Seminars and Colloquia by Series

Generalization and sampling from the dynamics perspective

Series
Applied and Computational Mathematics Seminar
Time
Monday, February 27, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005 and https://gatech.zoom.us/j/98355006347
Speaker
Prof. Nisha ChandramoorthyGT CSE

Please Note: Speaker will present in person

In this talk, we obtain new computational insights into two classical areas of statistics: generalization and sampling. In the first part, we study generalization: the performance of a learning algorithm on unseen data. We define a notion of generalization for non-converging training with local descent approaches via the stability of loss statistics. This notion yields generalization bounds in a similar manner to classical algorithmic stability. Then, we show that more information from the training dynamics provides clues to generalization performance.   

In the second part, we discuss a new method for constructing transport maps. Transport maps are transformations between the sample space of a source (which is generally easy to sample) and a target (typically non-Gaussian) probability distribution. The new construction arises from an infinite-dimensional generalization of a Newton method to find the zero of a "score operator". We define such a score operator that gives the difference of the score -- gradient of logarithm of density -- of a transported distribution from the target score. The new construction is iterative, enjoys fast convergence under smoothness assumptions, and does not make a parametric ansatz on the transport map.

Crossing the transcendental divide: from translation surfaces to algebraic curves

Series
Algebra Seminar
Time
Monday, February 27, 2023 - 10:20 for 1.5 hours (actually 80 minutes)
Location
Skiles 005
Speaker
Yelena MandelshtamUC Berkeley

A translation surface is obtained by identifying edges of polygons in the plane to create a compact Riemann surface equipped with a nonzero holomorphic one-form. Every Riemann surface can be given as an algebraic curve via its Jacobian variety. We aim to construct explicitly the underlying algebraic curves from their translation surfaces, given as polygons in the plane. The key tools in our approach are discrete Riemann surfaces, which allow us to approximate the Riemann matrices, and then, via theta functions, the equations of the curves. In this talk, I will present our algorithm and numerical experiments. From the newly found Riemann matrices and equations of curves, we can then make several conjectures about the curves underlying the Jenkins-Strebel representatives, a family of examples that until now, lived squarely on the analytic side of the transcendental divide between Riemann surfaces and algebraic curves.

Exploring global dynamics and blowup in some nonlinear PDEs

Series
CDSNS Colloquium
Time
Friday, February 24, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006 and Online
Speaker
Jonathan JaquetteBrown University

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

Conservation laws and Lyapunov functions are powerful tools for proving the global existence or stability of solutions to PDEs, but for most complex systems these tools are insufficient to completely understand non-perturbative dynamics. In this talk I will discuss a complex-scalar PDE which may be seen as a toy model for vortex stretching in fluid flow, and cannot be neatly categorized as conservative nor dissipative.

In a recent series of papers, we have shown (using computer-assisted-proofs) that this equation exhibits rich dynamical behavior existing globally in time: non-trivial equilibria, homoclinic orbits, heteroclinic orbits, and integrable subsystems foliated by periodic orbits. On the other side of the coin, we show several mechanisms by which solutions can blowup.

An Approximate Bayesian Computation Approach for Embryonic Cell Migration Model Selection

Series
Mathematical Biology Seminar
Time
Friday, February 24, 2023 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Tracy StepienUniversity of Florida - Department of Mathematics

Please Note: The classroom version of this event will be held in Skiles 005. Everyone on campus at Georgia Tech is highly encouraged to attend this version. The virtual version will be administered through Zoom. (Link: https://gatech.zoom.us/j/95527383236)

In embryonic development, formation of blood vessels in the retina of the eye is critically dependent on prior establishment of a mesh of astrocytes.  Astrocytes emerge from the optic nerve head and then migrate over the retinal surface in a radially symmetric manner and mature through differentiation.  We develop a PDE model describing the migration and differentiation of astrocytes and study the appropriateness of the model equation components that combines approximate Bayesian computation (ABC) and sensitivity analysis (SA). Comparing numerical simulations to experimental data, we identify model components that can be removed via model reduction using ABC+SA.

Lefschetz Fibrations and Exotic 4-Manifolds

Series
Dynamical Systems Working Seminar
Time
Friday, February 24, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Nur Saglam and Jon SimoneGeorgia Tech

Lefschetz fibrations are very useful in the sense that they have one-one correspondence with the relations in the Mapping Class Groups and they can be used to construct exotic (homeomorphic but not diffeomorphic) 4-manifolds. In this series of talks, we will first introduce Lefschetz fibrations and Mapping Class Groups and give examples. Then, we will dive more into 4-manifold world. More specifically, we will talk about the history of  exotic 4-manifolds and we will define the nice tools used to construct exotic 4-manifolds, like symplectic normal connect sum, Rational Blow-Down, Luttinger Surgery, Branch Covers, and Knot Surgery. Finally, we will provide various constructions of exotic 4-manifolds.

Online Covering: Prophets, Secretaries, and Samples

Series
ACO Student Seminar
Time
Friday, February 24, 2023 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Gregory KehneHarvard Computer Science

We give a polynomial-time algorithm for online covering IPs with a competitive ratio of O(\log mn) when the constraints are revealed in random order, essentially matching the best possible offline bound of \Omega(\log n) and circumventing the \Omega(\log m \log n) lower bound known in adversarial order. We then leverage this O(\log mn)-competitive algorithm to solve this problem in the prophet setting, where constraints are sampled from a sequence of known distributions. Our reduction in fact relies only on samples from these distributions, in a manner evocative of prior work on single-sample prophet inequalities for online packing problems. We present sample guarantees in the prophet setting, as well as in the setting where random samples from an adversarial instance are revealed at the outset.

This talk is based on joint work with Anupam Gupta and Roie Levin, part of which appeared at FOCS 2021. 

Symmetrically Hyperbolic Polynomials

Series
Algebra Student Seminar
Time
Friday, February 24, 2023 - 10:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Kevin ShuGeorgia Institute of Technology

We'll begin with a primer on hyperbolic and stable polynomials, which have been popular in recent years due to their many surprising appearances in combinatorics and algebra. We will cover a sketch of the famous Branden Borcea characterization of univariate stability preservers in the first part of the talk. We will then discuss more our recent work on multivariate hyperbolic polynomials which are invariant under permutations of their variables and connections to this Branden Borcea characterization.

 

Zoom Link: https://gatech.zoom.us/j/99596774152

Covariance Representations, Stein's Kernels and High Dimensional CLTs

Series
Stochastics Seminar
Time
Thursday, February 23, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Christian HoudréGeorgia Tech

In this continuing joint work with Benjamin Arras, we explore connections between covariance representations and Stein's method. In particular,  via Stein's kernels we obtain quantitative high-dimensional CLTs in 1-Wasserstein distance when the limiting Gaussian probability measure is anisotropic. The dependency on the parameters is completely explicit and the rates of convergence are sharp.

Certain aspects of the theory of Anderson Localization

Series
Time
Thursday, February 23, 2023 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Omar HurtadoGeorgia Tech and University of California, Irvine

The Anderson tight binding model describes an electron moving in a disordered material. Such models are, depending on various parameters of the system, either expected to or known to display a phenomenon known as Anderson localization, in which this disorder can "trap" electrons. Different versions of this phenomenon can be characterized spectrally or locally. We will review both the dominant methods and some of the foundational results in the study of these systems in arbitrary dimension, before shifting our focus to aspects of the one-dimensional theory.

 

Specifically, we will examine the transfer matrix method, which allows us to leverage the Furstenberg theory of random matrix products to understand the asymptotics of generalized eigenfunctions. From this, we will briefly sketch a proof of localization given originally in Jitomirskaya-Zhu (2019). Finally, we will discuss recent work of the speaker combining the argument in Jitomirskaya-Zhu with certain probabilistic results to prove localization for a broader class of models.

Convergence of discrete non-linear Fourier transform via spectral problems for canonical systems

Series
Analysis Seminar
Time
Wednesday, February 22, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Ashley ZhangUW Madison

This talk will be about connections between spectral problems for canonical systems and non-linear Fourier transforms (NLFTs). Non-linear Fourier transform is closely connected to Dirac systems, which form a subclass of canonical systems of differential equations. This connection allows one to find analogs of results on inverse spectral problems for canonical systems in the area of NLFT. In particular, NLFTs of discrete sequences, discussed in the lecture notes by Tao and Thiele, are related to spectral problems for periodic measures and the theory of orthogonal polynomials.

I will start the talk with the basics of non-linear Fourier transforms, then connect NLFTs to canonical systems. Then I will present an explicit algorithm for inverse spectral problems developed by Makarov and Poltoratski for locally-finite periodic spectral measures, as well as an extension of their work to certain classes of non-periodic spectral measures. Finally I will return to NLFT and translate the results for inverse spectral problems to results for NLFT.

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