# Seminars & Events for 2011-2012

##### Complexity theory applied to voting theory

As it will be shown with results and examples, the paradoxes associated with standard voting rules are surprisingly likely and are so complex that one must worry about the legitimacy of election outcomes.

##### Regularity of twisted Bergman projections - Part 1

The usual Bergman projection is not globally regular on a general (smoothly bounded) pseudoconvex domain, by a result of Christ. We will discuss a family of "twisted" projections, similar to the Bergman projection, and show they are regular on this class of domains.

##### Moduli spaces of higher dimensional varieties

The moduli spaces of smooth curves of genus at least two and its compactification, the space of stable curves, are one of the most investigated objects of algebraic geometry. In the past two decades, natural higher dimensional generalizations, the moduli space of canonically polarized manifolds and of stable schemes have been constructed.

##### Time Evolution and Stationary States of Classical and Quantum Systems

I will review both old and recent work about the time dependence and steady states of isolated macroscopic systems as well as those in contact with infinite thermal reservoirs. The emphasis will be on quantum systems and will include a discussion of the micro/macro connection in isolated ones and the derivation of a master equation for open oes.

##### Regularity of twisted Bergman projections - Part 2

The usual Bergman projection is not globally regular on a general (smoothly bounded) pseudoconvex domain, by a result of Christ. We will discuss a family of "twisted" projections, similar to the Bergman projection, and show they are regular on this class of domains.

##### Stability Theorems for some Sharp Inequalities and their Applications

We explain recent results on stability theorems for some classical functional and ge- ometric inequalities, along with two applications: one to evolution equations, and one to statistical mechanics. The inequalities in question include certain Gagliardo-Nirenberg- Sobolev inequalities, the Brun-Minkowski inequality, for example.

##### On the strong coloring of graphs with bounded degree

Let $G$ be a graph with $n$ vertices and let $r$ be a number dividing $n$. We say that $G$ is strongly $r$-colorable if for every partition of the vertices of $G$ into sets of size $r$, there exists a proper coloring of $G$ in which every set in the partition is colored in all colors.

##### Regularity of twisted Bergman projections - Part 3

The usual Bergman projection is not globally regular on a general (smoothly bounded) pseudoconvex domain, by a result of Christ. We will discuss a family of "twisted" projections, similar to the Bergman projection, and show they are regular on this class of domains.

##### Curves with many symmetries

By a celebrated theorem of Hurwitz, a curve $X/{\bf C}$ of genus $g>1$ has at most $84(g-1)$ points. Curves that attain or come close to this bound, such as the modular curves ${\rm X}(N)$, often have a rich structure with diverse connections to and near number theory.

##### Volume optimization on triangulated 3-manifolds

We propose a finite dimensional variational principle on triangulated 3-manifolds so that its critical points are related to solutions to Thurston's equation and Haken's normal surface equation. The action functional is the volume. This is a generalization of an earlier program by Casson and Rivin for compact 3-manifolds with torus boundary

##### Qualitative Properties of Gromov-Witten Invariants

Over 15 years ago, di Francesco and Itzykson gave an estimate on the growth (as the degree increases) of the number of plane rational curves passing through the appropriate number of points. This provides an example of an upper bound on (primary) Gromov-Witten invariants.

##### Gravitational Impulsive Waves

We consider spacetimes satisfying the vacuum Einstein equations with gravitational impulsive waves without symmetry assumptions. These are spacetimes such that some components of the Riemann curvature tensor have delta singularities on a null hypersurface.

##### A new model for self-organized dynamics: From particle to hydrodynamic descriptions

Self-organized dynamics is driven by "rules of engagement" which describe how each agent interacts with its neighbors. They consist of long-term attraction, mid-range alignment and short-range repulsion. Many self-propelled models are driven by the balance between these three forces, which yield emerging structures of interest.

##### Absence of mobility edge for the Anderson random potential on tree graphs at weak disorder

We discuss recently established criteria for the formation of extended states on tree graphs in the presence of disorder. These criteria have the surprising implication that for bounded random potentials, as in the Anderson model, in the weak disorder regime there is no transition to a spectral regime of Anderson localization in the form usually envisioned.

##### Toric mirror maps revisited

For a compact semi-Fano toric manifold $X$, Givental's mirror theorem says that a generating function of $1$-point genus $0$ descendant Gromov-Witten invariants, the J-function of $X$, coincides up to a mirror map with a function $I_X$ which is written using the combinatorics of $X$.

##### Resonaces and synchronization in a simple chaotic system

A chaotic system under periodic forcing can develop a periodically visited strange attractor. We discuss simple models in which the phenomenon, quite easy to see in numerical simulations, can be completely studied analytically.

##### Existence of small families of t-wise independent permutations and t-designs via local limit theorems

We show existence of rigid combinatorial objects that previously were not known to exist. Specifically, we consider two families of objects:

##### On real zeros of holomorphic Hecke cusp forms and sieving short intervals

A. Ghosh and P. Sarnak have recently initiated the study of so-called real zeros of holomorphic Hecke cusp forms, that is zeros on certain geodesic segments on which the cusp form (or a multiple of it) takes real values.

##### A rank inequality for the knot Floer homology of branched double covers

Given a knot $K$ in the three sphere, we compare the knot Floer homology of $(S3, K)$ with the knot Floer homology of $(\Sigma(K), K)$, where $\Sigma(K)$ is the double branched cover of the three-sphere over $K$.

##### Obstruction-Flat Asymptotically Locally Euclidean Metrics

Given an even dimensional Riemannian manifold $(M^{n},g)$ with $n\ge 4$, it was shown in the work of Charles Fefferman and Robin Graham on conformal invariants the existence of a non-trivial 2-tensor which involves $n$ derivatives of the metric, arises as the first variation of a conformally invariant functional and vanishes for metrics that are conformally Einstein.