# Seminars & Events for 2017-2018

##### Stability of some super-resolution problems

The problem of computational super-resolution asks to recover an object from its noisy and limited spectrum. In this talk, we consider two inverse problems of this flavor, mainly from the point of view of stability estimates.

##### Min-max theory for constant mean curvature hypersurfaces

We describe the construction of closed constant mean curvature (CMC) hypersurfaces using min-max methods. In particular, our theory allows us to show the existence of closed CMC hypersurfaces of any prescribed mean curvature in any closed Riemannian manifold. This work is joint with Xin Zhou.

##### Log term of the Bergman kernel and the deformation complex for CR structures

Fefferman showed that the Bergman kernel of strictly pseudoconvex domains admit logarithmic singularity. For the case of the ball, the log term vanishes and it is conjectured that the log term vanishes only for the ball. Robin Graham proved it in 2-dimensions, while for higher dimensions, counter examples were found for domains which are not Stein. However, I still believe that the conjectur

##### The Slow Bond Conjecture

The slow bond model is a variant of the 1-dimensional Totally Asymmetric Exclusion Process, where the bond at the origin rings at a slower rate r<1. Janowsky and Lebowitz conjectured that for all values of r<1, the single slow bond produces a macroscopic change in the current of the system. Equivalently, this can be viewed as a pinning problem in last passage

##### Bernoulli numbers

We discuss (and define) the Bernoulli numbers, a sequence of fractions remarkable from several points of view. The numerators and denominators are entries A027642 and A000367 in the On-Line Encyclopedia of Integer Sequences (http://oeis.org/A027642 and https://oeis.org/A000367).

##### Growth rates of unbounded orbits in non-periodic twist maps and a theorem by Neishtadt

We consider twist maps on the plane (like the ping-pong map) with non-periodic angles, where typically bounded and unbounded motions co-exist. For the latter case we prove a theorem which shows that in the analytic setting the growth rate is at most logarithmic, and furthermore an example of a system is given where all orbits grow at this rate.

##### Rainbow matchings.

Given a family F_1,...,F_m of sets of edges in a graph or hypergraph, a ``rainbow matching'' is a choice of disjoint edges

e_1 in F_1,..., e_m in F_m. The talk is on open problems regarding this notion, some old and some new results, and what topology has to contribute to the subject.

##### Complex curves through a contact lens

Every four-dimensional Stein domain has a height function whose regular level sets are contact three-manifolds. This allows us to study complex curves in the Stein domain via their intersection with these contact level sets, where we can comfortably apply three-dimensional tools.

##### Some recent results on wave turbulence and quantum kinetics theories.

Wave turbulence is a branch of science studying the out-of-equilibrium statistical mechanics of random nonlinear waves of all kinds and scales.

Despite the fact that wave fields in nature are enormous diverse; to describe the processes of random wave interactions, there is a common mathematical concept; the wave kinetic equations.

##### Kloosterman sums and Siegel zeros

Kloosterman sums arise naturally in the study of the distribution of various arithmetic objects in analytic number theory. The 'vertical' Sato-Tate law of Katz describes their distribution over a fixed field F_p, but the equivalent 'horizontal' distribution as the base field varies over primes remains open.

##### Some recent work on conformal biharmonic maps

Biharmonic maps are generalizations of harmonic maps and biharmonic functions. As solutions of a system of 4th order PDEs, examples and the general properties of biharmonic maps are hard to reveal. In this talk, we will talk about some recent work on the study of biharmonic maps among conformal maps.

##### Nonuniqueness of weak solutions to the Navier-Stokes equation

For initial datum of finite kinetic energy Leray has proven in 1934 that there exists at least one global in time finite energy weak solution of the 3D Navier-Stokes equations. In this talk, I will discuss very recent joint work with Vlad Vicol in which we prove that weak solutions of the 3D Navier-Stokes equations are not unique in the class of weak solutions with finite kinetic energy

##### Trace reconstruction for the deletion channel

In the trace reconstruction problem, an unknown string $x$ of $n$ bits is observed through the deletion channel, which deletes each bit with some constant probability $q$, yielding a contracted string. How many independent outputs (traces) of the deletion channel are needed to reconstruct $x$ with high probability?

##### Gravitational allocation to uniform points on the sphere

Given n uniform points on the surface of a two-dimensional sphere, how can we partition the sphere fairly among them ? "Fairly" means that each region has the same area. It turns out that if the given points apply a two-dimensional gravity force to the rest of the sphere, then the basins of attraction for the resulting gradient flow yield such a partition-with exactly equal areas, no matter

##### On boundedness of some algebraic fiber spaces

In this talk I will describe few examples where it is possible to obtain boundedness of the total space of a fibration from boundedness of the bases. The techniques presented will apply to certain Mori fiber spaces and to Calabi-Yau varieties with an elliptic fibration. If time permits, I will discuss some open problems.

##### TBA-Lionel Levine

##### Compactification of the configuration space for constant curvature conical metrics

In this joint work with Rafe Mazzeo, we would like to understand the deformation theory of constant curvature metrics with prescribed conical singularities on a compact Riemann surface. We construct a resolution of the configuration space, and prove a new regularity result that the family of constant curvature conical metrics has a nice compactification as the cone points coalesce.

##### New Faculty Talks, I

4:30 p.m. | Gabriele Di Cerbo, Assistant Professor |

4:50 p.m. | Tristan Buckmaster, Assistant Professor |

5:10 p.m. | Oanh Nguyen, Instructor |

5:30 p.m. | Yunqing Tang, Instructor |

##### Painlevé VI, dynamics, and beyond

We begin by discussing the Painlevé VI equation, a nonlinear second order ordinary differential equation discovered by R. Fuchs (1906), which has several beautiful properties and applications. We describe how the classification of its algebraic solutions, completed by Lisovyy-Tykhyy (2008), connects to mapping class group dynamics of the four punctured sphere on certain moduli spaces.

##### Mixing Times for the Ising model

I will discuss results on the mixing time for the Glauber dynamics for the Ising model and how it relates to the phase transitions of the model.