Seminars and Colloquia by Series

Structured multi-objective optimization: Optimization on dynamic graphs and multi-task learning

Series
Colloquia
Time
Thursday, November 17, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Justin RombergGeorgia Tech

We will discuss two types of structured multi-objective optimization programs.  In the first, the goal is to minimize a sum of functions described by a graph: each function is associated with a vertex, and there is an edge between vertices if two functions share a subset of their variables.   Problems of this type arise in state estimation problems, including simultaneous localization and mapping (SLAM) in robotics, tracking, and streaming reconstruction problems in signal processing.  We will show that under mild smoothness conditions, these types of problems exhibit a type of locality: if a node is added to the graph (changing the optimization problem), the optimal solution changes only for variables that are ``close’’ to the added node, immediately giving us a quick way to update the solution as the graph grows.

In the second part of the talk, we will consider a multi-task learning problem where the solutions are expected to lie in a low-dimensional subspace.  This corresponds to a low-rank matrix recover problem where the columns of the matrix have been ``sketched’’ independently.  We show that a novel convex relaxation of this problem results in optimal sample complexity bounds.  These bounds demonstrate the statistical leverage we gain by solving the problem jointly over solving each individually.

Coprime matchings and lonely runners

Series
Colloquia
Time
Thursday, November 10, 2022 - 11:00 for
Location
Skiles 006
Speaker
Tom BohmanCarnegie Mellon University

Suppose n runners are running on a circular track of circumference 1, with all runners starting at the same time and place. Each runner proceeds at their own constant speed. We say that a runner is lonely at some point in time if the distance around the track to the nearest other runner is at least 1/n. For example, if there two runners then there will come a moment when they are at anitpodal points on the track, and at this moment both runners are lonely. The lonely runner conjecture asserts that for every runner there is a point in time when that runner is lonely. This conjecture is over 50 years old and remains widely open.

A coprime matching of two sets of integers is a matching that pairs every element of one set with a coprime element of the other set. We present a recent partial result on the lonely runner conjecture. Coprime matchings of intervals of integers play an central role in the proof of this result.

Joint work with Fei Peng

Linear and nonlinear stability of shear flows and vortices

Series
Colloquia
Time
Thursday, October 27, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
Online: Zoom link: https://gatech.zoom.us/j/96410391996?pwd=VkQvcUdoREtsbUJPNVFTbzdKaC9TQT09
Speaker
Alexandru IonescuPrinceton University

I will talk about some recent work on the stability problem of shear flows and vortices as solutions of the Euler equations in 2D.  Our results include nonlinear stability theorems for monotonic shear  flows and point vortices, as well as linear stability theorems for more general flows. This is joint work with Hao Jia.

Learning to Solve Hard Minimal Problems

Series
Colloquia
Time
Thursday, October 13, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Anton LeykinGeorgia Tech

The main result in this talk concerns a new fast algorithm to solve a minimal problem with many spurious solutions that arises as a relaxation of a geometric optimization problem. The algorithm recovers relative camera pose from points and lines in multiple views. Solvers like this are the backbone of structure-from-motion techniques that estimate 3D structures from 2D image sequences.   

Our methodology is general and applicable in areas other than computer vision. The ingredients come from algebra, geometry, numerical methods, and applied statistics. Our fast implementation relies on a homotopy continuation optimized for our setting and a machine-learned neural network.

(This covers joint works with Tim Duff, Ricardo Fabbri, Petr Hruby, Kathlen Kohn, Tomas Pajdla, and others. The talk is suitable for both professors and students.)

Recurrent solutions and dynamics of turbulent flows

Series
Colloquia
Time
Thursday, September 22, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Predrag CvitanovićSchool of Physics, Georgia Tech

In the world of moderate, everyday turbulence of fluids flowing across planes and down pipes, a quiet revolution is taking place. Applied mathematicians can today compute 'exact coherent structures', i.e. numerically precise 3D, fully nonlinear Navier-Stokes solutions: unstable equilibria, traveling waves, and (relative) periodic orbits. Experiments carried out at Georgia Tech today yield measurements as detailed as the numerical simulations; our experimentalists measure 'exact coherent structures' and trace out their unstable manifolds. What emerges is a dynamical systems theory of low-Reynolds turbulence as a walk among sets of weakly unstable invariant solutions.

 

We take you on a tour of this newly breached, hitherto inaccessible territory. Mastery of fluid mechanics is no prerequisite, and perhaps a hindrance: the talk is aimed at anyone who had ever wondered why - if no cloud is ever seen twice - we know a cloud when we see one? And how do we turn that into mathematics?

Topological full groups

Series
Colloquia
Time
Tuesday, November 17, 2015 - 10:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Professor Volodymyr NekrashevychDepartment of Mathematics, Texas A&M

Please Note: This talk should interest people in Algebra, Dynamical Systems and Mathematical Physics in addition to Geometry and Topology. Volodia Nekrashevych will visit Atlanta from Sunday November 15th evening until Tuesday November 17th afternoon. He will be available for private talks on Monday November 14th after noon or on Tueasday morning before 10AM. Contact him directly by email or contact jeanbel@math.gatech.edu to schedule a meeting or to have a dinner with him.

Topological full groups are naturally associated with semigroups of local homeomorphisms: iterations of a single homeomorphism, holonomy groupoids of laminations, groupoids of local isomorphisms of quasiperiodic sets (for example Penrose tilings), etc. Some of these groups have interesting properties from the point of view of group theory. For instance, they provide first examples of amenable infinite simple finitely generated groups (by a result of K. Juschenko and N. Monod) and first examples of simple amenable groups of Burnside type. The full group of the Penrose tiling is another interesting example from the point of view of amenability.