# Seminars & Events for 2015-2016

##### Permutons

What do permutations of 1 through n, for large n, look like? For example, how can we generate a random permutation that inverts a third of its pairs? How many such permutations are there?

##### A tale of two norms

The first cohomology of a hyperbolic 3-manifold has two natural norms: the Thurston norm, which measure topological complexity of surfaces representing the dual homology class, and the harmonic norm, which is just the L^2 norm on the corresponding space of harmonic 1-forms.

##### The vanishing viscosity limit in porous media

We consider the flow of a viscous, incompressible, Newtonian fluid in a perforated domain in the plane. We study the simultaneous limit of vanishing pore size and inter-pore distance, and vanishing viscosity.

##### Generating series of arithmetic divisors in unitary Shimura varieties

In this talk, I will describe roughly how to define a generating function arithmetic divisors (in Arakelov sense) on a unitary Shimura variety of type (n-1,1). I will then briefly explain why it is modular. If time permits, I will also talk briefly about its application to Gross-Zagier-Zhang type formula and to Colmez conjecture.

##### Asymptotic shapes of neckpinch singularities in geometric flows

Geometric flows such as Ricci flow and mean curvature flow are nonlinear parabolic PDEs that tend to develop singularities in finite time.

##### Ergodic theorems beyond amenable groups

Let G be a locally compact group acting by measure-preserving transformations on a probability space (X,mu). To every probability measure on G there is an associated averaging operator on L^p(X,mu). Ergodic theorems describe the pointwise and norm limits of sequences of such operators.

##### Point-like bounding chains in open Gromov-Witten theory

Over a decade ago Welschinger defined invariants of real symplectic manifolds of complex dimensions 2 and 3, which count $J$-holomorphic disks with boundary and interior point constraints. Since then, the problem of extending the definition to higher dimensions has attracted much attention.

##### Arithmetic of Double Torus Quotients and the Distribution of Periodic Torus Orbits

**Special Number Theory / Ergodic Theory Seminar -- Note special time and place. **

##### Arithmetic of Double Torus Quotients and the Distribution of Periodic Torus Orbits

**Special Number Theory / Ergodic Theory Seminar -- Note special time and place. **

##### Tian's properness conjectures, the strong Moser-Trudinger inequality, and infinite-dimensional Finsler geometry

In the 90's, Tian introduced a notion of properness in the space of Kahler metrics in terms of Aubin's nonlinear Dirichlet energy and Mabuchi's K-energy and put forward several conjectures on the relation between properness and existence of Kähler-Einstein metrics. These can be viewed as the Kahler analog of the classical Moser-Trudinger inequality from conformal geometry.

##### Faster Convex Optimization - Simulated Annealing with an Efficient Universal Barrier

Interior point methods and random walk approaches have been long considered disparate approaches for convex optimization. We show how simulated annealing, one of the most common random walk algorithms, is equivalent, in a certain sense, to the central path interior point algorithm applied to the entropic universal barrier function.

##### Mori fibre spaces for 3-folds in positive characteristic

There has been much progress in recent years on the LMMP for 3-folds in characteristic p>5. In this talk I will discuss the proof of the base point free theorem and how it leads to termination of the LMMP with scaling and the existence of Mori fibre spaces. This is joint work with Caucher Birkar.

##### Number-theoretic algorithms in quantum computing

In quantum computation, one considers groups of unitary operators that are generated by some finite set of operators called "gates". Words in these generators are called "circuits". An important problem is the so-called approximate synthesis problem: to find a quantum circuit, preferably as short as possible, that approximates a given unitary operator up to given epsilon.

##### Calculus of variations, Sobolev spaces, and geometric measure theory

We will discuss how the theory of Sobolev spaces and geometric measure theory are frameworks that arise more or less in the same way in the study of certain variational problems. I will provide some concrete examples, of which the main ones will be minimal surfaces and some elliptic PDEs.

##### Random Real Algebraic Geometry

We discuss the work of Fedor Nazarov and Mikhail Sodin on zero sets of randomly generated functions of several real variables. They prove that there is an asymptotic formula for the number of connected components of such a set. The ability to handle functions of more than one variable is a major breakthrough and makes it possible to study many interesting questions.

##### Real rooted polynomials in graph theory

In this talk, I will discuss some recent applications of real rooted polynomials in graph theory. I will begin by discussing the more classical results concerning graph polynomials, including the real rootedness of the matching polynomial and the extension of Chudnovsky and Seymour to the independence polynomial of claw-free graphs.

##### Toric polynomial generators in the unitary cobordism ring

It is well-known that the unitary cobordism ring Omega^* is isomorphic to the polynomial ring with one generator in every even degree. However explicit description of 'nice' representatives of the generators turns out to be a difficult problem. In this talk we aim at constructing connected algebraic toric polynomial generators of Omega^*.

##### A nonlocal diffusion problem on manifolds

**Please note special day, time and location. ** We consider a nonlocal diffusion problem on a manifold. This kind of equations can model diffusions when there are long range effects and have been widely studied in Euclidean space.

##### The first order theory of meromorphic functions

By a result of Julia Robinson, we know that the first order theory of the field of rational numbers is undecidable, and in fact the same holds for any number field. In view of this, it is suggested by analogies studied by Vojta and others that the first order theory of meromorphic functions over a complete algebraically closed field should also be undecidable.

##### Harmonic Chern Forms on Polarized Kähler Manifolds

The higher K-energies are functionals whose critical points give Kähler metrics with harmonic Chern forms. In this talk, we relate the higher K-energies to discriminants and use the theory of stable pairs to obtain results on their boundedness and asymptotics.