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.

Thursday April 12, 2018 2:30 PM P131
 Beibei Liu, UC Davis
Geometric finiteness in negatively pinched Hadamard manifoldsWe generalize Bonahon’s characterization of geometrically infinite torsionfree discrete subgroups of PSL(2, C) to geometrically infinite discrete torsionfree 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 P131
 Chris Green, Macquarie University, Australia
Using the SchottkyKlein prime function to solve problems in multiply connected domains.The SchottkyKlein 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 closedform. In this talk, we will present the solutions to three different problems where it has been advantageous to employ the SchottkyKlein prime function and its associated function theory in order to construct closedform analytical solutions.

Friday April 20, 2018 11:00 AM P131
 Evita Nestoridi, Princeton University
Optimal strong stationary times for random walks on the chambers of a hyperplane arrangement.The BidigareHanlonRockmore 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.

