Abstracts

Week of November 27, 2022

November 28, 2022
Applied Mathematics A new version of the fast Gauss transform and the simulation of diffusion in complex geometries 3:00pm -
AKW 200

Abstract: We will review the state of the art in fast algorithms for the solution of the heat equation in moving geometries, using integral equation methods. Such methods achieve optimal complexity and, in the homogeneous case, require the discretization of the space-time boundary alone. They achieve high order accuracy with suitable quadratures and are straightforward to implement adaptively in space-time. This set of tools has direct application to biophysical modeling, reaction-diffusion systems, and computational fluid dynamics.
Group Actions, Geometry and Dynamics Approximate lattices in algebraic groups 4:00pm -
 
Approximate lattices in locally compact groups are approximate subgroups that are discrete and have finite co-volume. They provide natural examples of objects at the intersection of the theory of discrete subgroups of Lie groups, ergodic theory and additive combinatorics.
 
A central question arising from seminal work of Yves Meyer asks whether approximate lattices have an arithmetic origin. I will present a complete structure theorem for approximate lattices in linear algebraic groups in terms of bounded cohomology that, in particular, answers this question. I will pinpoint key instances where the interplay between additive combinatorics and other fields - e.g. ergodic theory and algebraic groups - is particularly fruitful. 
Geometry, Symmetry and Physics Seminar cancelled 4:30pm -
LOM 214
November 29, 2022
Geometry & Topology Minimal Surface Entropy for Negatively Curved N-manifolds and Rigidity 4:15pm -
LOM 214

We focus on an odd-dimensional closed manifold M that admits a hyperbolic metric. For any metric on M with sectional curvature less than or equal to -1, we introduce the minimal surface entropy to count the number of surface subgroups. It attains the minimum if and only if the metric is hyperbolic. This result is an extension of the work on 3-manifolds by Calegari-Marques-Neves. I will introduce their ideas and discuss the problems and solutions for higher dimensions. If time permits, I will mention the results for locally symmetric spaces and hyperbolic 3-manifolds of finite volume.
November 30, 2022
Colloquium Complex proofs of real theorems. 4:15pm -
LOM 214
The title refers to Painlevé’s and Hadamard’s famous dictum, “the shortest and best way between two truths of the real domain often passes through the imaginary one.” I’ll discuss examples from my research on this theme. By ‘complexifying’ a problem the goal is to make it, in fact, less complex. Often, one can achieve a surprising level of simplicity this way. Topics include quasiconvexity problem in calculus of variations, bilipschitz rotational multifractal spectra and limit shapes of random Young tableaux. 
December 2, 2022
Friday Morning Seminar Friday Morning Seminar 10:30am -
LOM 205

A relaxed-pace seminar on impromptu subjects related to the interests of the audience.

Everyone is welcome.

The subjects are geometry, probability, combinatorics, dynamics, and more!
Graduate Student Seminar Consensus on Dynamic Stochastic Block Models 12:00pm -
LOM

We introduce two models of consensus following a majority rule on time-evolving stochastic block models (SBM), in which the network evolution is Markovian or non-Markovian. Under the majority rule, in each round, each agent simultaneously updates his/her opinion according to the majority of his/her neighbors. Our network has a community structure and randomly evolves with time. In contrast to the classic setting, the dynamics is not purely deterministic, and reflects the structure of SBM by resampling the connections at each step, making agents with the same opinion more likely to connect than those with different opinions. In the Markovian model, connections between agents are resampled at each step according to the SBM law and each agent updates his/her opinion via the majority rule. In the non-Markovian model, a connection between two agents is resampled according to the SBM law only when some of the two changes opinion and is otherwise kept the same. We study the phase transition between the fast convergence to the consensus and a halt of the dynamics. Moreover, we establish thresholds of the initial lead for various convergence speeds. This is based on joint work with J. Wei and Z. Zhang.
Geometric Analysis and Application Existence of static metrics with prescribed Bartnik boundary data 2:00pm -

A static vacuum metric produces a Ricci flat manifold of one dimension higher and naturally arises on studying scalar curvature deformation and gluing. Originating from his quasi-local mass program in 1989, R. Bartnik conjectured that one can always find an asymptotically flat, static vacuum metric with quite arbitrarily prescribed Bartnik boundary data. I will discuss well-posedness of this geometric boundary value problem and the recent progress toward the conjecture for large classes of boundary data. It is based on joint work with Zhongshan An.