Seminars and Colloquia Schedule

Lie Groups and Applications to Multi-Orientation Image Analysis

Series
Algebra Seminar
Time
Monday, October 7, 2024 - 11:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Nicky J. van den BergEindhoven University of Technology

There will be a pre-talk at 10:55 am in Skiles 005.

Retinal images are often used to examine the vascular system in a non-invasive way. Studying the behavior of the vasculature on the retina allows for noninvasive diagnosis of several diseases as these vessels and their behavior are representative of the behavior of vessels throughout the human body. For early diagnosis and analysis of diseases, it is important to compare and analyze the complex vasculature in retinal images automatically.

During this talk, we will talk about a geodesic tracking approach that is better able to handle difficult structures, like high curvature and crossings. Additionally, we discuss how one can identify connected components in images that allow for small interruptions within the same component. Both methods takes place in the lifted space of positions and orientations SE(2), which allows us to differentiate between crossings and bifurcations.

A Lorentzian manifold-with-boundary where causality breaks down due to shock singularities

Series
Geometry Topology Seminar
Time
Monday, October 7, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Leo AbbresciaGeorgia Tech

We present a novel example of a Lorentzian manifold-with-boundary featuring a dramatic degeneracy in its deterministic and causal properties known as “causal bubbles” along its boundary. These issues arise because the regularity of the Lorentzian metric is below Lipschitz and fit within the larger framework of low regularity Lorentzian geometry. Although manifolds with causal bubbles were recently introduced in 2012 as a mathematical curiosity, our example comes from studying the fundamental equations of fluid mechanics and shock singularities which arise therein. No prior knowledge of Lorentzian geometry or fluid mechanics will be assumed for this talk.

Data-driven model discovery meets mechanistic modeling for biological systems

Series
Applied and Computational Mathematics Seminar
Time
Monday, October 7, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005 and https://gatech.zoom.us/j/98355006347
Speaker
Niall M ManganNorthwestern University
Abstract: Building models for biological, chemical, and physical systems has traditionally relied on domain-specific intuition about which interactions and features most strongly influence a system.  Alternatively, machine-learning methods are adept at finding novel patterns in large data sets and building predictive models but can be challenging to interpret in terms of or integrate with existing knowledge. Our group balances traditional modeling with data-driven methods and optimization to get the best of both worlds.  Recently developed for and applied to dynamical systems, sparse optimization strategies can select a subset of terms from a library that best describes data, automatically interfering potential model structures from a broad but well-defined class. I will discuss my group's application and development of data-driven methods for model selection to 1) recover chaotic systems models from data with hidden variables,  2) discover models for metabolic and temperature regulation in hibernating mammals, and 3) model selection for differential-algebraic-equations. I'll briefly discuss current preliminary work and roadblocks in developing new methods for model selection of biological metabolic and regulatory networks.
 
Short Bio: Niall M. Mangan received the Dual BS degrees in mathematics and physics, with a minor in chemistry, from Clarkson University, Potsdam, NY, USA, in 2008, and the PhD degree in systems biology from Harvard University, Cambridge, MA, USA, in 2013. Dr. Mangan worked as a postdoctoral associate in the Photovoltaics Lab at MIT from 2013-2015 and as an Acting Assistant Professor at the University of Washington, Seattle from 2016-2017. She is currently an Assistant Professor of engineering sciences and applied mathematics with Northwestern University, where she works at the interface of mechanistic modeling, machine learning, and statistical inference. Her group applies these methods to many applications including metabolic and regulatory networks to accelerate engineering.

Non-measurable colourings avoiding large distances (James Davies)

Series
Graph Theory Seminar
Time
Tuesday, October 8, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
James DaviesCambridge University

 In 1983, Furstenberg, Katznelson, and Weiss proved that for every finite measurable colouring of the plane, there exists a $d_0$ such that for every $d\geq d_0$ there is a monochromatic pair of points at distance $d$. In contrast to this, we show that there is a finite colouring avoiding arbitrarily large distances. Joint work with Rutger Campbell.

Sparsity of Fourier mass of passively advected scalars in the Batchelor regime

Series
PDE Seminar
Time
Tuesday, October 8, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Manh Khang HuynhGeorgia Tech

In this paper we propose a general dynamical mechanism that can lead to the failure of the Batchelor's mode-wise power spectrum law in passive scalar turbulence and hyperbolic dynamics, while the cumulative law remains true. Of technical interest, we also employ a novel method of power spectral variance to establish an exponential radial shell law for the Batchelor power spectrum. An accessible explanation of the power spectrum laws via harmonic analysis is also given.

On non-resonant planar Carleson-Radon operator along homogeneous curves

Series
Analysis Seminar
Time
Wednesday, October 9, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Martin HsuPurdue University

We go over some relevant history and related problems to motivate the study of the Carleson-Radon operator and the difficulty exhibiting in the planar case. Our main result confirms that the planar Carleson-Radon operator along homogenous curve with general monomial \(t^\alpha\) term modulation admits full range \(L^p\) bound assuming the natural non-resonant condition. In the talk, I'll provide a brief overview of the three key ingredients of the LGC based proof:

 

  1. A sparse-uniform dichotomy of the input function adapted to appropriate time-frequency foliation of the phase-space;
  2. A joint structural analysis of the linearizing stopping-time function in the phase in relation to the Gabor coefficients of the input;
  3. A level set analysis on the time-frequency correlation set.
 

Limiting Distributions of Conjugate Algebraic Integers

Series
Number Theory
Time
Wednesday, October 9, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Bryce OrloskiPenn State

A recent advance by Smith establishes a quantitative converse (conjectured by Smyth and Serre) to Fekete's celebrated theorem for compact subsets of $\mathbb{R}$. Answering a basic question raised by Smith, we formulate and prove a quantitative converse of Fekete for general symmetric compact subsets of $\mathbb{C}$. We highlight and exploit the algorithmic nature of our approach to give concrete applications to abelian varieties over finite fields and to extremal problems in algebraic number theory.

CANCELLED - Domino tilings beyond 2D

Series
School of Mathematics Colloquium
Time
Thursday, October 10, 2024 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Caroline KlivansBrown University

There is a rich history of domino tilings in two dimensions.  Through a variety of techniques we can answer questions such as: how many tilings are there of a given region or what does a random tiling look like?  These questions and their answers become significantly more difficult in dimension three and above.  Despite this curse of dimensionality, I will discuss recent advances in the theory.  I will also highlight problems that still remain open. 

Improved performance guarantees for Tukey’s median

Series
Stochastics Seminar
Time
Thursday, October 10, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Stanislav MinskerUniversity of Southern California

Is there a natural way to order data in dimension greater than one? The approach based on the notion of data depth, often associated with the name of John Tukey, is among the most popular. Tukey’s depth has found applications in robust statistics, graph theory, and the study of elections and social choice. 

We will give an introduction to the topic, describe the properties of Tukey’s depth, and introduce some remaining open questions as well as our recent progress towards the solutions.

The talk is based on a joint work with Yinan Shen.

A Roth type result for dense subsets of the integer lattice

Series
Additional Talks and Lectures
Time
Friday, October 11, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Akos MagyarThe University of Georgia

 Let A be a subset of the integer lattice of positive upper density. Roth' theorem in this setting states that there are points x,x+y,x+2y in A with the length of the gap y arbitrary large. We show that the lengths of the gaps y contain an infinite arithmetic progression, as long as one measures the length in lp for p>2 even, while this not true for the Euclidean distance.

 

Such results have been previously obtained in the continuous settings for measurable subsets of Euclidean spaces using methods of time-frequency analysis, as opposed our approach is based on some ideas from additive combinatorics such as uniformity norms and arithmetic regularity lemmas. If time permits, we discuss some other results that can be obtained similarly.

On the number of error correcting codes

Series
Combinatorics Seminar
Time
Friday, October 11, 2024 - 15:15 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Nitya ManiMIT

We show that for a fixed $q$, the number of $q$-ary $t$-error correcting codes of length $n$ is at most $2^{(1 + o(1)) H_q(n,t)}$ for all $t \leq (1 - q^{-1})n - 2\sqrt{n \log n}$, where $H_q(n, t) = q^n / V_q(n,t)$ is the Hamming bound and $V_q(n,t)$ is the cardinality of the radius $t$ Hamming ball. This proves a conjecture of Balogh, Treglown, and Wagner and makes progress towards a 2005 question of Sapozhenko. 

Generic dynamics of the mean curvature flows

Series
CDSNS Colloquium
Time
Friday, October 11, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 314
Speaker
Jinxin XueTsinghua University

Talk is in-person. Zoom-link available as well: https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT09 <br />

The mean curvature flow is to evolve a hypersurface in Euclidean space using the mean curvatures at each point as the velocity field. The flow has good smoothing property, but also develops singularities. The singularities are modeled on an object called shrinkers, which give homothetic solutions to the flows. As there are infinitely many shrinkers that seem impossible to classify, it is natural to explore the idea of generic mean curvature flows that is to introduce a generic perturbation of the initial conditions. In this talk, we shall explain our work on this topic, including perturbing away nonspherical and noncylindrical shrinkers, and generic isolatedness of cylindrical singularities. The talk is based on a series of works jointly with Ao Sun.