TBA
We will consider the inverse problem of determining the sound
speed or index of refraction of a medium by measuring the travel times of
waves going through the medium. This problem arises in global seismology
in an attempt to determine the inner structure of the Earth by measuring
travel times of earthquakes. It also has several applications in optics
and medical imaging among others.
The problem can be recast as a geometric problem: Can one determine
the Riemannian metric of a Riemannian manifold with boundary by
measuring the distance function between boundary points? This is the
boundary rigidity problem.
We will also describe some recent results, joint with Plamen Stefanov
and Andras Vasy, on the partial data case, where you are making
measurements on a subset of the boundary.
Please Note: Special date and special room
We shall explain a simple remarkable stability phenomenon regarding the centers of the group algebras of the symmetric groups in n letters, as n goes to infinity. The same type of stability phenomenon extends to a wide class of finite groups including wreath products and finite general linear groups. Such stability has connections and applications to the cohomology rings of Hilbert schemes of n points on algebraic surfaces.
A fundamental result in Asymptotic Geometric Analysis is Dvoretzky’s theorem, which asserts that almost euclidean structure is locally present in any high-dimensional normed space. V. MIlman promoted the random version of the “Dvoretzky Theorem” by introducing the “concentration of measure Phenomenon.” Quantifying this phenomenon is important in theory as well as in applications. In this talk I will explain how techniques from High-dimensional Probability can be exploited to obtain optimal bounds on the randomized Dvoretzky theorem. Based on joint work(s) with Petros Valettas.
I will give a hopefully accessible introduction to some work on
tropical moduli spaces of curves and abelian varieties. I will report
on joint work with Madeline Brandt, Juliette Bruce, Margarida Melo,
Gwyneth Moreland, and Corey Wolfe, in which we find new rational
cohomology classes in the moduli space A_g of abelian varieties using
tropical techniques. And I will try to touch on a new point of view on
this topic, namely that of differential forms on tropical moduli
spaces, following the work of Francis Brown.
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.
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
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.
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.)
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?
