2023. In this project (joint w/ Theresa Anderson), we ask the same question for reciprocal (a.k.a. palindromic) polynomials, which arise for instance as the characteristic polynomials of symplectic matrices. Using a suitably modified variant of the Fourier-analytic methods of Bhargava and others, we find that polynomials with non-generic Galois group appear with frequency O(log H/H) and, unlike in van der Waerden's setting, almost all of these polynomials are irreducible.

 On orientations of graphs with forbidden out-degrees (Owen Henderschedt, Auburn)

Series
Graph Theory Seminar
Time
Tuesday, September 17, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Owen HenderschedtAuburn University

 When does a graph admit an orientation with some desired properties? This question has been studied extensively for many years and across many different properties. Specifically, I will talk about properties having to do with degree restrictions, and progress towards a conjecture of Akbari, Dalirrooyfard, Ehsani, Ozeki, and Sherkati dealing with a list-type of degree restriction. This is all joint work with my PhD advisor Jessica McDonald.

Finite-time blowup for the Fourier-restricted Euler and hypodissipative Navier-Stokes model equations

Series
PDE Seminar
Time
Tuesday, September 17, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Evan MillerUniversity of Alabama in Huntsville

In this talk, I will introduce the Fourier-restricted Euler and hypodissipative Navier–Stokes equations. These equations are analogous to the Euler equation and hypodissipative Navier–Stokes equation, respectively, but with the Helmholtz projection replaced by a projection onto a more restrictive constraint space. The nonlinear term arising from the self-advection of velocity is otherwise unchanged. I will prove finite time-blowup when the dissipation is weak enough, by making use of a permutation symmetric Ansatz that allows for a dyadic energy cascade of the type found in the Friedlander-Katz-Pavlović dyadic Euler/Navier–Stokes model equation.

Maximal volume matrix cross approximation for image compression and least squares solution

Series
Applied and Computational Mathematics Seminar
Time
Monday, September 16, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Zhaiming ShenGeorgia Tech

We study the classic matrix cross approximation based on the maximal volume submatrices. Our main results consist of an improvement of the classic estimate for matrix cross approximation and a greedy approach for finding the maximal volume submatrices. More precisely, we present a new proof of the classic estimate of the inequality with an improved constant. Also, we present a family of greedy maximal volume algorithms to improve the computational efficiency of matrix cross approximation. The proposed algorithms are shown to have theoretical guarantees of convergence. Finally, we present two applications: image compression and the least squares approximation of continuous functions. Our numerical results demonstrate the effective performance of our approach.

More homology cobordism invariants

Series
Geometry Topology Seminar
Time
Monday, September 16, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jen HomGeorgia Tech

We begin with a survey of some Floer-theoretic knot concordance and homology cobordism invariants. Building on these ideas, we describe a new family of homology cobordism invariants and give a new proof that there are no 2-torsion elements with Rokhlin invariant 1. This is joint work in progress with Irving Dai, Matt Stoffregen, and Linh Truong.

Orlik-Terao algebras and internal zonotopal algebras

Series
Algebra Seminar
Time
Monday, September 16, 2024 - 11:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Colin CrowleyUniversity of Oregon

Please Note: There will be a pre-seminar at 10:50am in Skiles 005.

In 2017 Moseley, Proudfoot, and Young conjectured that the reduced Orlik-Terao algebra of the braid matroid was isomorphic as a symmetric group representation to the cohomology of a certain configuration space. This was proved by Pagaria in 2022. We generalize Pagaria's result from the braid arrangement to arbitrary hyperplane arrangements and recover a new proof in the case of the braid arrangement. Along the way, we give formulas for several other invariants of a hyperplane arrangement. Joint with Nick Proudfoot.

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