Seminars and Colloquia by Series

Bertini theorems, connectivity of tropical varieties, and multivariate Puiseux series

Series
Algebra Seminar
Time
Monday, March 11, 2024 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Josephine YuGeorgia Tech

A theorem of Bertini says that an irreducible algebraic variety remains irreducible after intersecting with a generic hyperplane.  We will discuss toric Bertini theorems for intersections with generic algebraic subtori (defined by generic binomial equations) instead of hyperplanes. As an application, we obtain a tropical Bertini theorem and a strengthening of the Structure Theorem of tropical algebraic geometry, by showing that irreducible tropical varieties remain connected through codimension one even after removing some facets.  As part of the proof of the Toric Bertini over prime characteristics, we constructed a new algebraically closed field containing the multivariate rational functions, which is smaller than previously known constructions.  This is based on joint works with Diane Maclagan, Francesca Gandini, Milena Hering, Fatemeh Mohammadi, Jenna Rajchgot, and Ashley Wheeler.

ε-series by Corrine Yap, Jing Yu, and Changxin Ding

Series
Graph Theory Seminar
Time
Friday, March 8, 2024 - 15:15 for 1 hour (actually 50 minutes)
Location
Skiles 308
Speaker
Corrine Yap, Jing Yu, and Changxin DingGeorgia Tech

Corrine Yap:  The Ising model is a classical model originating in statistical physics; combinatorially it can be viewed as a probability distribution over 2-vertex-colorings of a graph. We will discuss a fixed-magnetization version—akin to fixing the number of, say, blue vertices in every coloring—and a natural Markov chain sampling algorithm called the Kawasaki dynamics. We show some surprising results regarding the existence and location of a fast/slow mixing threshold for these dynamics. (joint work with Aiya Kuchukova, Marcus Pappik, and Will Perkins)


Changxin Ding: For trees on a fixed number of vertices, the path and the star are two extreme cases. Many graph parameters attain its maximum at the star and its minimum at the path among these trees. A trivial example is the number of leaves. I will introduce more interesting examples in the mini talk.

Jing Yu: We show that all simple outerplanar graphs G with minimum degree at least 2 and positive Lin-Lu-Yau Ricci curvature on every edge have maximum degree at most 9. Furthermore, if G is maximally outerplanar, then G has at most 10 vertices. Both upper bounds are sharp.

Riemannian geometry and contact topology

Series
Geometry Topology Working Seminar
Time
Friday, March 8, 2024 - 14:00 for 2 hours
Location
Skiles 006
Speaker
John EtnyreGeorgia Tech

This series of talks will discuss connections between Riemannian geometry and contact topology. Both structures have deep connections to the topology of 3-manifolds, but there has been little study of the interactions between them (at least the implications in contact topology). We will see that there are interesting connections between curvature and properties of contact structures. The talks will give a brief review of both Riemannian geometry and contact topology and then discuss various was one might try to connect them. There will be many open problems discussed (probably later in the series). 

Critical phenomena through the lens of the Ising model

Series
School of Mathematics Colloquium
Time
Friday, March 8, 2024 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Hugo Duminil-CopinIHES and Université de Genève

The Ising model is one of the most classical lattice models of statistical physics undergoing a phase transition. Initially imagined as a model for ferromagnetism, it revealed itself as a very rich mathematical object and a powerful theoretical tool to understand cooperative phenomena. Over one hundred years of its history, a profound understanding of its critical phase has been obtained. While integrability and mean-field behavior led to extraordinary breakthroughs in the two-dimensional and high-dimensional cases respectively, the model in three and four dimensions remained mysterious for years. In this talk, we will present recent progress in these dimensions based on a probabilistic interpretation of the Ising model relating it to percolation models.

From Coffee to Mathematics: Making Connections and Finding Unexpected Links

Series
Stelson Lecture Series
Time
Thursday, March 7, 2024 - 16:30 for 1 hour (actually 50 minutes)
Location
Howey-Physics L3
Speaker
Hugo Duminil-CopinUniversité de Genève and IHES Université Paris-Saclay

The game of HEX has deep mathematical underpinnings despite its simple rules.  What could this game possibly have to do with coffee?!  And how does that connection, once identified, lead to consideration of ferromagnetism and even to the melting polar ice caps?  Join Hugo Duminil-Copin, Professor of Mathematics at IHES and the University of Geneva, for an exploration of the way in which mathematical thinking can help us make some truly surprising connections.

Large deviations for the top eigenvalue of deformed random matrices

Series
Stochastics Seminar
Time
Wednesday, March 6, 2024 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Benjamin McKennaHarvard University

In recent years, the few classical results in large deviations for random matrices have been complemented by a variety of new ones, in both the math and physics literatures, whose proofs leverage connections with Harish-Chandra/Itzykson/Zuber integrals. We present one such result, focusing on extreme eigenvalues of deformed sample-covariance and Wigner random matrices. This confirms recent formulas of Maillard (2020) in the physics literature, precisely locating a transition point whose analogue in non-deformed models is not yet fully understood. Joint work with Jonathan Husson.

Slow subgraph bootstrap percolation (Tibor Szabó, Freie Universität Berlin)

Series
Graph Theory Seminar
Time
Tuesday, March 5, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Tibor SzabóFreie Universität Berlin

 For a graph $H$ and an $n$-vertex graph $G$, the $H$-bootstrap percolation process on $G$ is the process which starts with $G$ and, at every time step, adds any missing edges on the vertices of $G$ that complete a copy of $H$. This process eventually stabilises and we are interested in the extremal question raised by Bollob\'as, of determining the maximum \emph{running time} (number of time steps before stabilising) of this process, over all possible choices of $n$-vertex graph $G$. We initiate a systematic study of this parameter, denoted $M_H(n)$, and its dependence on properties of the graph $H$. In a series of works we determine the precise running time for cycles and asymptotic running time for several other important classes. In general, we study necessary and sufficient conditions on $H$ for fast, i.e. sublinear or linear $H$-bootstrap percolation, and in particular explore the relationship between running time and minimum vertex degree and connectivity. Furthermore we also obtain the running time of the process for typical $H$ and discover several graphs exhibiting surprising behavior.  The talk represents joint work with David Fabian and Patrick Morris.

Viscosity solutions for Mckean-Vlasov control on a torus

Series
PDE Seminar
Time
Tuesday, March 5, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Online: https://gatech.zoom.us/j/95574359880?pwd=cGpCa3J1MFRkY0RUeU1xVFJRV0x3dz09
Speaker
Qinxin YanPrinceton University

An optimal control problem in the space of probability measures, and the viscosity solutions of the corresponding dynamic programming equations defined using the intrinsic linear derivative are studied. The value function is shown to be Lipschitz continuous with respect to a novel smooth Fourier Wasserstein metric. A comparison result between the Lipschitz viscosity sub and super solutions of the dynamic programming equation is proved using this metric, characterizing the value function as the unique Lipschitz viscosity solution. This is joint work with Prof. H. Mete Soner. 

Diffusion Models for Arbitrary Discrete Markov Processes

Series
Applied and Computational Mathematics Seminar
Time
Monday, March 4, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005 and https://gatech.zoom.us/j/98355006347
Speaker
Zachary FoxOak Ridge National Laboratory

Please Note: Speaker will present in person.

Diffusion models have become ubiquitous for image generation and are increasingly being used for scientific applications. To date, many flavors of diffusion models have been developed by varying the stochastic process that noises data, but also the domain on which these processes act. Typically, generative diffusion models rely on a Gaussian diffusion process for training the backward transformations, which can then be used to generate samples from Gaussian noise. However, real world data often takes place in discrete-state spaces, including many scientific applications. Here we develop a theoretical formulation for arbitrary discrete-state Markov processes in the forward diffusion process using exact analysis. We relate the theory to the existing continuous-state Gaussian diffusion in discrete and continuous time. This approach is validated using a simple stochastic decay process, in which the reverse process generates images from a single all-black image, rather than a noisy prior distribution.

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