# Seminars & Events for 2013-2014

##### Genus of abstract modular curves with level l structure

To any bounded family of \F_l-linear representations of the etale fundamental of a curve X one can associate families of abstract modular curves which, in this setting, generalize the `usual' modular curves with level l structure (Y_0(l), Y__1(l), Y(l) etc.). Under mild hypotheses, it is expected that the genus (and even the geometric gonality) of these curves goes to infty with l.

##### Gopakumar-Vafa conjecture for symplectic manifolds

The Gopakumar-Vafa conjecture predicts that the Gromov-Witten invariants of a Calabi-Yau 3-fold can be canonically expressed in terms of integer invariants called BPS numbers. In this talk, based on joint work with Tom Parker, I describe the proof of this conjecture, coming from a more general structure theorem for the Gromov-Witten invariants of symplectic 6-manifolds.

##### Morse index and uniqueness results for free boundary problems

##### Connected sum construction of constant Q-curvature manifolds in higher dimensions

**This is a special talk in addition to the 3:00 pm talk on the same date. **In geometric analysis, gluing constructions are well-known methods to create new solutions to nonlinear PDEs from existing ones.

##### Continued fraction digit averages and MacLaurin's Inequalities

**Please note different day and location. ** A classical result of Kinchin says that for almost every real number x, the geometric mean of the first n digits in the continued fraction expansion of x converges to a number K=2.685... as n tends to infinity. On the other hand, for almost every x, the arithmetic mean of the first n digits tends to infinity.

##### Global well-posedness and scattering for the defocusing, mass critical generalized KdV problem

The mass - critical gKdV has many features in common with the mass critical NLS problem. In particular, we show that scattering for the quintic problem implies a concentration compactness result for the gKdV problem. We then define an interaction Morawetz estimate that implies scattering.

##### Geometric methods in image processing, networks, and machine learning

Geometric methods have revolutionized the field of image processing and image analysis. I will review some of these classical methods including image snakes, total variation minimization, image segmentation methods based on curve minimization, diffuse interface methods, and state of the art fast computational algorithms that use ideas from compressive sensing.

##### Mildly singular anticanonical divisors and K-stability (joint work with Y.Sano, T.Okada)

K-stability of Fano manifold is recently proved to be equivalent to the existence of Einstein-Kahler metric. After brifely explaining the recent developement of the differential geometric background by Chen-Donaldson-Sun and Tian, I would like to revisit my algebro-geometric works done a few years ago about the following question: "When a Fano manifold really is K-stable?".

##### Regularity theory for a class of fully nonlinear integral operators

We consider a class of non-linear integral variational problems involved in nonlocal image and signal processing. We show the existence of global in time classical solutions for those problems. The method is based on the De Giorgi method applied to nonlocal operators. It is an extension of a similar result we first obtained for the the so-called Surface Geostrophic Equation.

##### Analysis and Algorithms for the Phase Retrieval Problem

The phase retrieval problem presents itself in many applications is physics and engineering. Recent papers on this topic present examples ranging from X-Ray crystallography to audio and image signal processing, classification with deep networks, quantum information theory, and fiber optics data transmission.

##### Nonarchimedean methods for multiplication maps

Multiplication maps on linear series are among the most basic structures in algebraic geometry, encoding, for instance, the product structure on the graded pieces of the homogeneous coordinate ring of a projective variety.

##### On the Bogolubov-Hartree-Fock (BHF) Approximation for the Pauli-Fierz Model

The minimal energy of the nonrelativistic one-electron Pauli-Fierz model within the class of quasifree states is studied. It is shown that this minimum is unchanged if one restricts the variation to pure quasifree states, which simplifies the variational problem considerably.

##### The logarithmic Minkowski problem

The logarithmic Minkowski problem asks for necessary and sufficient conditions in order that a nonnegative finite Borel measure in (n-1)-dimensional projective space be the cone-volume measure of the unit ball of an n-dimensional Banach space. The solution to this problem is presented.

##### On the Reality of Black Holes

##### Picard-Lefschetz theory and hidden symmetries

Picard-Lefschetz theory studies algebraic varieties by induction on their dimension. It can be used to determine their topology, and in more modern terms their symplectic geometry. We will apply this theory to describe extra structure which appears for a special class of algebraic varieties.

##### Super-diffusion in the periodic Lorentz gas

I report on recent work with Balint Toth on the proof of a super-diffusive CLT for the periodic Lorentz gas in the limit of small scatterers. This complements work by Bunimovich & Sinai (CMP 1980), Bleher (JSP 1992) and Szasz & Varju (JSP 2007) in the case of scatterers with fixed radii.

##### Sum-free subgroups

The study of sum and product problems in finite fields motivates the investigation of additive structures in multiplicative subgroups of such fields. It turns out that such subgroups that are sufficiently large with respect to the size of the field must contain rich structures of additive relations, while even prime fields may contain sum-free multiplicative subgroups of substantial size.

##### Generating the Fukaya category

The Fukaya category is an interesting invariant of a symplectic manifold. It is, at first sight, a rather fearsome thing: its objects are all Lagrangian submanifolds, an enormous and unruly set. Nevertheless, in certain circumstances one can find a finite set of Lagrangians which `generate' the category in an appropriate sense.

##### Patching and p-adic local Langlands

The p-adic local Langlands correspondence is well understood for GL_2(Q_p), but appears much more complicated when considering GL_n(F), where either n>2 or F is a finite extension of Q_p.

##### Feynman categories, universal operations and master equations

Feynman categories are a new universal categorical framework for generalizing operads, modular operads and twisted modular operads. The latter two appear prominently in Gromov-Witten theory and in string field theory respectively. Feynman categories can also handle new structures which come from different versions of moduli spaces with different markings or decorations, e.g.