Thursday January 25, 2018 2:30 PM P131
 Leonid Kovalev, Syracuse University
Metrically removable setsA 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 ab. Every set of zero (n1)dimensional measure is metrically removable, but not conversely. Metrically removable sets can even have positive ndimensional measure.
I will describe some properties of metrically removable sets and outline a proof of the following fact: totally disconnected sets of finite (n1)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 P131
 Frank Thorne, South Carolina
Levels of Distribution for Prehomogeneous Vector SpacesThis 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 P131
 Max Engelstein, MIT
An Epiperimetric approach to singular points in the AltCaffarelli functionalWe prove a uniqueness of blowups result for isolated singular points in the free boundary of minimizers to the AltCaffarelli 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 areaminimizing hypersurfaces.

Thursday March 01, 2018 2:30 PM  03:30 AM P131
 Ewain Gwynne, MIT
Tutte embeddings of random planar maps via random walk in inhomogeneous random environmentsWe 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 matedCRT 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 spacefilling path on the embedded map converges to spacefilling 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 twodimensional 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 P131

CANCELED Harrison Pugh, Stony Brook University
Algebraic Structures on Currents

Thursday March 29, 2018 2:30 PM P131
 Silvia Ghinassi, Stony Brook University
Sufficient conditions for $C^{1,α}$ rectifiabilityWe 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 BishopJones 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.

Friday April 20, 2018 11:00 PM P131
 Evita Nestoridi, Princeton University
TBATBA

