# Seminars & Events for 2012-2013

##### Compressive imaging: Sampling strategies and reconstruction guarantees

In many applications such as Magnetic Resonance Imaging, images are acquired using Fourier transform measurements. Such measurements can be expensive, and it is of interest to exploit the wavelet domain sparsity of natural images to reduce the number of measurements without destroying reconstruction quality. Much work in compressed sensing has been devoted to this problem in recent years.

##### Hassett-Keel Program in Genus Four

The Hassett-Keel program aims to give modular interpretations of log canonical models for the moduli spaces of curves. The program, while relatively new, has attracted the attention of a number of researchers, and has rapidly become one of the most active areas of research concerning the moduli of curves.

##### From the Jones polynomial to Khovanov homotopy

In the early 1980's, Jones introduced a new knot invariant, now called the Jones polynomial. Roughly ten years ago, Khovanov gave a refinement -- or categorification -- of the Jones polynomial; this refinement is now called Khovanov homology. In this talk we will sketch definitions of the Jones polynomial and Khovanov homology, and mention some of their most spectacular applications.

##### The Bieberbach Conjecture

The Bieberbach conjecture was a mirage, but it has become solid reality. Who knows when the next mirage will become an oasis of heavenly delight?" -- Lars Ahlfors

##### On the Loss of Regularity for the Three-Dimensional Euler Equations

A basic example of shear flow was introduced by DiPerna and Majda to study the weak limit of oscillatory solutions of the Euler equations of incompressible ideal fluids. In particular, they proved by means of this example that weak limit of solutions of Euler equations may, in some cases, fail to be a solution of Euler equations.

##### What have we learned about graph partitioning?

The graph partitioning problem consists of cutting a graph into two or more large pieces so as to minimize the number of edges between them. Since the problem is NP-hard, one needs to resort to approximation. This talk will give a quick survey of techniques for graph partitioning, focusing on work in the past 7-8 years that uses semidefinite programming and spectral techniques.

##### Poincare and Hodge rings and their applications

We determine the structure of the rings of Poincare and Hodge polynomials, and analyze the tautological comparison map between them. This leads to interesting results about the Hodge numbers of Kaehler manifolds and of algebraic varieties, and in particular to the complete solution of a classical problem of Hirzebruch's.

##### A Converse Theorem for SL_2

We'll prove a converse theorem for forms forms on SL_2. While the theorem is easy to prove once it has been formulated, the number-theoretic considerations leading to its' formulation nevertheless pose some interesting and apparently unsolved questions.

##### TQFT, Hochschild homology and localization

Hochschild homology is a categorification of the trace. We will start by explaining what this means, and how both the trace and Hochschild homology are relevant to topological field theories. We will then explain an algebraic result for Hochschild homology, and how it can sometimes be used to study TQFT invariants of covering spaces.

##### positivity questions in K\"ahler-Einstein theory

In this talk, I describe how the existence of both complete and incomplete K"ahler-Einstein(KE) metrics on quasi-projective varieties motivates a number of questions around the theme of positivity in complex geometry. After describing in some details the case of complex surfaces, I discuss some higher dimensional results.

##### Symplectic Geometry of Quantum Noise

We discuss a quantum counterpart, in the sense of the Berezin-Toeplitz quantization, of certain constraints on Poisson brackets coming from "hard" symplectic geometry. It turns out that they can be interpreted in terms of the quantum noise of observables in operational quantum mechanics.

##### Breakdown criterion in general relativity: spherically symmetric spacetimes

At the heart of the (weak and strong) cosmic censorship conjectures is a statement regarding singularity formation in general relativity. Even in spherical symmetry, cosmic censorship seems, at the moment, mathematically intractable.

##### Data-driven modeling and dimensionality reduction

Dimensionality reduction is a common method for rendering tractable a host of problems arising in the physical, engineering and biological sciences. In recent years, methods from data analysis have started playing critical roles in more traditional applied mathematics problems typically analyzed with dynamical systems and PDE techniques.

##### Functoriality in Gromov--Witten theory

I will discuss the functoriality problem of Gromov-Witten theory, including the past works on crepant transformation conjecture for ordinary flops. The tools used include standard techniques like localization, as well as novel ones (in GWT) like algebraic cobordism. If time allows, I will mention possible functoriality under extremal transitions. This is a joint project with H.-W.

##### Inaugural Minerva Lectures I: Equidistribution

Equidistribution

##### Five hundred years of differentiation in fifty minutes

We will cover the history of the derivative and its many historical variants from the late Renaissance through early twentieth century generalized derivatives, highlighting the problems, politics, and institutions that drove changing mathematical conceptions of differentiation in the modern world.

##### Geodesics in 2D First-Passage Percolation

I will discuss geodesics in first-passage percolation, a model for fluid flow in a random medium. C. Newman and collaborators have studied questions related to existence and coalescence of infinite geodesics under strong assumptions.

##### Extremal problems in Eulerian digraphs

Graphs and digraphs behave quite differently, and many classical results for graphs are often trivially false when extended to general digraphs. Therefore it is usually necessary to restrict to a smaller family of digraphs to obtain meaningful results. One such very natural family is Eulerian digraphs, in which the in-degree equals out-degree at every vertex.

##### Modular forms modulo 2

##### A Transcendental Invariant of Pseudo-Anosov Maps

For each pseudo-Anosov map, we will associate it with a $\mathbb{Q}$-submodule of $\mathbb{R}$. This invariant is defined by interaction between Thurston norm and dilatation of pseudo-Anosov map. We will develop a few nice properties of our invariant and give a few examples to show it can be nontrivial. These nontrivial examples give negative answer to a question asked by McMullen.