Seminars and Colloquia by Series

Sqrt and Levers

Series
Athens-Atlanta Number Theory Seminar
Time
Tuesday, October 24, 2023 - 17:15 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Vesselin DimitrovGeorgia Tech

If the formal square root of an abelian surface over Q looks like an elliptic curve, it has to be an elliptic curve."


We discuss what such a proposition might mean, and prove the most straightforward version where the precise condition is simply that the L-function of the abelian surface possesses an entire holomorphic square root. The approach follows the Diophantine principle that algebraic numbers or zeros of L-functions repel each other, and is in some sense similar in spirit to the Gelfond--Linnik--Baker solution of the class number one problem.

We discuss furthermore this latter connection: the problems that it raises under a hypothetical presence of Siegel zeros, and a proven analog over finite fields. The basic remark that underlies and motivates these researches is the well-known principle (which is a consequence of the Deuring--Heilbronn phenomenon, to be taken with suitable automorphic forms $f$ and $g$): an exceptional character $\chi$ would cause the formal $\sqrt{L(s,f)L(s,f \otimes \chi)}L(s,g)L(s, g \otimes \chi)$ to have a holomorphic branch on an abnormally big part of the complex plane, all the while enjoying a Dirichlet series formal expansion with almost-integer coefficients. This leads to the kind of situation oftentimes amenable to arithmetic algebraization methods. The most basic (qualitative) form of our main tool is what we are calling the "integral converse theorem for GL(2)," and it is a refinement of a recent Unbounded Denominators theorem that we proved jointly with Frank Calegari and Yunqing Tang. 

 

Moments of Dirichlet L-functions

Series
Athens-Atlanta Number Theory Seminar
Time
Tuesday, October 24, 2023 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Caroline Turnage-ButterbaughCarleton College

In recent decades there has been much interest and measured progress in the study of moments of the Riemann zeta-function and, more generally, of various L-functions. Despite a great deal of effort spanning over a century, asymptotic formulas for moments of L-functions remain stubbornly out of reach in all but a few cases. In this talk, we consider the problem for the family of all Dirichlet L-functions of even primitive characters of bounded conductor. I will outline how to harness the asymptotic large sieve to prove an asymptotic formula for the general 2kth moment of an approximation to this family. The result, which assumes the generalized Lindelöf hypothesis for large values of k, agrees with the prediction of Conrey, Farmer, Keating, Rubenstein, and Snaith. Moreover, it provides the first rigorous evidence beyond the so-called “diagonal terms” in their conjectured asymptotic formula for this family of L-functions

On the hardness of finding balanced independent sets in random bipartite graphs

Series
Graph Theory Seminar
Time
Tuesday, October 24, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Clough Commons room 102
Speaker
Yuzhou WangGeorgia Tech

We consider the algorithmic problem of finding large balanced independent sets in sparse random bipartite graphs, and more generally the problem of finding independent sets with specified proportions of vertices on each side of the bipartition. In a bipartite graph it is trivial to find an independent set of density at least half (take one of the partition classes). In contrast, in a random bipartite graph of average degree d, the largest balanced independent sets (containing equal number of vertices from each class) are typically of density (2 + od(1)) log d/d . Can we find such large balanced independent sets in these graphs efficiently? By utilizing the overlap gap property and the low-degree algorithmic framework, we prove that local and low-degree algorithms (even those that know the bipartition) cannot find balanced independent sets of density greater than (1 + ε) log d/d for any ε > 0 fixed and d large but constant.

Flexible Krylov methods for advanced regularization

Series
Applied and Computational Mathematics Seminar
Time
Monday, October 23, 2023 - 14:00 for
Location
Skiles 005 and https://gatech.zoom.us/j/98355006347
Speaker
Malena Landman Sabate Emory University

Inverse problems involve the reconstruction of hidden objects from possibly noisy indirect measurements and are ubiquitous in a variety of scientific and engineering applications. This kind of problems have two main features that make them interesting yet challenging to solve. First, they tend to be ill-posed: the reconstruction is very sensitive to perturbations in the measurements. Second, real-world applications are often large-scale: resulting in computationally demanding tasks. In this talk I will focus on discrete linear problems: giving a general overview of the well-established class of solvers called Krylov subspace methods and its regularizing properties; as well as flexible variants that make them suitable to solve more challenging optimization tasks. I will show results and examples in different imaging applications.

Standard monomials and Gröbner bases for positroid varieties

Series
Algebra Seminar
Time
Monday, October 23, 2023 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ayah AlmousaUniversity of Minnesota - Twin Cities

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

Positroid varieties are subvarieties of the Grassmannian that arise in the study of total positivity. Knutson, Lam, and Speyer described a certain type of Gröbner degeneration called the Hodge degeneration as projections of order complexes of intervals in the Bruhat order, but their description does not give an explicit Gröbner basis nor initial ideal. We give an explicit, combinatorial description of the Gröbner basis and initial ideal corresponding to the Hodge degeneration for an arbitrary positroid variety. As an application, we show that promotion on rectangular-shaped semistandard tableaux gives a bijection between standard monomials of a positroid variety and its cyclic shifts. This is joint work with Shiliang Gao (UIUC) and Daoji Huang (Minnesota).

Computation of high-order normal forms in diffeomorphisms

Series
CDSNS Colloquium
Time
Friday, October 20, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 249
Speaker
Joan GimenoGeorgia Tech

This talk will delve into a method specifically designed for
constructing high-order normal forms in Poincaré maps with high-order
precision and without any major assumption or structure of the
dynamical system itself. We will use the result to generate explicit
twist maps, calculating invariant tori, and determining the flying
time expansions around an elliptic fixed point of a Poincaré map. In
particular, this approach is able to check some non-degenerate
conditions in perturbation theory.

Electromagnetism and Falling Cats II

Series
Geometry Topology Working Seminar
Time
Friday, October 20, 2023 - 14:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 006
Speaker
Daniel IrvineGeorgia Institute of Technology

In this talk I will continue to develop a parallel between the classical field theory of electromagnetism and geometric mechanics of animal locomotion. The focus of the previous talk was on electromagnetism, and the focus of this talk will be on the geometric mechanics of animal locomotion. We will investigate the aphorism that a cat dropped (from a safe height) upside-down always lands on her feet. I will explain how non-trivial topology of the configuration space of the cat can act as a "source" of locomotion.

No prior knowledge of classical field theory will be assumed. I will rely on some results from part 1, but I will review the relevant definitions.

On the spectral synthesis for the unit circle in ${\mathcal F} L_s^q ({\mathbf R}^2)$

Series
Analysis Seminar
Time
Wednesday, October 18, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
TBA
Speaker
Masaharu KobayashiHokkaido University

Let ${\mathcal F}L^q_s ({\mathbf R}^2)$ denote the set of all tempered distributions $f \in {\mathcal S}^\prime ({\mathbf R}^2)$ such that the norm $ \| f \|_{{\mathcal F}L^q_s} = (\int_{{\mathbf R}^2}\, ( |{\mathcal F}[f](\xi)| \,( 1+ |\xi| )^s )^q\, d \xi )^{ \frac{1}{q} }$ is finite, where ${\mathcal F}[f]$ denotes the Fourier transform of $f$. We investigate the spectral synthesis for the unit circle $S^1 \subset {\mathbf R}^2$ in ${\mathcal F}L^q_s ({\mathbf R}^2)$ with $1\frac{2}{q^\prime}$, where $q^\prime$ denotes the conjugate exponent of $q$. This is joint work with Prof. Sato (Yamagata University).

An algorithm for comparing Legendrian links

Series
Geometry Topology Seminar
Time
Wednesday, October 18, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ivan DynnikovSteklov Mathematical Institute

The talk is based on my joint works with Maxim Prasolov and Vladimir Shastin, where we studied the relation between rectangular diagrams of links and Legendrian links. This relation allows for a complete classification of exchange classes of rectangular diagrams in terms of equivalence classes of Legendrian links and their symmetry groups. Since all rectangular diagrams of given complexity can be searched, this yields a method to algorithmically compare Legendrian links. Of course, the general algorithm has too high complexity for a practical implementation, but in some situations, the most time consuming parts can be bypassed, which allows us to confirm the non-equivalence of Legendrian knots in several previously unresolved cases.

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