Seminars and Colloquia by Series

Generalized Olson-Zalik Conjecture

Series
Analysis Seminar
Time
Wednesday, April 2, 2025 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Pu-Ting YuUniversity of Oregon

In 1992, Olson and Zalik conjectured that no system of translates can be a Schauder basis for L^2(R). This conjecture remains open as of the time of writing. Although some partial results regarding Olson-Zalik conjecture have been proved to be true, a characterization of subspaces of L^2(R) that do not admit a Schauder basis, or an unconditional basis is still unknown. 

In this talk, we will begin with a brief introduction to Olson-Zalik conjecture including its recent development. Then we will show that a family of modulation spaces do not admit unconditional bases formed by a system of translates. This observation led us to the following generalized Olson-Zalik conjecture ``Assume X is a separable Banach space that is continuously embedded into L^2(R). Then X does not admit a Schauder basis of translates if it is closed under Fourier transform". Finally, we close this talk by showing that if a closed subspace of L^2(R) is closed under Fourier transform, then it does not admit a Schauder basis of certain translates.

Manin's conjecture for Châtelet surfaces

Series
Athens-Atlanta Number Theory Seminar
Time
Tuesday, April 1, 2025 - 17:15 for 1 hour (actually 50 minutes)
Location
Skiles 314
Speaker
Katherine WooPrinceton University

We resolve Manin's conjecture for all Châtelet surfaces over $\mathbb{Q}$ (surfaces given by equations of the form x^2 + ay^2 = f(z)) -- we establish asymptotics for the number of rational points of increasing height. The key analytic ingredient is estimating sums of Fourier coefficients of modular forms along polynomial values.

Hilbert 10 via additive combinatorics

Series
Athens-Atlanta Number Theory Seminar
Time
Tuesday, April 1, 2025 - 16:00 for 1 hour (actually 50 minutes)
Location
314 Skiles
Speaker
Carlo Pagano Concordia University

 

In 1970 Matiyasevich, building on earlier work of Davis--Putnam--Robinson, proved that every enumerable subset of Z is Diophantine, thus showing that Hilbert's 10th problem is undecidable for $\mathbb{Z}$. The problem of extending this result to the ring of integers of number fields (and more generally to finitely generated infinite rings) has attracted significant attention and, thanks to the efforts of many mathematicians, the task has been reduced to the problem of constructing, for certain quadratic extensions of number fields $L/K$, an elliptic curve $E/K$ with $rk(E(L))=rk(E(K))>0$. 

In this talk I will explain joint work with Peter Koymans, where we use Green--Tao to construct the desired elliptic curves, settling Hilbert 10 for every finitely generated infinite ring.

Accelerated materials innovation using AI/ML and Digital Twins

Series
GT-MAP Seminar
Time
Tuesday, April 1, 2025 - 10:00 for 2 hours
Location
Skiles 006
Speaker
Prof. Surya R. KalidindiGeorgia Tech ME, CSE & MSE

Please Note: In person

This presentation will expound the challenges involved in the generation of digital twins (DT) as valuable tools for supporting innovation and providing informed decision support for the optimization of properties and/or performance of advanced material systems. This presentation will describe the foundational AI/ML (artificial intelligence/machine learning) concepts and frameworks needed to formulate and continuously update the DT of a selected material system. The central challenge comes from the need to establish reliable models for predicting the effective (macroscale) functional response of the heterogeneous material system, which is expected to exhibit highly complex, stochastic, nonlinear behavior. This task demands a rigorous statistical treatment (i.e., uncertainty reduction, quantification and propagation through a network of human-interpretable models) and fusion of insights extracted from inherently incomplete (i.e., limited available information), uncertain, and disparate (due to diverse sources of data gathered at different times and fidelities, such as physical experiments, numerical simulations, and domain expertise) data used in calibrating the multiscale material model. This presentation will illustrate with examples how a suitably designed Bayesian framework combined with emergent AI/ML toolsets can uniquely address this challenge.

TBD

Series
Geometry Topology Seminar
Time
Monday, March 31, 2025 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ryan DickmannVanderbilt

TBD

TBD by Jonathan Boretsky

Series
Algebra Seminar
Time
Monday, March 31, 2025 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Jonathan BoretskyCentre de Recherches Mathématiques, Montreal

Universality for graphs of bounded degeneracy

Series
Combinatorics Seminar
Time
Friday, March 28, 2025 - 15:15 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Anita LiebenauUNSW Sydney

What is the smallest number of edges that a graph can have if it contains all $D$-degenerate graphs on $n$ vertices as subgraphs? A counting argument shows that this number is at least of order $n^{2−1/D}$, assuming n is large enough. We show that this is tight up to a polylogarithmic factor.

Joint work with Peter Allen and Julia Böttcher.

Randomized Iterative Sketch-and-Project Methods as Efficient Large-Scale Linear Solvers

Series
Stochastics Seminar
Time
Thursday, March 27, 2025 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Elizaveta RebrovaPrinceton

Randomized Kaczmarz methods — popular special case of the sketch-and-project optimization framework — solve linear systems through iterative projections onto randomly selected equations, resulting in exponential expected convergence via cheap, local updates. While known to be effective in highly overdetermined problems or under the restricted data access, identifying generic scenarios where these methods are advantageous compared to classical Krylov subspace solvers (e.g., Conjugate Gradient, LSQR, GMRES) remained open. In this talk, I will present our recent results demonstrating that properly designed randomized Kaczmarz (sketch-and-project) methods can outperform Krylov methods for both square and rectangular systems complexity-wise. In addition, they are particularly advantageous for approximately low-rank systems common in machine learning (e.g., kernel matrices, signal-plus-noise models) as they quickly capture the large outlying singular values of the linear system. Our approach combines novel spectral analysis of randomly sketched projection matrices with classical numerical analysis techniques, such as including momentum, adaptive regularization, and memoization.

Local-to-global in thin orbits

Series
School of Mathematics Colloquium
Time
Thursday, March 27, 2025 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Kate StangeUniversity of Colorado, Boulder

Primitive integral Apollonian circle packings are fractal arrangements of tangent circles with integer curvatures.  The curvatures form an orbit of a 'thin group,' a subgroup of an algebraic group having infinite index in its Zariski closure.  The curvatures that appear must fall into a restricted class of residues modulo 24. The twenty-year-old local-global conjecture states that every sufficiently large integer in one of these residue classes will appear as a curvature in the packing. We prove that this conjecture is false for many packings, by proving that certain quadratic and quartic families are missed. The new obstructions are a property of the thin Apollonian group (and not its Zariski closure), and are a result of quadratic and quartic reciprocity, reminiscent of a Brauer-Manin obstruction. Based on computational evidence, we formulate a new conjecture.  This is joint work with Summer Haag, Clyde Kertzer, and James Rickards.  Time permitting, I will discuss some new results, joint with Rickards, that extend these phenomena to certain settings in the study of continued fractions.

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