Stony Brook Analysis Seminar, Fall 2017
Thursday 2.30  3.30 pm
Room P131
Schedule

November 2
Nina Holden (MIT)
Convergence of percolationdecorated triangulations to SLE and LQG
Abstract.
The SchrammLoewner evolution (SLE) is a family of random fractal curves, which is the proven or conjectured scaling limit of a variety of twodimensional lattice models in statistical mechanics. Liouville quantum gravity (LQG) is a model for a random surface which is the proven or conjectured scaling limit of discrete surfaces known as random planar maps (RPM). We consider scaling limits results for percolationdecorated RPM to SLEdecorated LQG. Based on joint works with Bernardi, Garban, Gwynne, Miller, Sepulveda, Sheffield, and Sun.
Slides from the talk are available here.

November 16
Megan Bernstein (Georgia Tech)
Progress in showing cutoff for random walks on the symmetric group
Abstract. Cutoff is a remarkable property of many Markov chains in which they rapidly transition from an unmixed to a mixed distribution. Most random walks on the symmetric group, also known as card shuffles, are believed to mix with cutoff, but we are far from being able to proof this. We will survey existing cutoff results and techniques for random walks on the symmetric group, and present three recent results: cutoff for a biased transposition walk, cutoff for the randomtorandom card shuffle (answering a 2001 conjecture of Diaconis), and precutoff for the involution walk, generated by permutations with a binomially distributed number of twocycles. The results use either probabilistic techniques such as strong stationary times or diagonalization through algebraic combinatorics and representation theory of the symmetric group.
Includes joint work with Nayantara Bhatnagar, Evita Nestoridi, and Igor Pak.

November 30
Paul Bourgade (Courant Institute)
Random matrices, the Riemann zeta function and trees
Abstract. Fyodorov, Hiary & Keating have conjectured that the maximum of the characteristic polynomial of random unitary matrices behaves like extremes of logcorrelated Gaussian fields. This allowed them to predict the typical size of local maxima of the Riemann zeta function along the critical axis. I will first explain the origins of this conjecture, and then outline the proof for the leading order of the maximum, for unitary matrices and the zeta function. This talk is based on joint works with Arguin, Belius, Radziwill and Soundararajan.

January 26
Frank Thorne (South Carolina)
Levels of Distribution for Prehomogeneous Vector Spaces
Abstract. This will be a continuation of Thursday's colloquium, 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.

February 15
Max Engelstein (MIT)
An Epiperimetric approach to singular points in the AltCaffarelli functional
Abstract. We 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.

March 1
Ewain Gwynne (MIT)
Tutte embeddings of random planar maps via random walk in inhomogeneous random environments
Abstract. We prove that the Tutte embeddings (a.k.a. harmonic/barycentric embeddings) of certain random planar maps converge to γLiouville quantum gravity (γLQG). 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 SLEκ for κ=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.

March 29
Silvia Ghinassi (Stony Brook)
Sufficient conditions for C^{1,α} rectifiability
Abstract. We provide sufficient conditions for a set or measure in Rn to be C^{1,α} drectifiable, 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.

April 12
Beibei Liu (UC Davis)
Geometric finiteness in negatively pinched Hadamard manifolds.
Abstract. We generalize Bonahon's characterization of geometrically infinite torsionfree discrete subgroups of PSL(2, C) to geometrically infinite discrete torsionfree subgroups $\Gamma$ of isometries of negatively pinched Hadamard manifolds $X$. We then generalize a theorem of Bishop to prove that every such geometrically infinite isometry subgroup $\Gamma$ has a set of nonconical limit points with cardinality of continuum.

April 19
Chris Green (Macquarie University)
Using the SchottkyKlein prime function to solve problems in multiply connected domains.
Abstract. 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.

April 20
Evita Nestoridi (Princeton)
Optimal strong stationary times for random walks on the chambers of a hyperplane arrangement.
Abstract. 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.