Stony Brook Analysis Seminar, 2018-2019

Organizers

Chris Bishop, Robert Hough, and Raanan Schul.

Schedule

• September 27 Boris Bukh (Carnegie Mellon)
  
  Nearly orthogonal vectors
  
  Abstract. 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.
  
  
  
• October 4 Nicholas Edelen (MIT)
  
  Effective Reifenberg theorems for measures
  
  Abstract. 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.
  
  
  
• October 11 Dan Jerison (Tel Aviv, Israel) (Canceled)
  
  TBA
  
  Abstract. TBA
  
  
  
• October 18 Matthew Badger (University of Connecticut)
  
  Traveling along Holder curves
  
  Abstract. 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) Holder 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 Hlder curves and their subsets.
  
  
  
• October 25 Philippe Sosoe (Cornell University)
  
  A sharp transition for Gibbs measures associated to the nonlinear Schroedinger equation
  
  Abstract. 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)
  
  
  
• November 15 Vyron Vellis (UConn)
  
  Quasisymmetric uniformization of metric spheres
  
  Abstract. 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 $n \geq 2$. 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.
  
  
  
• February 7 Eun Hye Lee (U. Illinois Chicago)
  
  On Certain Multiple Dirichlet Series
  
  Abstract. In this talk, I will be talking about the analytic properties of multiple Dirichlet series defined using the space of binary cubic forms. First I will construct the double zeta function from the 2 (out of 4) semi-invariants of the binary cubic forms, and then I will prove its meromorphic continuation to the whole $\mathbb{C}^2$. This work is joint with Ramin Takloo-Bighash.
  
  
  
• March 7 Guy David (Ball State University)
  
  Lipschitz differentiability, embeddings, and rigidity for group actions
  
  Abstract. We discuss a class of metric spaces that, despite being non-Euclidean, support a first-order calculus for Lipschitz functions developed by Cheeger. After introducing these spaces, we will survey some of their embedding properties and explain a theorem of the speaker and Kyle Kinneberg concerning embeddings in Carnot groups. Then we will explain an application of this last result to a problem on group actions in hyperbolic geometry.
  
  
  
• March 14 Lutz Warnke (Georgia Tech)
  
  Large girth approximate Steiner triple systems
  
  Abstract. In 1973 Erdos asked whether there are n-vertex partial Steiner triple systems with arbitrary high girth and quadratically many triples. (Here girth is defined as the smallest integer g \ge 4 for which some g-element vertex-set contains at least g-2 triples.) We answer this question, by showing existence of approximate Steiner triple systems with arbitrary high girth. More concretely, for any fixed \ell \ge 4 we show that a natural constrained random process typically produces a partial Steiner triple system with (1/6-o(1))n^2 triples and girth larger than \ell. The process iteratively adds random triples subject to the constraint that the girth remains larger than \ell. Our result is best possible up to the o(1)-term, which is a negative power of n. This is joint work with Tom Bohman, see https://arxiv.org/abs/1808.01065.
  
  
  
• March 28 Sean Li (UConn)
  
  Traveling salesman in Carnot groups
  
  Abstract. The analyst's traveling salesman problem asks when a set E lies on a finite length rectifiable curve. In 1990, Peter Jones gave a solution for sets in R2 in terms of beta numbers. Given a set E and a ball B(x,r), the beta-number of E in B(x,r) is a scale-invariant measure of how much E deviates from an affine line in B(x,r). Jones showed that E lies on a rectifiable curve if and only if a weighted L2 integral of the beta numbers converge. Since then, the traveling salesman has been generalized to Rn and Hilbert spaces. We discuss recent work on generalizing the TSP to Carnot groups, a nonabelian generalization of Euclidean spaces. We show that a Jones-like traveling salesman theorem exists in this setting if one generalizes the beta-numbers in a correct way. The talk includes multiple joint works with Raanan Schul, Vasilis Chousionis, and Scott Zimmerman.
  
  
  
• April 4 Michael Damron (Georgia Tech University)
  
  Majority vote model on the 3-regular tree
  
  Abstract. In the majority vote model, each vertex of a graph is initially assigned a spin value of +1 or -1. At exponential times, vertices update their values by assuming the majority value of their neighbors. I will review some of the major questions and conjectures on lattices, and then explain some new work with Arnab Sen (Minnesota) on the 3-regular tree. We relate the majority vote model to a new model, which we call the median process, and use this process to answer questions about the limiting state of spins. For example, we show that when the initial state is given by a Bernoulli(p) product measure, the probability that a vertex's limiting spin is +1 is a continuous function of p.
  
  
  
• April 11 Li-Cheng Tsai (Columbia University)
  
  TBA
  
  Abstract. TBA.
  
  
  
• April 25 Vladimir Bozin (University of Belgrade)
  
  TBA
  
  Abstract. TBA.
  
  
  
• May 2 Hoi Nguyen (Ohio State University)
  
  TBA
  
  Abstract. TBA.