Seminars and Colloquia Schedule

Certified computation in algebraic geometry using interval arithmetic

Series
Algebra Seminar
Time
Monday, December 4, 2023 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Kisun LeeClemson University

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

The homotopy continuation is a widely recognized method for finding solutions to polynomial systems by tracking the homotopy paths of solutions. However, the current implementation of homotopy continuation relies on heuristics, and hence it requires certification to verify its correctness. We discuss two modalities of certification in algebraic geometry exploiting interval arithmetic. The first is certified homotopy tracking using the Krawczyk method which guarantees correct tracking without path jumping. The second is Smale’s alpha theory over regions for faster certification. We discuss experimental results to demonstrate the effectiveness of these new methods. This talk is a preliminary report of two separate ongoing works.

Long simple curves on hyperbolic surfaces and the geometry of their complements by Aaron Calderon

Series
Geometry Topology Seminar
Time
Monday, December 4, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Speaker
Aaron Calderon

In her thesis, Maryam Mirzakhani counted the number of simple closed geodesics of bounded length on a (real) hyperbolic surface. This breakthrough theorem and the subsequent explosion of related results use techniques and draw inspiration from Teichmüller theory, symplectic geometry, surface topology, and homogeneous dynamics. In this talk, I’ll discuss some of these connections and a qualitative strengthening of her theorem, describing what these curves, and their complements, actually (generically) look like. This is joint work with Francisco Arana-Herrera.

Sparse Solution Technique for Local Clustering and Function Approximation

Series
Applied and Computational Mathematics Seminar
Time
Monday, December 4, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005 and https://gatech.zoom.us/j/98355006347
Speaker
Zhaiming ShenUniversity of Georgia

The sparse solution obtained from greedy-based optimization approach such as orthogonal matching pursuit can be very useful and have many applications in different directions. In this talk, I will present two research projects, one is about semi-supervised local clustering, and the other is about function approximation, which make use of the sparse solution technique. We will show that the target cluster can be effectively retrieved in the local clustering task and the curse of dimensionality can be overcome for a dense subclass of the space of continuous functions via Kolmogorov superposition theorem. Both the theoretical and numerical results will be discussed.

Growth of cohomology in towers of manifolds: a topological application of the Langlands program

Series
Job Candidate Talk
Time
Tuesday, December 5, 2023 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Mathilde Gerbelli-GauthierMcGill University

https://gatech.zoom.us/j/94328087718

How complicated can successive manifolds get in a tower of covering
spaces? Specifically, how large can the dimension of the first
cohomology get? We will begin with a tour of possible behaviors for
low-dimensional spaces, and then focus on arithmetic manifolds.
Specifically, for towers of complex-hyperbolic manifolds, I will
describe how to bound the rates of growth using known instances of
Langlands functoriality.

Quantitative acceleration of convergence to invariant distribution by irreversibility in diffusion processes

Series
PDE Seminar
Time
Tuesday, December 5, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Yuqing WangGeorgia Tech

Sampling from the Gibbs distribution is a long-standing problem studied across various fields. Among many sampling algorithms, Langevin dynamics plays a crucial role, particularly for high-dimensional target distributions. In practical applications, accelerating sampling dynamics is always desirable. It has long been studied that adding an irreversible component to reversible dynamics, such as Langevin, can accelerate convergence. Concrete constructions of irreversible components have also been explored in specific scenarios. However, a general strategy for such construction is still elusive. In this talk, I will introduce the concept of leveraging irreversibility to accelerate general dynamics, along with the quantification of irreversible dynamics. Our theory is mainly based on designing a modified entropy functional originally developed for linear kinetic equations (Dolbeault et al., 2015).

Subsquares in random Latin squares and rectangles

Series
Graph Theory Seminar
Time
Tuesday, December 5, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Alex DivouxGeorgia Tech

A $k \times n$ partial Latin rectangle is \textit{$C$-sparse} if the number of nonempty entries in each row and column is at most $C$ and each symbol is used at most $C$ times. We prove that the probability a uniformly random $k \times n$ Latin rectangle, where $k < (1/2 - \alpha)n$, contains a $\beta n$-sparse partial Latin rectangle with $\ell$ nonempty entries is $(\frac{1 \pm \varepsilon}{n})^\ell$ for sufficiently large $n$ and sufficiently small $\beta$. Using this result, we prove that a uniformly random order-$n$ Latin square asymptotically almost surely has no Latin subsquare of order greater than $c\sqrt{n\log n}$ for an absolute constant $c$. This is joint work with Tom Kelly, Camille Kennedy, and Jasdeep Sidhu.

Critical points of high-dimensional random functions

Series
Job Candidate Talk
Time
Tuesday, December 5, 2023 - 16:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Benjamin McKennaHarvard University

How many critical points does a random function from R^N to R have for large N? Such functions appear naturally in probability, data science, and mathematical physics. Questions like this one, which have attracted longstanding interest from both physicists and mathematicians, can help explain both physical phase transitions and algorithmic thresholds. I will give an overview of this "landscape complexity" program, its motivations, and recent progress coming from random matrices.

Spectral monotonicity under Gaussian convolution

Series
Analysis Seminar
Time
Wednesday, December 6, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Eli PuttermanTel Aviv University

The Poincaré constant of a body, or more generally a probability density, in $\mathbb R^n$ measures how "spread out" the body is - for instance, this constant controls how long it takes heat to flow from an arbitrary point in the body to any other. It's thus intuitively reasonable that convolving a "sufficiently nice" measure with a Gaussian, which tends to flatten and smooth out the measure, would increase its Poincaré constant ("spectral monotonicity"). We show that this is true if the original measure is log-concave, via two very different strategies - a dynamic variant of Bakry-Émery's $\Gamma$-calculus, and a mass-transportation argument. Moreover, we show that the dynamic $\Gamma$-calculus argument can also be extended to the discrete setting of measures on $\mathbb Z$, and that spectral monotonicity holds in this setting as well. Some results joint with B. Klartag.  

Spectral monotonicity under Gaussian convolution

Series
Analysis Seminar
Time
Wednesday, December 6, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Eli PuttermanTel Aviv University

The Poincaré constant of a body, or more generally a probability density, in $\mathbb R^n$ measures how "spread out" the body is - for instance, this constant controls how long it takes heat to flow from an arbitrary point in the body to any other. It's thus intuitively reasonable that convolving a "sufficiently nice" measure with a Gaussian, which tends to flatten and smooth out the measure, would increase its Poincaré constant ("spectral monotonicity"). We show that this is true if the original measure is log-concave, via two very different strategies - a dynamic variant of Bakry-Émery's $\Gamma$-calculus, and a mass-transportation argument. Moreover, we show that the dynamic $\Gamma$-calculus argument can also be extended to the discrete setting of measures on $\mathbb Z$, and that spectral monotonicity holds in this setting as well. Some results joint with B. Klartag.

Asymmetric Distribution of Extreme Values of Cubic L-functions on the 1-line

Series
Number Theory
Time
Wednesday, December 6, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Chantal DavidConcordia University

A fundamental problem in analytic number theory is to calculate the maximal value of L-functions at a given point. For L-functions associated to quadratic Dirichlet characters at s = 1, the upper bounds and Ω-results of Littlewood differ by a factor of 2, and it is a long-standing (and still unsolved) problem to find out which one is closer to the truth. The important work of Granville and Soundararajan, who model the distribution of L(1, χ) by the distribution of random Euler products L(1, X) for random variables X(p) attached to each prime, shed more light to the question. We use similar techniques to study the distribution of L(1, χ) for cubic Dirichlet characters. Unlike the quadratic case, there is an asymmetry between lower and upper bounds for the cubic case, and small values are less probable than large values. This is a joint work with P. Darbar, M. Lalin and A. Lumley.

The Poisson point process and an application to semisimple symmetric spaces

Series
Job Candidate Talk
Time
Thursday, December 7, 2023 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006; Streaming available via zoom
Speaker
Amanda WilkensUT Austin

Link to join via Zoom: https://gatech.zoom.us/j/93394018195?pwd=MGJZaWIwQUhVYW9ZZDFoWWFOc29EZz0... />
<br />
Meeting ID: 933 9401 8195<br />
Passcode: SoM

We define and motivate the Poisson point process, which is, informally, a “maximally random” scattering of points in some locally compact, second countable space. We introduce the ideal Poisson--Voronoi tessellation (IPVT), a new random object with intriguing geometric properties when considered on a semisimple symmetric space (the hyperbolic plane, for example). In joint work with Mikolaj Fraczyk and Sam Mellick, we use the IPVT to prove the minimal number of generators of a torsion-free lattice in a higher rank, semisimple Lie group is sublinear in the co-volume of the lattice. We give some intuition for the proof. No prior knowledge on Poisson point processes or symmetric spaces will be assumed.

"SAM as an Optimal Relaxation of Bayes" and "Lie Group updates for Learning Distributions on Machine Learning Parameters"

Series
Applied and Computational Mathematics Seminar
Time
Friday, December 8, 2023 - 11:00 for 1 hour (actually 50 minutes)
Location
https://gatech.zoom.us/j/98355006347
Speaker
Dr. Thomas Moellenhoff and Dr. Eren Mehmet KıralRIKEN

Note special time, due to time zone difference from Japan.<br />
Joint with SIAM GT Student Chapter Seminar

Part I (SAM as an Optimal Relaxation of Bayes) Dr. Thomas Moellenhoff

Sharpness-aware minimization (SAM) and related adversarial deep-learning methods can drastically improve generalization, but their underlying mechanisms are not yet fully understood. In this talk, I will show how SAM can be interpreted as optimizing a relaxation of the Bayes objective where the expected negative-loss is replaced by the optimal convex lower bound, obtained by using the so-called Fenchel biconjugate. The connection enables a new Adam-like extension of SAM to automatically obtain reasonable uncertainty estimates, while sometimes also improving its accuracy.

Part II (Lie Group updates for Learning Distributions on Machine Learning Parameters) Dr. Eren Mehmet Kıral

I will talk about our recent paper https://arxiv.org/abs/2303.04397 with Thomas Möllenhoff and Emtiyaz Khan, and other related results. Bayesian Learning learns a distribution over the model parameters, allowing for different descriptions of the same data. This is (contrary to classical learning which "bets-it-all" on a single set of parameters in describing a given dataset and making predictions. We focus on classes of distributions which have a transitive Lie group action on them given by pushforwards of an action on the parameter space. I will also specialize to a few concrete Lie groups and show distinct learning behavior.

Staircases and cuspidal curves in symplectic four manifolds

Series
School of Mathematics Colloquium
Time
Friday, December 8, 2023 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Dusa McDuffBarnard College, Columbia

This colloquium will also be the staring talk for the 2023 Tech Topology Conference.

This talk will give an elementary introduction to my joint work with Kyler Siegel that shows how cuspidal curves in a symplectic manifold X such as the complex projective plane determine when an ellipsoid can be symplectically embedded into X.