# Seminars & Events for 2013-2014

##### Singular perturbation of minimal surfaces

(w/ N. Kapouleas and N.M. M\o ller) I discuss recent work in which we use singular perturbation techniques to show that the space of complete embedded minimal surfaces with four ends and genus $k$ ($\mathcal{M}(k,4)$) is non-empty and non-compact for large $k$.

##### Codimension one stability of the catenoid under the vanishing mean curvature flow in Minkowski space

We study time-like hypersurfaces with vanishing mean curvature in the $(3+1)$ dimensional Minkowski space, which are the hyperbolic counterparts to minimal embeddings of Riemannian manifolds. The catenoid is a stationary solution of the associated Cauchy problem.

##### Coherent vortices in 2D turbulence

Two-dimensional turbulence generated in a finite box produces large-scale coherent flow with relatively weak fluctuations on its ground. The coherent flow contains vortices with velocity much greater than the typical coherent flow velocity. The vortices are isotropic and have some scaling structure with a number of different subregions.

##### Feynman categories: Universal operations and Hopf algebras

After briefly giving the definition of Feynman categories -a toy example being finite sets and surjections- we will consider other algebraic structures that can be derived for them. The first are universal operations,

##### Volume in Seiberg-Witten theory and the existence of two Reeb orbits

I will discuss recent joint work with Vinicius Gripp and Michael Hutchings relating the volume of any contact three-manifold to the length of certain finite sets of Reeb orbits. I will also explain why this result implies that any closed contact three-manifold has at least two embedded Reeb orbits.

##### Optimizing to Optimize

Parsimony, including sparsity and low rank, has been shown to successfully model data in numerous machine learning and signal processing tasks. Traditionally, such modeling approaches rely on an iterative algorithm that minimizes an objective function with parsimony-promoting terms. PLEASE CLICK ON COLLOQUIUM TITLE FOR COMPLETE ABSTRACT.

##### An invariant-theoretic view of the Cox ring and effective cone of \bar{M}_{0,n}

I'll discuss joint work with Brent Doran and Dave Jensen in which we use an "algebraic uniformization" of \bar{M}_{0,n} to study the Cox ring and effective cone. This construction exhibits this moduli space as a non-reductive GIT quotient of affine space and reveals a precise sense in which it is "one G_a away" from being a toric variety.

##### Ill-posedness / Well-posedness Results for a Class of Active Scalar Equations.

**Please note special day (Tuesday). **We discuss a class of active scalar equations where the transport velocities are more singular than the active scalar. There is a significant difference in the well-posedness properties of the problem depending on whether the Fourier multiplier symbol for the velocity is even or odd.

##### Multiple Dirichlet Series

We review the theory of multiple Dirichlet series which are Dirichlet series in several complex variables having analytic continuation with finitely many polar divisors and satisfying a finite group of functional equations.

##### Complex compact manifolds with maximal torus action

**Please note different time and location. **In the talk we describe a class of manifolds Z constructed via certain combinatorial data, called a complete simplicial fan. In the case of rational fans, the manifold Z is the total space of a holomorphic bundle over a toric variety with fibres compact complex tori.

##### On sparse block models

Block models are random graph models which have been extensively studied in statistics and theoretical computer science as models of communities and clustering. A conjecture from statistical physics by Decelle et. al predicts an exact formula for the location of the phase transition for statistical detection for this model. I will discuss recent progress towards a proof of the conjecture.

##### Simple eigenvalues of vertex-transitive graphs

A simple eigenvalue of a graph is an eigenvalue of the adjacency matrix with multiplicity 1. It has been observed that graphs having many simple eigenvalues tend to have small automorphism groups. The only vertex-transitive graph with all eigenvalues simple is K_2 and it is well-known that a k-regular vertex-transitive graph will have at most k+1 simple eigenvalues.

##### Heegner points and a B-SD conjecture

We prove a B-SD conjecture for elliptic curves (for the p^infinity Selmer groups with arbitrary rank) a la Mazur-Tate and Darmon in anti-cyclotomic setting, for certain primes p. This is done, among other things, by proving a conjecture of Kolyvagin in 1991 on p-indivisibility of (derived) Heegner points over ring class fields.

##### Plane Floer Homology and Knot Concordance Group

Plane Floer Homology defines a functor from the category of 3-manifolds and cobordisms to the category of vector spaces over an appropriate Novikov field N. Like other Floer homologies assigned to 3-manifolds, this homology theory carries one important property, i.e., surgery exact triangle or more generally ``cubical surgery relation''.

##### Tori in four-dimensional Milnor fibres

The Milnor fibre of any isolated hypersurface singularity contains exact Lagrangian spheres: the vanishing cycles associated to a Morsification of the singularity. Moreover, for simple singularities, it is known that the only possible exact Lagrangians are spheres. I will explain how to construct exact Lagrangian tori in the Milnor fibres of all non-simple singularities of real dimension four.

##### Lengths of Monotone Subsequences in a Mallows Permutation

**Please note special day and location. **The longest increasing subsequence (LIS) of a uniformly random permutation is a well studied problem. Vershik-Kerov and Logan-Shepp first showed that asymptotically the typical length of the LIS is 2sqrt(n).

##### Mean curvature flow of mean convex hypersurfaces

In the last 15 years, White and Huisken-Sinestrari developed a far-reaching structure theory for the mean curvature flow of mean convex hypersurfaces. Their papers provide a package of estimates and structural results that yield a precise description of singularities and of high curvature regions in a mean convex flow.

##### Global solutions and asymptotic behavior for two dimensional gravity water waves

The main result of this talk is a global existence theorem for the water waves equation with smooth, small, and decaying at infinity Cauchy data. We obtain moreover an asymptotic description of the solution which shows that modified scattering holds.

##### Hilbert transform and image reconstruction from incomplete data in tomography

In conventional Computer Tomography (CT), reconstruction of even a small region of interest inside an object requires irradiating the entire object with x-rays. Recently a new group of algorithms has emerged that reconstructs an image from interior data, i.e. the data that is based only on x-rays intersecting the region of interest.

##### Structure of large bosonic systems: the mean-field approximation and the quantum de Finetti theorem

I will discuss a general strategy to derive Hartree's theory for the ground state of a generic interacting many-bosons system with mean-field scaling. The validity of the mean-field approximation is interpreted as a consequence of the structure of the set of bosonic density matrices with large number of particles, in particular of the so-called quantum de Finetti theorem.