# Seminars & Events for 2007-2008

##### Wave Functions in Thermal Equilibrium—GAP Measures and Canonical Typicality

I will talk about the claim that a quantum system in thermal equilibrium at temperature $1/\beta$ has a random wave function whose distribution is a particular probability measure on Hilbert space called $GAP(\beta )$.

##### Contact structures in dimension 3 and the Seiberg-Witten equations

I hope to give some indication of how the Seiberg-Witten equations are used to study the dynamics of vector fields on 3-dimensional manifolds. One result of this research is a proof of Alan Weinstein's conjecture about the existence of closed integral curves of the Reeb vector field for a contact 1-form.

##### Mirror symmetry for Gromov-Witten invariants of a quintic threefold

The mirror symmetry principle of string theory provides closed formulas for GW-invariants, with special attention devoted to a quintic threefold, $Q3$. The genus $0$ mirror prediction for $Q3$ was verified 12 years ago by using the Atiyah-Bott localization theorem.

##### Critical triangle-free graphs with lots of edges

How close are the triangle-free graphs to the bipartite graphs? The different ways of asking this question give different answers. For example: on the one hand, triangle-free graphs can have arbitrarily large chromatic number; on the other, natural constructions to demonstrate this fact are typically very sparse.

##### Comultiplication and the Ozsvath-Szabo contact invariant

Let $S$ be a surface with boundary and suppose that $g$ and $h$ are diffeomorphisms of $S$ which restrict to the identity on the boundary. I'll describe how the Ozsvath-Szabo contact invariants associated to the open books $(S,g)$, $(S,h)$, and $(S,hg)$ are natural with respect to a comultiplication on the corresponding Heegaard-Floer homology groups.

##### Heegner Divisors, L-Functions and Harmonic Weak Maass Forms

##### Gluing Monopoles

A canonical method is established to glue Seiberg-Witten monopoles over a closed 4-manifold split along a 3-dimensional submanifold. The method is quite generic and only requires mild conditions.

##### Collective motion and decision-making in animal groups

Animal groups such as bird flocks, insect swarms and fish schools are spectacular, ecologically important and sometimes devastating features of the biology of various species. Outbreaks of the desert locust, for example, can invade approximately one fifth of the Earth's land surface and are estimated to affect the livelihood of one in ten people on the planet.

##### Optimality in Large-Scale Multiple Testing

##### Random matrices, statistical mechanics and hyperbolic supersymmetry

We present a statistical mechanics model with a hyperbolic supersymmetry. This model is expected to qualitatively describe properties of random band matrices in N dimensions eg localization and delocalization. The "spins" in this model may be thought of taking values in a Poincare super-disc. In three dimensions we show that this model has a diffusive phase.

##### Attractors with Large Invisible Parts

Philosophically, an attractor of a dynamical system is a closed subset of the phase space which orbits "approach" as time goes to infinity. Different meanings of the word "approach" produce different versions of attractors: maximal, Milnor, statistical, minimal etc. The question of generic non-coincidence between these various types of attractors has not yet been answered.

##### Components of Springer fibers and Khovanov's arc algebra

Using the structure of certain Springer fibers and their components, I'll describe a geometric construction of an algebra which is painfully close to being isomorphic to the arc algebra defined by Khovanov, but in fact, isn't. I'll then hopefully explain why this is actually a good thing.

##### On the $\sigma_2$-scalar curvature and its application

In this talk, we establish an analytic foundation for a fully non-linear equation $\frac{\sigma_2}{\sigma_1}=f$ on manifolds with positive scalar curvature. This equation arises from conformal geometry.

##### Subdirect products of surfaces, homological finiteness, and residually-free groups

##### Hidden Markov models, Markov chains in random environments, and systems theory

An essential ingredient of the statistical inference theory for hidden Markov models is the nonlinear filter. The asymptotic properties of nonlinear filters have received particular attention in recent years, and their characterization has significant implications for topics such as the convergence of approximate filtering algorithms, maximum likelihood estimation, and stochastic control.

##### Projective geometry on manifolds

Rich classes of geometric structures on manifolds are defined by coordinate atlases taking values in a fixed homogeneous space. The existence and classification of such structures leads to a moduli space, which itself is modelled on the algebraic variety of representations of the fundamental group in the automorphism group of the geometry.

##### The Probabilistic Method and A Classical Result of Erdos

The basic idea behind the 'probabilistic method' in combinatorics is as follows: in order to prove the existence of an object with a certain property, one defines an appropriate (non-empty) probability space and shows that a randomly chosen object from this space has the desired property with positive probability.

##### The circular chromatic index of graphs of high girth

Colorings of graphs form a prominent topic in graph theory. Several relaxations of usual colorings have been introduced and intensively studied. In this talk, we focus on circular colorings.

##### The subconvexity bounds for L-functions

For a general L-function, the bound on its critical line obtained by applying the Phragmen-Lindeloff interpolation method is called the convexity bound. Any bounds with a power saving of the convexitybound are called subconvexity bounds. In this talk we will give the first subconvexity bounds for GL(3) L-functions as well as for GL(3) x GL(2) L-functions.