Analysis Seminar

from Friday
June 01, 2018 to Monday
December 31, 2018
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Thursday
September 27, 2018

2:30 PM - 03:30 AM
P-131
Boris Bukh, Carnegie Mellon University
Nearly orthogonal vectors

How can d+k vectors in R^d be arranged so that they are as close to orthogonal as possible? We show intimate connection of this problem to the problem of equiangular lines, and to the problem of bounding the first moment of isotropic measures. Using these connections, we pin down the answer precisely for several values of k and establish asymptotics for all k. Joint work with Chris Cox.


Thursday
October 04, 2018

2:30 PM
P-131
Nicholas Edelen, MIT
Effective Reifenberg theorems for measures

The Jones' $β$-numbers quantify how ``linear'' is the support of a measure. These have important uses in singularity analysis of solutions to PDE and harmonic analysis. In this talk, I explain joint work with Aaron Naber and Daniele Valtorta which gives quantitative control and Lipschitz structure on measures satisfying natural conditions on the $β$-numbers, and generalizations of our results to infinite-dimensional spaces. Our work can be viewed as an ``analyst's traveling-salesman'' type theorem.


Thursday
October 18, 2018

2:30 PM
P-131
Matthew Badger, University of Connecticut
Traveling along Hölder curves

One goal of geometric measure theory is to understand a measure through its interaction with canonical lower dimension sets. The interaction of Radon measures in the plane or a higher-dimensional Euclidean space with finite sets or rectifiable curves is now completely understood. However, with respect to any other elementary family of sets, we only know how measures behave under additional regularity hypotheses. To make progress towards understanding the structure of Radon measures, we need to first understand the geometry of more classes of sets.

I will describe my latest work with L. Naples and V. Vellis, in which we find sufficient conditions to identify (subsets of) Hölder continuous curves of Hausdorff dimension $s>1$. Our conditions are related to the Analyst's Traveling Salesman Theorem, which characterizes subsets of rectifiable curves. On the other hand, standard self-similar sets such as the Sierpinski carpet show that our sufficient condition is not necessary. I will discuss this and other obstructions to the problem of characterizing Hölder curves and their subsets.


Thursday
October 25, 2018

2:30 PM
P-131
Philippe Sosoe, Cornell University
A sharp transition for Gibbs measures associated to the nonlinear Schroedinger equation

In 1987, Lebowitz, Rose and Speer (LRS) showed how to
construct formally invariant measures for the nonlinear Schroedinger
equation on the torus. This seminal contribution spurred a large amount
of activity in the area of partial differential equations with random
initial data. In this talk, I will explain LRS's result, and discuss a
sharp transition in the construction of the Gibbs-type invariant
measures considered by these authors. (Joint work with Tadahiro Oh and
Leonardo Tolomeo)


Thursday
November 15, 2018

2:30 PM
P-131
Vyron Vellis, University of Connecticut
Quasisymmetric uniformization of metric spheres

One of the biggest problems in Quasiconformal Analysis is the classification of metric spaces which are quasisymmetric (or quasiconformal) to the unit sphere $\mathbb{S}^n$. While settled for $n=1$, the problem is completely open for $n ≥ 2$. In this talk we present a survey on this question and some recent results regarding on how bad (geometrically and analytically) quasisymmetric spheres can be and, similarly, how good non-quasisymmetric spheres can be.


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