# Seminars & Events for 2008-2009

##### Computational Astrophysics and the Dynamics of Accretion Disks

The ever increasing performance of computer hardware and improvements to the accuracy of numerical algorithms are revolutionizing scientific research in many disciplines, but perhaps none more so than astrophysics.

##### Towards a classification of modular compactifications of the moduli space of curves

The class of stable curves is deformation-open and satisfies the unique limit property, hence gives rise to the modular Deligne-Mumford compactification of $M_{g,n}$.

##### The geometry underlying Donaldson-Thomas theory

Donaldson-Thomas invariants are algebraic analogues of Casson invariants. They are virtual counts of stable coherent sheaves on Calabi-Yau threefolds.

##### Packing cycles with modularity

Erdős and Posa proved that a there exists a function $f$ such that any graph either has $k$ disjoint cycles or there exists a set of $f(k)$ vertices that intersects every cycle. The analogous statement is not true for odd cycles - there exist numerous examples of graphs that do not have two disjoint odd cycles, and yet no bounded number of vertices intersects every odd cycle.

##### Primitive-stable representations of the free group

Automorphisms of the free group $F_n$ act on its representations into a given group $G$. When $G$ is a simple compact Lie group and $n>2$, Gelander showed that this action is ergodic. We consider the case $G=PSL(2,C)$, where the variety of (conjugacy classes of) representations has a natural invariant decomposition, up to sets of measure $0$, into discrete and dense representations.

##### Langlands functoriality and the inverse problem in Galois theory

In a couple of recent works with C. Khare and M. Larsen we contruct finte groups of Lie type $B_n$, $C_n$ and $G_2$ as Galois groups over rational numbers. The method combines some established, special cases of the functoriality principle with $l$-adic representations attached to self-dual automrophic representations of $GL(n)$.

##### The subconvexity problem for ${GL}_2$

The subconvexity problem consist in providing non-trivial upper bounds for central values of $L$-function. In recent years, this has been recognized as a central point to many arithmetic problems which could be related to the analytic theory of automorphic forms (like tha arithmetic quantum unique ergodicity conjecture or the study of representations of integers by ternary quadratic forms).

##### Global Existence for Nonlinear Dispersive Equation

Starting from small data, when does a nonlinear dispersive PDE have global solutions? A classical approach, just like for ODE, is to study resonances. But I will show that for PDE a new kind of resonances arises, that I call space resonances.

##### Systems Engineering for Water Management

It is estimated that we harvest and utilize about 65% of the readily available fresh water resources of the world. In general, perhaps because water is perceived as an abundantly available resource, we use water rather poorly. Typically less than half the water taken from the environment serves the objective for which it was intended.

##### Automorphism groups of curves

Hurwitz proved that a complex curve of genus $g>1$ has at most $84(g-1)$ automorphisms. In case equality holds, the automorphism group has a quite special structure.

##### The Nielsen Realization Problem

The purpose of this talk will be to describe some problems in topology whose solutions rely on geometric techniques. I will introduce the mapping class group of a space $X$, which is the group of homeomorphisms of $X$ up to isotopy.

##### On simple additive configurations in random sets

We show that with high probability a random subset of $[n]$ of size $\theta(n^{1-1/k})$ contains two elements $a$ and $a+d^k$, where $d$ is a positive integer. As a consequence, we prove an analogue of the Sarkozy-Furstenberg theorem for a random subset of $[n]$.

##### A quadratic bound on the number of boundary slopes of essential surfaces

##### On the Andre-Oort conjecture

Let S be a Shimura variety and L a set of special points on S. Andre and Oort conjecture that any irreducible component of the Zariki-closure of L is a subvariety of Hodge type of S. I will indicate a proof of this conjecture under GRH (this is joint work with A. Yafaev, relying on some work by Ullmo-Yafaev).

##### Ends of locally symmetric spaces

We intend to explain joint work with Lizhen Ji and Peter Li on relating the size of the bottom spectrum to the number of ends for locally symmetric spaces.

##### Hyperdiscriminant polytopes, Chow Polytopes, and K-energy asymptotics

Let (X,L) be a polarized algebraic manifold. I have recently proved that the Mabuchi energy of (X,L) is bounded from below along any degeneration if and only if the Hyperdiscriminant polytope contains the Chow polytope (with respect to the various Kodaira embeddings).

##### Stable Internet Routing Without Global Coordination

Global Internet connectivity results from a competitive cooperation of tens of thousands of independently-administered networks (called Autonomous Systems), each with their own preferences for how traffic should flow. The responsibility for reconciling these preferences falls to interdomain routing, realized today by the Border Gateway Protocol (BGP).

##### The cubic fourth order Schrödinger equation

We will discuss on which dimensions the cubic fourth-order Schrödinger equation is globally wellposed in the natural energy space. We will mainly concentrate on the case when the equation becomes energy-critical.

##### Limiting Distribution of Large Frobenius Numbers

##### Square-paths and square-cycles in graphs with high minimum degree

We investigate under which minimum-degree condition does a graph G contain a square-path and a square-cycle of a given length. We give precise thresholds, assuming that the order of G is large. This extends results of Fan and Kierstead [J. Combin. Theory Ser.