Seminars and Colloquia by Series

Improving and maximal inequalities in discrete harmonic analysis

Series
Dissertation Defense
Time
Wednesday, June 7, 2023 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 006 & online
Speaker
Christina Giannitsi – Georgia Tech –

►Presentation will be in hybrid format. Zoom link: https://gatech.zoom.us/j/99128737217?pwd=dllnNE1kSW1DZURrY1UycGxrazJtQT09

►Abstract: We study various averaging operators of discrete functions, inspired by number theory, in order to show they satisfy  $\ell^p$ improving and maximal bounds. The maximal bounds are obtained via sparse domination results for $p \in (1,2)$, which imply boundedness on $\ell ^p (w)$ for $p \in (1, \infty )$, for all weights $w$ in the Muckenhoupt $A_p$ class. 

We start by looking at averages along the integers weighted by the divisor function $d(n)$, and obtain a uniform, scale free $\ell^p$-improving estimate for $p \in (1,2)$. We also show that the associated maximal function satisfies $(p,p)$ sparse bounds for $p \in (1,2)$. We move on to study averages along primes in arithmetic progressions, and establish improving and maximal inequalities for these averages, that are uniform in the choice of progression. The uniformity over progressions imposes several novel elements on our approach. Lastly, we generalize our setting in the context of number fields, by considering averages over the Gaussian primes.

Finally, we explore the connections of our work to number theory:   Fix an interval $\omega \subset \mathbb{T}$. There is an integer $N_\omega $, so that every odd integer $n$ with $N(n)>N_\omega $ is a sum of three Gaussian primes with arguments in $\omega $.  This is the weak Goldbach conjecture. A density version of the strong Goldbach conjecture is proved, as well.

                                                   

►Members of the committee:
· Michael Lacey (advisor)
· Chris Heil
· Ben Krause
· Doron Lubinsky
· Shahaf Nitzan

Two Phases of Scaling Laws for Nearest Neighbor Classifiers

Series
Applied and Computational Mathematics Seminar
Time
Thursday, May 25, 2023 - 10:30 for 1 hour (actually 50 minutes)
Location
https://gatech.zoom.us/j/98355006347
Speaker
Jingzhao Zhang – Tsinghua University

Please Note: Special time & day. Remote only.

A scaling law refers to the observation that the test performance of a model improves as the number of training data increases. A fast scaling law implies that one can solve machine learning problems by simply boosting the data and the model sizes. Yet, in many cases, the benefit of adding more data can be negligible. In this work, we study the rate of scaling laws of nearest neighbor classifiers. We show that a scaling law can have two phases: in the first phase, the generalization error depends polynomially on the data dimension and decreases fast; whereas in the second phase, the error depends exponentially on the data dimension and decreases slowly. Our analysis highlights the complexity of the data distribution in determining the generalization error. When the data distributes benignly, our result suggests that nearest neighbor classifier can achieve a generalization error that depends polynomially, instead of exponentially, on the data dimension.

Symmetric nonnegative polynomials and sums of squares: mean roads to infinity

Series
Dissertation Defense
Time
Wednesday, May 24, 2023 - 11:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 006
Speaker
Jose Acevedo – Georgia Tech –
We study the limits of cones of symmetric nonnegative polynomials and symmetric sums of squares of fixed degree, when expressed in power-mean or monomial-mean basis. These limits correspond to forms with stable expression in power-mean polynomials that are globally nonnegative (resp. sums of squares) regardless of the number of variables. Using some elements of the representation theory of the symmetric group we introduce partial symmetry reduction to describe the limit cone of symmetric sums of squares, which simultaneously allows us to tropicalize its dual cone. Using tropical convexity to describe the tropicalization of the dual cone to symmetric nonnegative forms we then compare both tropicalizations, which turn out to be convex polyhedral cones. We then show that the cones are different for all degrees larger than 4. For even symmetric forms we show that the cones agree up to degree $8$, and are different starting at degree 10. We also find, via tropicalization, explicit examples of symmetric forms that are nonnegative but not sums of squares at the limit.

Quantitative Generalized CLT with Self-Decomposable Limiting Laws by Spectral Methods

Series
Stochastics Seminar
Time
Thursday, May 18, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Benjamin Arras – Université de Lille –

In this talk, I will present new stability results for non-degenerate centered self-decomposable laws with finite second moment and for non-degenerate symmetric alpha-stable laws with alpha in (1,2). These stability results are based on Stein's method and closed forms techniques. As an application, explicit rates of convergence are obtained for several instances of the generalized CLTs. Finally, I will discuss the standard Cauchy case.

Dynamics of excitable cells: neurons and cardiomyocytes

Series
Other Talks
Time
Wednesday, May 10, 2023 - 11:00 for 1 hour (actually 50 minutes)
Location
PLOS (second floor of Howey)
Speaker
Roberto Barrio – Univ. of Zaragoza
In recent years, much attention has been paid to the description of excitable media,
such as the dynamics of the brain and heart.
In both cases, the building blocks are excitable cells, neurons, and cardiomyocytes,
and a detailed look at the mathematics behind some of their mathematical models provides
a good starting point for answering some observed phenomena.
In this talk we show how some apparently  simple phenomena,
such as the spike-adding process,
have important consequences in the models and how various elements intervene behind their formation,
such as homoclinic bifurcations, fast-slow decompositions, "canards",
the completion of the Smale topological template, the formation of Morse surfaces
creating geometric bifurcations, etc.
Finally, we will illustrate its relevance in insect gait patterns and in the formation of cardiac arrhythmias.
 

Some Global Relaxation Methods for Quadratic and Semidefinite Programming

Series
Dissertation Defense
Time
Tuesday, May 9, 2023 - 11:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 005 and ONLINE
Speaker
Shengding Sun – Georgia Tech –

Zoom link: https://gatech.zoom.us/meeting/96948840253

Quadratic programming and semidefinite programming are vital tools in discrete and continuous optimization, with broad applications. A major challenge is to develop methodologies and algorithms to solve instances with special structures. For this purpose, we study some global relaxation techniques to quadratic and semidefinite programming, and prove theoretical properties about their qualities. In the first half we study the negative eigenvalues of $k$-locally positive semidefinite matrices, which are closely related to the sparse relaxation of semidefinite programming. In the second half we study aggregations of quadratic inequalities, a tool that can be leveraged to obtain tighter relaxation to quadratic programming than the standard Shor relaxation. In particular, our results on finiteness of aggregations can potentially lead to efficient algorithms for certain classes of quadratic programming instances with two constraints.

A deter-mean-istic description of Stochastic Oscillators

Series
CDSNS Colloquium
Time
Friday, May 5, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Online
Speaker
Alberto Pérez-Cervera – Universidad Complutense de Madrid, Spain –

Link: https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT09

Abstract: The Parameterisation Method is a powerful body of theory to compute the invariant manifolds of a dynamical system by looking for a parameterization of them in such a way that the dynamics on this manifold expressed in the coordinates of such parameterization writes as simply as possible. This methodology was foreseen by Guillamon and Huguet [SIADS, 2009 & J. Math. Neurosci, 2013] as a possible way of extending the domain of accuracy of the phase-reduction of periodic orbits. This fruitful approach, known as phase-amplitude reduction, has been fully developed during the last decade and provides an essentially complete understanding of deterministic oscillatory dynamics.
In this talk, we pursue the "simpler as possible" philosophy underlying the Parameterisation Method to develop an analogous phase-amplitude approach to stochastic oscillators. Main idea of our approach is to find a change of variables such that the system, when transformed to these variables, expresses in the mean as the deterministic phase-amplitude description. Then, we take advantage of the simplicity of this approach, to develop interesting objects with the aim of further clarifying the stochastic oscillation.

Extension of homeomorphisms and vector fields of the circle: From Anti-de Sitter to Minkowski geometry.

Series
Geometry Topology Seminar
Time
Monday, May 1, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Farid Diaf – Université Grenoble Alpes –

In 1990, Mess gave a proof of Thurston's earthquake theorem using the Anti-de Sitter geometry. Since then, several of Mess's ideas have been used to investigate the correspondence between surfaces in 3-dimensional Anti de Sitter space and Teichmüller theory.

In this spirit, we investigate the problem of the existence of vector fields giving infinitesimal earthquakes on the hyperbolic plane, using the so-called Half-pipe geometry which is the dual of Minkowski geometry in a suitable sense. In particular, we recover Gardiner's theorem, which states that any Zygmund vector field on the circle can be represented as an infinitesimal earthquake. Our findings suggest a connection between vector fields on the hyperbolic plane and surfaces in 3-dimensional Half-pipe space, which may be suggestive of a bigger picture.

 

Free energy and uniqueness in 1D spin systems with random Hamiltonians

Series
CDSNS Colloquium
Time
Friday, April 28, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Online
Speaker
Cesar Octavio Maldonado Ahumada – IPICYT –


Link: https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT09

Abstract: In this talk, I will discuss problems and results in the rigorous statistical mechanics of particle systems in a one-dimensional lattice.
I will briefly describe the classical examples, such as the Ising model and its various generalizations concerning the
existence of the free energy, thermodynamic limit and the phase transition phenomenon.
Towards the end of the talk, I will talk about a recent work in collaboration with Jorge Littin, on a generalization of the
Khanin and Sinai model with random interactions for which one can prove that there exists a critical behavior in the free
energy for some parameters of the model and on the other side one can also have uniqueness of the equilibrium state.


 

An approach to universality using canonical systems

Series
Math Physics Seminar
Time
Thursday, April 27, 2023 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Milivoje Lukic – Rice University –

 It is often expected that the local statistical behavior of eigenvalues of some system depends only on its local properties; for instance, the local distribution of zeros of orthogonal polynomials should depend only on the local properties of the measure of orthogonality. This phenomenon is studied using an object called the Christoffel-Darboux kernel. The most commonly studied case is known as bulk universality, where the rescaled limit of Christoffel-Darboux kernels converges to the sine kernel. We will present a new approach which gives for the first time a completely local sufficient condition for bulk universality. This approach is based on a matrix version of the Christoffel-Darboux kernel and the de Branges theory of canonical systems, and it applies to other self-adjoint systems with 2x2 transfer matrices such as continuum Schrodinger and Dirac operators. The talk is based on joint work with Benjamin Eichinger (Technical University Wien) and Brian Simanek (Baylor University).

Pages