Analysis Seminar

from Monday
January 01, 2018 to Thursday
May 31, 2018
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Instructions for subscribing to Stony Brook Math Department Calendars

Thursday
January 25, 2018

2:30 PM
P-131
Leonid Kovalev, Syracuse University
Metrically removable sets

A compact subset K of Euclidean space R^n is called metrically removable if any two points a,b of its complement can be joined by a curve that is disjoint from K and has length arbitrarily close to |a-b|. Every set of zero (n-1)-dimensional measure is metrically removable, but not conversely. Metrically removable sets can even have positive n-dimensional measure.
I will describe some properties of metrically removable sets and outline a proof of the following fact: totally disconnected sets of finite (n-1)-dimensional measure are metrically removable. This answers a question raised by Hakobyan and Herron in 2008.
Joint work with Sergei Kalmykov and Tapio Rajala.


Friday
January 26, 2018

11:00 AM - 12:00 AM
P-131
Frank Thorne, South Carolina
Levels of Distribution for Prehomogeneous Vector Spaces

This will be a continuation of Thursday's talk, where I will explain multiple approaches to the lattice point counting problem. The quantitatively strongest estimates all involve Fourier analysis in some guise, which turns out to have nice interplay with the action of the group.


Thursday
February 15, 2018

2:30 PM
P-131
Max Engelstein, MIT
An Epiperimetric approach to singular points in the Alt-Caffarelli functional

We prove a uniqueness of blowups result for isolated singular points in the free boundary of minimizers to the Alt-Caffarelli functional. The key tool is a (log-)epiperimetric inequality, which we prove by means of two Fourier expansions (one on the function and one on its free boundary).

If time allows we will also explain how this approach can be adapted to (re)prove old and new regularity results for area-minimizing hypersurfaces.


Thursday
March 01, 2018

2:30 PM - 03:30 AM
P-131
Ewain Gwynne, MIT
Tutte embeddings of random planar maps via random walk in inhomogeneous random environments

We prove that the Tutte embeddings (a.k.a. harmonic/barycentric embeddings) of certain random planar maps converge to $γ$-Liouville quantum gravity ($γ$-LQG). Specifically, we treat mated-CRT maps, which are discretized matings of correlated continuum random trees, and $γ$ ranges from $0$ to $2$ as one varies the correlation parameter. We also show that the associated space-filling path on the embedded map converges to space-filling SLE$_{\kappa}$ for $\kappa =16/γ^2$ (in the annealed sense) and that the embedded random walk converges to Brownian motion (in the quenched sense). This constitutes the first proof that discrete conformal embeddings of random planar maps converge to LQG.

To accomplish this, we establish a very general quenched invariance principle for random walk in a two-dimensional random environments in which the length scale is allowed to vary from place to place, so that the environment is only translation invariance modulo scaling. This result contains a number of existing results concerning random walk in random environments, and we expect that it will be useful in other settings as well, including other random environments related to random planar maps and LQG.

Based on joint work with Jason Miller and Scott Sheffield.


Thursday
March 08, 2018

2:30 PM
P-131
CANCELED- Harrison Pugh, Stony Brook University
Algebraic Structures on Currents


Thursday
March 29, 2018

2:30 PM
P-131
Silvia Ghinassi, Stony Brook University
Sufficient conditions for $C^{1,α}$ rectifiability

We provide sufficient conditions for a set or measure in $\mathbb{R}^n$ to be $C^{1,α}$ $d$-rectifiable, with $α ∈ [0,1]$. The conditions use a Bishop-Jones type square function and all statements are quantitative in that the $C^{1,α}$ constants depend on such a function. Key tools for the proof come from Guy David and Tatiana Toro's parametrization of Reifenberg flat sets (with holes) in the Hölder and Lipschitz categories.


Thursday
April 12, 2018

2:30 PM
P-131
Beibei Liu, UC Davis
Geometric finiteness in negatively pinched Hadamard manifolds

We generalize Bonahonís characterization of geometrically infinite torsion-free discrete subgroups of PSL(2, C) to geometrically infinite discrete torsion-free subgroups $Γ$ of isometries of negatively pinched Hadamard manifolds $X$. We then generalize a theorem of Bishop to prove that every such geometrically infinite isometry subgroup $Γ$ has a set of nonconical limit points with cardinality of continuum.


Thursday
April 19, 2018

2:30 PM
P-131
Chris Green, Macquarie University, Australia
Using the Schottky-Klein prime function to solve problems in multiply connected domains.

The Schottky-Klein prime function is a special transcendental function which plays a central role in problems involving multiply connected domains (i.e. domains with multiple boundary components). Despite this, it has been scarcely used by pure and applied mathematicians since it was originally written down (independently by both Schottky and Klein towards the end of the 19th century). It turns out this function can be used to solve many mathematical problems set in multiply connected domains, exactly or in closed-form. In this talk, we will present the solutions to three different problems where it has been advantageous to employ the Schottky-Klein prime function and its associated function theory in order to construct closed-form analytical solutions.


Friday
April 20, 2018

11:00 AM
P-131
Evita Nestoridi, Princeton University
Optimal strong stationary times for random walks on the chambers of a hyperplane arrangement.

The Bidigare-Hanlon-Rockmore random walk on the chambers of real hyperplane arrangements is a Markov chain that generalizes famous examples, such as the Tsetlin library and riffle shuffles. We will introduce lower bounds for the separation distance and a strong stationary time, which allow for the first time to study cutoff for hyperplane arrangement walks under certain conditions. We will also discuss how the method for the lower bound can be used to prove a uniform lower bound for the mixing time of Glauber dynamics on a monotone system, such as the Ising model.


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