# Seminars & Events for 2016-2017

##### On Noether's inequality for stable log surfaces

In this talk I report on some recent progress on the geography problem of stable log surfaces. This is about restrictions on their holomorphic invariants, such as the volume K^2 and the geometric genus p_g. Compared to the case of surfaces of general type, a new feature here is that the volume of a stable log surface is not necessarily an integer.

##### Uniqueness of weak solutions to the Ricci flow

In his resolution of the Poincaré and Geometrization Conjectures, Perelman constructed Ricci flows in which singularities are removed by a surgery process. His construction depended on various auxiliary parameters, such as the scale at which surgeries are performed.

##### The spectral gap of dense random regular graphs

Let G be uniformly distributed on the set of all simple d-regular graphs on n vertices, and assume d is bigger than some (small) power of n.

##### Energy Identity for Stationary Yang Mills

The first part of this talk will introduce and discuss the basics of Yang Mills connections, which are connections over a principle bundle which are critical points of the energy functional \int |F|^2, the L^2 norm of the curvature of A, and thus A may be viewed as a solution to a nonlinear pde. In many problems, e.g.

##### Capsets, Sunflower-free sets in {0,1}^n, and the slice rank method

In this talk we will look at the recent breakthrough of Ellenberg and Gijswijt and Croot, Lev and Pach, which used polynomial method to obtain exponential upper bounds for the Capset problem, that is upper bounds for the size of the largest set in F_3^n which contains no three term arithmetic progressions.

##### Quasi-periodic solutions to nonlinear PDE

We discuss time quasi-periodic solutions to nonlinear Schroedinger (NLS) and nonlinear wave equations (NLW) on the torus in arbitrary dimensions. The latter is hyperbolic and uses additionally, a Diophantine property of algebraic numbers. We mention also a work in progress, on space-time quasi-periodic solutions to non-integrable NLS, whose analysis rests on semi-algebraic geometry.

##### Non-Convex Phase Retrieval from STFT Measurements

The problem of recovering a one-dimensional signal from its Fourier transform magnitude, called phase retrieval, is ill-posed in most cases. We consider the closely-related problem of recovering a signal from its phaseless short-time Fourier transform (STFT) measurements. This problem arises naturally in several applications, such as ultra-short pulse characterization and ptychography.

##### Improperly coloring K_t minor-free graphs

We show that for every t > 0 there exists a constant c=c(t) such that, if a graph G does not contain K_t as a minor, then its vertex set can be partitioned into at most t-1 parts such that every part induces a subgraph with maximum component of size at most c.

##### Global-in-x Steady Prandtl Expansion over a Moving Boundary

I will outline the proof that steady, incompressible Navier-Stokes flows posed over the moving boundary, y = 0, can be decomposed into Euler and Prandtl flows globally in the tangential variable, assuming a sufficiently small velocity mismatch. The main obstacles in the analysis center around obtaining sharp decay rates for the linearized profiles and the remainders.

##### Arithmetic and Geometry of Picard modular surfaces

Of interest are (i) the conjecture of Bombieri (and Lang) that for any smooth projective surface X of general type over a number field k, the set X(k) of k-rational points is not Zariski dense, and (ii) the conjecture of Lang that X(k) is even finite if in addition X is hyperbolic, i.e., there is no non-constant holomorphic map from the complex line C into X(C).

##### A monopole invariant for foliations without transverse invariant measure

The question about existence and flexibility of taut foliations on a three manifold has been studied for decades. Floer-theoretical obstructions for the existence of taut foliations on rational homology spheres have been obtained by Kronheimer, Mrowka, Ozsvath, and Szabo by perturbing of the foliation to contact structures.

##### New Junior Faculty Lectures II

The Department of Mathematics is holding the second of two events where instructors and assistant professors wh joined the department this fall will speak briefly about their research.

**3:30 p.m. — Otis Chodosh, Veblen Research Instructor**

“Global geometric questions involving scalar curvature”

##### Maximizers for the Stein–Tomas Inequality

We give a necessary and sufficient condition for the precompactness of all optimizing sequences for the Stein–Tomas inequality. In particular, if a well-known conjecture about the optimal constant in the Strichartz inequality is true, we obtain the existence of an optimizer in the Stein–Tomas inequality. Our result is valid in any dimension.

##### Quantum Oracle Classification: The Case of Group Structure

The Quantum Oracle Classification (QOC) problem is to classify a function, given only quantum black box access, into one of several classes without necessarily determining the entire function. Generally, QOC captures a very wide range of problems in quantum query complexity.

##### Positive loops of loose Legendrians and applications

In this talk, I will give a simple and geometrical proof of the following theorem from my thesis: Every loose Legendrian is in a positive loop amongst Legendrian embeddings. The idea is that we add wrinkles to a loose Legendrian and rotate the wrinkles positively then resolve the wrinkles without changing the isotopy class of the initial Legendrian.

##### Log geometric techniques for open invariants in mirror symmetry

**This is a joint Algebraic Geometry and Symplectic Geometry seminar. Please note different room (322) and start time (4:30). **We would like to discuss an algebraic-geometric approach to some open invariants arising naturally on the A-model side of mirror symmetry.

##### Log geometric techniques for open invariants in mirror symmetry

**This is a joint Algebraic Geometry and Symplectic Geometry seminar. Please note different room (322) and start time (4:30). **We would like to discuss an algebraic-geometric approach to some open invariants arising naturally on the A-model side of mirror symmetry.

##### Properly immersed CMC surfaces in hyperbolic 3-manifolds of finite volume

**Please note special time: 2:00.** If $N$ is a noncompact hyperbolic 3-manifold of finite volume and $\Sigma$ is a properly immersed surface of finite topology with nonnegative constant mean curvature less than 1, then we prove that each end of $\Sigma$ is asymptotic (with finite positive multiplicity) to a totally umbilic annulus, properly embedded in $N$.

##### Alternating projections for phase retrieval with random sensing vectors

Phase retrieval problems are a subclass of low-rank matrix recovery problems, that have been studied for a long time because of their important applications in physics. They have traditionally been solved with non-convex algorithms, that came with no theoretical convergence guarantees, and were known to sometimes get stuck in local optima.

##### Gaussian complex zeros on the hole event: the emergence of a forbidden region

Consider the Gaussian Entire Function (GEF) whose Taylor coefficients are independent complex-valued Gaussian variables, and the variance of the k-th coefficient is 1/k!. This random Taylor series is distinguished by the invariance of its zero set with respect to the isometries of the complex plane.