# Seminars & Events for 2012-2013

##### Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time I

In connection to the theory of hydrodynamic turbulence, Onsager conjectured that solutions to the incompressible Euler equations with Holder regularity below 1/3 may dissipate energy. Recently, DeLellis and Székelyhidi have adapted the method of convex integration to construct energy-dissipating solutions with regularity up to 1/10.

##### Min-max methods, Willmore conjecture and the energy of links

The idea of finding closed geodesics in surfaces by using sweep-outs goes back to the work of Birkhoff in the 1920s. Minimal surfaces in three-manifolds can also be constructed that way. This was the main accomplishment of Almgren and Pitts (1981), using powerful tools of Geometric Measure Theory.

##### Embeddings of Rational Homology Balls

We will begin with a description of the rational homology balls appearing in Fintushel and Stern's rational blow-down procedure for smooth 4-manifolds, a generalization of the standard blow-down operation. We will then discuss various smooth and symplectic embedding results of these rational homology balls, as well as a description of a symplectic rational blow-up operation. THIS IS A JOINT TOPOLOGY/ALGEBRAIC TOPOLOGY SEMINAR.

##### Embeddings of Rational Homology Balls

We will begin with a description of the rational homology balls appearing in Fintushel and Stern's rational blow-down procedure for smooth 4-manifolds, a generalization of the standard blow-down operation. We will then discuss various smooth and symplectic embedding results of these rational homology balls, as well as a description of a symplectic rational blow-up operation. THIS IS A JOINT TOPOLOGY/ALGEBRAIC TOPOLOGY SEMINAR. PLEASE NOTE DIFFERENT TIME AND LOCATION.

##### Sato-Tate distributions in genus 2

For an abelian surface A over a number field k, we study the limiting distribution of the normalized Euler factors of the L-function of A. Under the generalized Sato-Tate conjecture, this is equal to the distribution of characteristic polynomials of random matrices in a closed subgroup ST(A) of USp(4). The Sato-Tate group ST(A) may be defined in terms of the Galois action on any Tate module of A, and must satisfy a certain set of constraints (the Sato-Tate axioms). Up to conjugacy, we find that there are exactly 55 subgroups of USp(4) that satisfy these axioms. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### A reverse isoperimetric inequality for J-holomorphic curves.

I'll discuss a bound on the length of the boundary of a J-holomorphic curve with Lagrangian boundary conditions by a constant times its area. The constant depends on the symplectic form, the almost complex structure, the Lagrangian boundary conditions and the genus. (CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.)

##### Gromov-Witten theory and cycle-valued modular forms

A remarkable phenomenon in Gromov-Witten theory is the appearance of (quasi)-modular forms. For example, Gromov-Witten generating functions for elliptic curve, local $\mathbb{P}^2$, elliptic orbifold $\mathbb{P}^1$ are all (quasi)-modular forms. In this talk, we will discuss modularity property of the Gwomov-Witten cycles of elliptic orbifold $\mathbb{P}^1$. (CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.)

##### Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time II

In connection to the theory of hydrodynamic turbulence, Onsager conjectured that solutions to the incompressible Euler equations with Holder regularity below 1/3 may dissipate energy. Recently, DeLellis and Székelyhidi have adapted the method of convex integration to construct energy-dissipating solutions with regularity up to 1/10.

##### Minimal and constant mean curvature surfaces in quotients of H^2xR

In this talk we will report on two types of questions related to the geometry of surfaces in quotients of H^2xR. We will talk about the classification of compact embedded constant mean curvature surfaces, and about the description of the asymptotic behavior of ends of proper minimal surfaces with finite total curvature. These theorems are inspired by classical results in Euclidean space.

##### The Calabi homomorphism, Lagrangian paths and special Lagrangians

The talk will have two parts. First, I'll discuss a generalization of the Calabi homomorphism to a functional on Lagrangian paths. Then I'll explain how the functional relates to special Lagrangian geometry, and as time permits, how it fits into the framework of mirror symmetry.

##### On inverse spectral problem for quasi-periodic Schrödinger equation

In this talk I will discuss the main ideas and new technology of our recent joint work with D.Damanik. We study the quasi-periodic Schr\"{o}dinger equation $$-\psi''(x) + V(x) \psi(x) = E \psi(x), \qquad x \in \mathbb{R} $$ in the regime of ``small'' $V$. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Solving the GPS Problem in Almost Linear Time

A client on the earth surface wants to know her geographical location. The Global Positioning System (GPS) was built to fulfill this task. It works as follows. Satellites send to earth their location. For simplicity, the location of a satellite is a bit b=1,-1. The satellite transmits to the earth a sequence of N>1000 complex numbers S[0],S[1],...,S[N-1] multiplied by its location b.

##### The open Gromov-Witten-Welschinger theory of blowups of the projective plane

I'll explain how to compute the Welschinger invariants of blowups of the projective plane at an arbitrary conjugation invariant configuration of points. Specifically, open analogues of the WDVV equation and Kontsevich-Manin axioms lead to a recursive algorithm that reconstructs all the invariants from a small set of known invariants. This result is joint work with Asaf Horev.

##### A mirror theorem for the mirror quintic

The celebrated Mirror Theorem of Givental and Lian-Liu-Yau states that the A model (quantum cohomology, rational curve counting) of the Fermat quintic threefold is equivalent to the B model (complex deformations, period integrals) of its mirror dual, the mirror quintic orbifold.

##### Asymptotics of Eigenvalues and Eigenfunctions in Periodic Homogenization

Understanding asymptotics of eigenvalues and eigenfunctions in homogenization is an important problem and it has attracted a great deal of attentions. One of the practical reasons is that it may be applied in the studies of some inverse problems and problems of uniform boundary controllability.

##### Higher Composition Laws

In 1798, Gauss described a composition law on the space of SL_2(Z) equivalence classes of binary quadratic forms with fixed discriminant D.

##### Cover-decomposition and polychromatic numbers

In the cover-decomposition problem we are given a ground set of points and a collection of its subsets. Suppose that we want to colour some subsets blue and the others red, so that every point lies in both a blue subset and a red subset. An obvious necessary condition is that every point lies in at least two subsets; is there an exact or approximate converse?

##### Toric Structures on Symplectic Bundles of Projective Spaces

Recently, extending work by Karshon, Kessler and Pinsonnault, Borisov and McDuff showed that a given symplectic manifold $(M,\omega)$ has a finite number of distinct toric structures. Moreover, McDuff also showed a product of two projective spaces $\mathbb{C} P^r\times \mathbb{C} P^s$ with any given symplectic form has a unique toric structure provided that $r,s\geq 2$.

##### Monodromy and arithmetic groups

Monodromy groups arise naturally in algebraic geometry and in differential equations, and often preserve an integral lattice. It is of interest to know whether the monodromy groups are arithmetic or thin.

In this talk we review the Deligne-Mostow theory and show that for cyclic coverings of degree $d$ of the projective line, with a prescribed number $m$ of branch points and prescribed ramification indices, the monodromy is an arithmetic group provided $m\geq 2d-2$. PLEASE CLICK ON THE SEMINAR TITLE FOR THE FULL ABSTRACT.

##### Fundamental groups of Kähler manifolds and combinatorial group theory

The study of fundamental groups of Kähler manifolds is a fascinating enterprise at the crossroads of various branches of geometry and topology, with strong relations to algebra and analysis as well. I will discuss some results in this area pertaining to groups of interest in low-dimensional topology and in combinatorial and geometric group theory.