Seminars and Colloquia by Series

Dynamical critical 2d first-passage percolation

Series
Stochastics Seminar
Time
Thursday, March 31, 2022 - 15:30 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
David HarperGeorgia Tech

In first-passage percolation (FPP), we let \tau_v be i.i.d. nonnegative weights on the vertices of a graph and study the weight of the minimal path between distant vertices. If F is the distribution function of \tau_v, there are different regimes: if F(0) is small, this weight typically grows like a linear function of the distance, and when F(0) is large, the weight is typically of order one. In between these is the critical regime in which the weight can diverge, but does so sublinearly. This talk will consider a dynamical version of critical FPP on the triangular lattice where vertices resample their weights according to independent rate-one Poisson processes. We will discuss results which show that if sum of F^{-1}(1/2+1/2^k) diverges, then a.s. there are exceptional times at which the weight grows atypically, but if sum of k^{7/8} F^{-1}(1/2+1/2^k) converges, then a.s. there are no such times. Furthermore, in the former case, we compute the Hausdorff and Minkowski dimensions of the exceptional set and show that they can be but need not be equal. These results show a wider range of dynamical behavior than one sees in subcritical (usual) FPP. This is a joint work with M. Damron, J. Hanson, W.-K. Lam.

This talk will be given on Bluejeans at the link https://bluejeans.com/547955982/2367

The mathematical theory of wave turbulence

Series
PDE Seminar
Time
Thursday, March 31, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 114
Speaker
Zaher HaniUniversity of Michigan

Please Note: Meeting also available online: https://gatech.zoom.us/j/92742811112

Wave turbulence is the theory of nonequilibrium statistical mechanics for wave systems. Initially formulated in pioneering works of Peierls, Hasselman, and Zakharov early in the past century, wave turbulence is widely used across several areas of physics to describe the statistical behavior of various interacting wave systems. We shall be interested in the mathematical foundation of this theory, which for the longest time had not been established.

The central objects in this theory are: the "wave kinetic equation" (WKE), which stands as the wave analog of Boltzmann’s kinetic equation describing interacting particle systems, and the "propagation of chaos” hypothesis, which is a fundamental postulate in the field that lacks mathematical justification. Mathematically, the aim is to provide a rigorous justification and derivation of those two central objects; This is Hilbert’s Sixth Problem for waves. The problem attracted considerable interest in the mathematical community over the past decade or so. This culminated in recent joint works with Yu Deng (University of Southern California), which provided the first rigorous derivation of the wave kinetic equation, and justified the propagation of chaos hypothesis in the same setting.

Meeting also available online: https://gatech.zoom.us/j/92742811112

Competition, Phenotypic Adaptation, and the Evolution of a Species' Range

Series
Mathematical Biology Seminar
Time
Wednesday, March 30, 2022 - 10:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Farshad ShiraniSchool of Mathematics, Georgia Institute of Technology

Please Note: Please Note: Meeting Link: https://bluejeans.com/426529046/8775

Why is a species’ geographic range where it is? Immediate thoughts such as penguins cannot climb steep cliffs or colonize deserts are often not the answer. In fact, identifying causes of species’ range limits is a fundamental problem in evolutionary ecology that has crucial implications in conservation biology and understanding mechanisms of speciation.

In this talk, I will briefly introduce some of the biotic, genetic, and environmental processes that can determine a species’ range. I will then focus on two of such processes, competition and (mal)adaptation to heterogeneous environments, that are commonly thought to halt  species’ range expansion and stabilize their range boundary. I will present a model of species range dynamics that incorporates these eco-evolutionary processes in a community of biologically related species. I will discuss biologically plausible ranges of values for the parameters of this model, and will demonstrate its dynamic behavior in a number of different evolutionary regimes.

Recent advances in Ramsey theory

Series
Graph Theory Seminar
Time
Tuesday, March 29, 2022 - 15:45 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Dhruv MubayiUniversity of Illinois at Chicago

Ramsey theory studies the paradigm that every sufficiently large system contains a well-structured subsystem. Within graph theory, this translates to the following statement: for every positive integer $s$, there exists a positive integer $n$ such that for every partition of the edges of the complete graph on $n$ vertices into two classes, one of the classes must contain a complete subgraph on $s$ vertices. Beginning with the foundational work of Ramsey in 1928, the main question in the area is to determine the smallest $n$ that satisfies this property.

For many decades, randomness has proved to be the central idea used to address this question. Very recently, we proved a theorem which suggests that "pseudo-randomness" and not complete randomness may in fact be a more important concept in this area. This new connection opens the possibility to use tools from algebra, geometry, and number theory to address the most fundamental questions in Ramsey theory. This is joint work with Jacques Verstraete.

Complex Ball Quotients and New Symplectic 4-Manifolds with Nonnegative Signatures

Series
Geometry Topology Seminar
Time
Tuesday, March 29, 2022 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Sumeyra SakalliUniversity of Arkansas

Please Note: Note this talk is at a different time and day

We first construct a complex surface with positive signature, which is a ball quotient. We obtain it as an abelian Galois cover of CP^2 branched over the Hesse arrangement. Then we analyze its fibration structure, and by using it we build new symplectic and also non-symplectic exotic 4-manifolds with positive signatures.

 

In the second part of the talk, we discuss Cartwright-Steger surfaces, which are also ball quotients. Next, we present our constructions of new symplectic and non-symplectic exotic 4-manifolds with non-negative signatures that have the smallest Euler characteristics in the so-called ‘arctic region’ on the geography chart.

 

More precisely, we prove that there exist infinite families of irreducible symplectic and infinite families of irreducible non-symplectic, exotic 4-manifolds that have the smallest Euler characteristics among the all known simply connected 4-manifolds with nonnegative signatures and with more than one smooth structures. This is a joint work with A. Akhmedov and S.-K. Yeung.

Image formation ideals

Series
Algebra Seminar
Time
Tuesday, March 29, 2022 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Tim DuffUniversity of Washington

Projective space, rational maps, and other notions from algebraic geometry appear naturally in the study of image formation and various camera models in computer vision. Considerable attention has been paid to multiview ideals, which collect all polynomial constraints on images that must be satisfied by a given camera arrangement. We extend past work on multiview ideals to settings where the camera arrangement is unknown. We characterize various "image formation ideals", which are interesting objects in their own right. Some nice previous results about multiview ideals also fall out from our framework. We give a new proof of a result by Aholt, Sturmfels, and Thomas that the multiview ideal has a universal Groebner basis consisting of k-focals (also known as k-linearities in the vision literature) for k in {2,3,4}. (Preliminary report based on ongoing joint projects with Sameer Agarwal, Max Lieblich, Jessie Loucks Tavitas, and Rekha Thomas.)

How Differential Equations Insight Benefit Deep Learning

Series
Applied and Computational Mathematics Seminar
Time
Monday, March 28, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
https://gatech.zoom.us/j/96551543941 (note: Zoom, not Bluejeans)
Speaker
Prof. Bao WangUniversity of Utah

We will present a new class of continuous-depth deep neural networks that were motivated by the ODE limit of the classical momentum method, named heavy-ball neural ODEs (HBNODEs). HBNODEs enjoy two properties that imply practical advantages over NODEs: (i) The adjoint state of an HBNODE also satisfies an HBNODE, accelerating both forward and backward ODE solvers, thus significantly accelerate learning and improve the utility of the trained models. (ii) The spectrum of HBNODEs is well structured, enabling effective learning of long-term dependencies from complex sequential data.

Second, we will extend HBNODE to graph learning leveraging diffusion on graphs, resulting in new algorithms for deep graph learning. The new algorithms are more accurate than existing deep graph learning algorithms and more scalable to deep architectures, and also suitable for learning at low labeling rate regimes. Moreover, we will present a fast multipole method-based efficient attention mechanism for modeling graph nodes interactions.

Third, if time permits, we will discuss proximal algorithms for accelerating learning continuous-depth neural networks.

Quasi-morphisms on Surface Diffeomorphism groups

Series
Geometry Topology Seminar
Time
Monday, March 28, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Speaker
Jonathan Bowden

We discuss the problem of constructing quasi-morphisms on the group of diffeomorphisms of a surface that are isotopic to the identity, thereby resolving a problem of Burago-Ivanov-Polterovich from the mid 2000’s. This is achieved by considering a new kind of curve graph, in analogy to the classical curve graph first studied by Harvey in the 70’s, on which the full diffeomorphism group acts isometrically. Joint work with S. Hensel and R. Webb. 

Application of optimal transport theory on numerical computation, analysis, and dynamical systems on graph

Series
Dissertation Defense
Time
Wednesday, March 23, 2022 - 14:00 for
Location
ONLINE
Speaker
Shu LiuGeorgia Institute of Technology

Abstract: 

In this talk, we mainly focus on the applications of optimal transport theory from the following two aspects:

(1)Based on the theory of Wasserstein gradient flows, we develop and analyze a numerical method proposed for solving high-dimensional Fokker-Planck equations (FPE). The gradient flow structure of FPE allows us to derive a finite-dimensional ODE by projecting the dynamics of FPE onto a finite-dimensional parameter space whose parameters are inherited from certain generative model such as normalizing flow. We design a bi-level minimization scheme for time discretization of the proposed ODE. Such algorithm is sampling-based, which can readily handle computations in high-dimensional space. Moreover, we establish theoretical bounds for the asymptotic convergence analysis as well as the error analysis for our proposed method.

(2)Inspired by the theory of Wasserstein Hamiltonian flow, we present a novel definition of stochastic Hamiltonian process on graphs as certain kinds of inhomogeneous Markov process. Such definition is motivated by lifting to the probability space of the graph and considering the Hamiltonian dynamics on this probability space. We demonstrate some examples of the stochastic Hamiltonian process in classical discrete problems, such as the optimal transport problems and Schrödinger bridge problems (SBP).

The Bluejeans link: https://bluejeans.com/982835213/2740

Definable combinatorics in hyperfinite graphs

Series
Combinatorics Seminar
Time
Friday, March 18, 2022 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Matthew BowenMcGill University

We discuss a few new results concerning the descriptive combinatorics of bounded degree hyperfinite Borel graphs. In particular, we show that the Baire measurable edge chromatic number of $G$ is at most $\lceil\frac{3}{2}\Delta(G)\rceil+6$ when G is a multigraph, and for bipartite graphs we improve this bound to $\Delta(G)+1$ and show that degree regular one-ended bipartite graphs have Borel perfect matchings generically. Similar results hold in the measure setting assuming some hyperfiniteness conditions. This talk is based on joint work with Kun and Sabok, Weilacher, and upcoming work with Poulin and Zomback.

Pages