# Seminars & Events for 2016-2017

##### Global Stability of Solutions to a Beta-Plane Equation

We study the motion of an incompressible, inviscid two-dimensional fluid in a rotating frame of reference. There the fluid experiences a Coriolis force, which we assume to be linearly dependent on one of the coordinates. This is a common approximation in geophysical fluid dynamics and is referred to as $\beta$-plane.

##### Bordered Floer homology via immersed curves II

I'll describe parts of an ongoing project with Jonathan Hanselman and Jake Rasmussen, which interprets bordered Floer homology for manifolds with torus boundary in terms of systems of immersed curves in the punctured torus.

##### Albanese of Picard modular surfaces, and rational points

This is a report on a work in progress in collaboration with Dinakar Ramakrishnan. A celebrated result of Mazur states that open modular curves of large enough level do not have rational points. We study analogous questions for the Picard modular surfaces, which are Shimura varieties for a unitary group in 3 variables defined over an imaginary quadratic field M.

##### New Junior Faculty Lectures I

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

**3:30 p.m. — Nathaniel Bottman, Postdoctoral Research Fellow**

"Witch trees and the Fulton-MacPherson operad"

##### Symmetries of linearized Einstein equations on Kerr spacetime

Motivated by the black hole stability problem, we discuss the structure of linear test fields on Kerr spacetime. The dynamics of the linearized gravitational field on a Kerr background is governed by certain curvature components solving the Teukolsky master equations (TME) and Teukolsky-Starobinski identities (TSI).

##### How Well Do Local Algorithms Solve Semidefinite Programs?

Several probabilistic models from high-dimensional statistics and machine learning reveal an intriguing -and yet poorly understood- dichotomy. Either simple local algorithms succeed in estimating the object of interest, or even sophisticated semi-definite programming (SDP) relaxations fail.

##### Symmetries and Critical Phenomena in Fluids

We describe recent results on studying the dynamics of fluid equations in critical spaces.

##### The gauged symplectic sigma-model

I will recall the construction of the space of states in a gauged topological A-model. Conjecturally, this gives the quantum cohomology of Fano symplectic quotients: in the toric case, this is Batyrev's presentation of quantum cohomology of toric varieties.

##### Ward and Belavin-Polyakov-Zamolodchikov (BPZ) identities for Liouville quantum field theory on the Riemann sphere

The foundations of modern conformal field theory (CFT) were introduced in a 1984 seminal paper by Belavin, Polyakov and Zamolodchikov (BPZ). Though the CFT formalism is widespread in the physics literature, it remains a challenge for mathematicians to make sense out of it.

##### Classical invariant theory and birational geometry of moduli spaces

**PLEASE NOTE SPECIAL START TIME: 5:00. **Invariant theory is a study of the invariant subring of a given ring equipped with a linear group action. Describing the invariant subring was one of the central mathematical problems in the 19th century and many great algebraists such as Cayley, Clebsch, Hilbert, and Weyl had contributed to it.

##### Maclaurin Lectures: Siegel's problem on small volume lattices

We outline in very general terms the history and the proof of the iIdentification of the minimal covolume lattice of hyperbolic 3-space as the 3-5-3 Coxeter group extended by the involution preserving the symmetry of this diagram.

##### Kahler-Einstein metrics and volume minimization

Futaki invariant is an obstruction to the existence of Kahler-Einstein metrics on Fano manifolds. Martelli-Sparks-Yau showed that the Futaki invariant is the derivative of a normalized volume functional on the space of Reeb vector fields of associated affine cones and derived the volume minimization principle for more general Sasaki-Einstein metrics.

##### Coincidences in homological densities

For certain natural sequences of topological spaces, the kth homology group stabilizes once you go far enough out in the sequence of spaces. This phenomenon is called homological stability. Two classical examples of homological stability are the configuration space of n unordered distinct points in the plane, studied in the 60's by Arnold' and the space of (based) algebraic maps from CP^1 to

##### A survey on the h-cobordism theorem and its applications

Smale's h-cobordism theorem has been one of the most influential results in topology, so much that the conventional line of separation between "low" and "high" dimensions is drawn between dimensions 4 and 5 because the h-cobordism holds true in dimension greater or equal to 5.

##### Strong stationary times for small shuffles

Consider shuffling a deck of cards by at each step, choosing two random cards and either swapping them or not. This random walk is well understood. A natural generalisation is to choose three (or more) cards at each step, and shuffle them amongst themselves. I will show how one may construct a strong stationary time for such a random walk, giving an asymptotic upper bound on the mixing time.

##### Matrices with large permanent

The permanent of a matrix has long been an important quantity in combinatorics and computer science, and more recently it has also had applications to physics and linear-optical quantum computing. A result of Gurvits bounds the permanent of an n by n matrix in terms of its operator norm via |perm(A)| \leq |A|^n, and he characterized the matrices for which this is tight (they are essentially j

##### Nonabelian Cohen-Lenstra Heuristics and Function Field Theorems

The Cohen-Lenstra Heuristics conjecturally give the distribution of class groups of imaginary quadratic fields.

##### Higher Order Corks

In 1960, Mazur constructed a contractible 4-manifold W with non-simply connected boundary whose product with an interval is the 5-ball. Thirty years later, Akbulut showed that ∂W supports an involution T that does not extend to a diffeomorphism of W, thus producing the first nontrivial *involutory cork *(W,T). Akbulut's proof was to embed W in a smooth 4-manifold so that the resulting *cork t

##### Blowup for model equations of fluid mechanics

In this talk, I discuss recent progress towards proving a finite time blowup for the 2D inviscid Boussinesq equations, inspired by the hyperbolic flow scenario.I will introduce various model equations for the Boussinesq system that isolate and capture possible mechanisms for singularity formation. An important theme is to achieve finite-time blowup in a controlled manner.

##### Global well posedness and scattering for the cubic nonlinear wave equation

In this talk we discuss the defocusing, cubic nonlinear wave equation. We prove scattering for radially symmetric, nearly sharp initial data. We do not assume an a priori bound on the critical norm. We prove this by studying the wave equation in hyperbolic coordinates.