Seminars and Colloquia by Series

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. 

Monopole Floer spectra of Seifert spaces

Series
Geometry Topology Seminar
Time
Monday, March 4, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Matt StoffregenMSU

We'll give a short description of what exactly monopole Floer spectra are, and then explain how to calculate them for AR plumbings, a class of 3-manifolds including Seifert spaces.  This is joint work with Irving Dai and Hirofumi Sasahira.

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.

Effective Whitney Stratification of Real Algebraic Varieties and Applications

Series
Algebra Seminar
Time
Monday, March 4, 2024 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Martin HelmerNorth Carolina State University

Please Note: There will be a pre-seminar from 11 am to 11:30 am in Skiles 005.

We describe an algorithm to compute Whitney stratifications of real algebraic varieties, and of polynomial maps between them, by exploiting the algebraic structure of certain conormal spaces.  One of the map stratification algorithms described here yields a new method for solving the real root classification problem. We also explore applications of this new map stratification algorithm to the study of the singularities of Feynman integrals; understanding and evaluating these integrals is a fundamental component in a wide variety of problems arising in quantum field theory. 

Some sketches of Floer homotopy

Series
Geometry Topology Seminar Pre-talk
Time
Monday, March 4, 2024 - 12:45 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Matt StoffregenMSU

In this talk, we'll sketch how one might hope to construct spaces (or spectra) from Floer theories, including framed flow categories and finite-dimensional approximation.  If time allows, we'll talk about some questions Floer spaces (or spectra) can be useful for.

Essentially tight bounds for rainbow cycles in proper edge-colourings (Matija Bucic, Princeton)

Series
Combinatorics Seminar
Time
Friday, March 1, 2024 - 15:15 for 1 hour (actually 50 minutes)
Location
Skiles 308
Speaker
Matija BucicPrinceton University

An edge-coloured graph is said to be rainbow if it uses no colour more than once. Extremal problems involving rainbow objects have been a focus of much research as they capture the essence of a number of interesting problems in a variety of areas. A particularly intensively studied question due to Keevash, Mubayi, Sudakov and Verstraëte from 2007 asks for the maximum possible average degree of a properly edge-coloured graph on n vertices without a rainbow cycle. Improving upon a series of earlier bounds, Tomon proved an upper bound of (log n)^(2+o(1)) for this question. Very recently, Janzer-Sudakov and Kim-Lee-Liu-Tran independently removed the o(1) term in Tomon's bound. We show that the answer to the question is equal to (log n)^(1+o(1)).  
Joint work with: Noga Alon, Lisa Sauermann, Dmitrii Zakharov and Or Zamir.

Load Balancing under Data Locality: Extending Mean-Field Framework to Constrained Large-Scale Systems

Series
Stochastics Seminar
Time
Thursday, February 29, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Debankur MukherjeeGeorgia Tech

Large-scale parallel-processing infrastructures such as data centers and cloud networks form the cornerstone of the modern digital environment. Central to their efficiency are resource management policies, especially load balancing algorithms (LBAs), which are crucial for meeting stringent delay requirements of tasks. A contemporary challenge in designing LBAs for today's data centers is navigating data locality constraints that dictate which tasks are assigned to which servers. These constraints can be naturally modeled as a bipartite graph between servers and various task types. Most LBA heuristics lean on the mean-field approximation's accuracy. However, the non-exchangeability among servers induced by the data locality invalidates this mean-field framework, causing real-world system behaviors to significantly diverge from theoretical predictions. From a foundational standpoint, advancing our understanding in this domain demands the study of stochastic processes on large graphs, thus needing fundamental advancements in classical analytical tools.

In this presentation, we will delve into recent advancements made in extending the accuracy of mean-field approximation for a broad class of graphs. In particular, we will talk about how to design resource-efficient, asymptotically optimal data locality constraints and how the system behavior changes fundamentally, depending on whether the above bipartite graph is an expander, a spatial graph, or is inhomogeneous in nature.

Smooth Fine Curve Graphs

Series
Geometry Topology Student Seminar
Time
Wednesday, February 28, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Speaker
Katherine BoothGeorgia Tech

The curve graph provides a combinatorial perspective to study surfaces. Classic work of Ivanov showed that the automorphisms of this graph are naturally isomorphic to the mapping class group. By dropping isotopies, more recent work of Long-Margalit-Pham-Verberne-Yao shows that there is also a natural isomorphism between the automorphisms of the fine curve graph and the homeomorphism group of the surface. Restricting this graph to smooth curves might appear to be the appropriate object for the diffeomorphism group, but it is not. In this talk, we will discuss why this doesn’t work and some progress towards describing the group of homeomorphisms that is naturally isomorphic to automorphisms of smooth fine curve graphs.

Recent advances on extremal problems of k-critical graphs (Jie Ma, University of Science and Technology of China)

Series
Graph Theory Seminar
Time
Tuesday, February 27, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jie MaUniversity of Science and Technology of China

 A graph is called k-critical if its chromatic number is k but any proper subgraph has chromatic number less than k. There have been extensive reseaches on k-critical graphs over the past decades, yet several basic problems remain widely open. One of such problems is to determine the maximum number of edges in an n-vertex k-critical graph. In this talk, we will discuss some recent results on extremal aspects of k-critical graphs, including improvments on the extremal number of edges/cliques/critical subgraphs in k-critical graphs.  This is based on some joint works with Jun Gao, Cong Luo and Tianchi Yang. 

Pages