# Seminars & Events for 2008-2009

##### State-of-the-art Computer Simulations of Supernova Explosions

To simulate supernova explosions, one must solve simultaneously the non-linear, coupled partial differential equations of radiation hydrodynamics. What's more, due to a variety of instabilities and asymmetries, this must eventually be accomplished in 3D. The current state-of-the-art is 2D, plus rotation and magnetic fields (assuming axisymmetry).

##### Mapping class groups, relative hyperbolicity and rigidity

##### Eigenvalue Statistics for Random CMV Matrices

CMV matrices are the unitary analogues of one dimensional discrete Schrödinger operators. We consider CMV matrices with random coefficients and we study the statistical distribution of their eigenvalues. For slowly decreasing random coefficients, we show that the eigenvalues are distributed according to a Poisson process.

##### Calabi-Yau threefolds with vanishing third Betti number

Smooth, projective, three dimensional, algebraic varieties with trivial canonical sheaf and vanishing third etale Betti number do not exist over fields of characteristic zero. In the past few years a number of examples have been found in positive characteristic. Some of these examples and questions they raise will be discussed.

##### Lee-Yang zeros for the Diamond Hierarchical Lattice and 2D rational dynamics

In a classical work of 1950's, Lee and Yang proved that zeros of the partition functions of the Ising models on graphs always lie on the unit circle. Distribution of these zeros is physically important as it controls phase transitions in the model.

##### On the geometry of space-time

In the relativistic point of view, the geometry of space should evolve with time, in a manner directed by Einstein equations. I will briefly summarize two interesting aspects with open questions:

##### Symplectic Galois representations over totally real fields

We associate $p$-adic Galois representations to globally generic cusp forms on $GSp(4)$, over a totally real field, with a Steinberg component at some finite place. At places $v$ not dividing $p$ one has local-global compatibility, the local correspondence being that defined by Gan and Takeda.

##### The Kähler-Ricci flow and canonical measures

We define and prove the existence of canonical measures of Einstein type on algebraic manifolds of nonnegative Kodaira dimension. We also show that the Kähler-Ricci flow can be uniquely defined on algebraic varieties with log terminal singularities. It reveals the deep connection between the Ricci flow and the classification of algebraic varieties.

##### An Extension of the Stability Theorem of the Minkowski Space in General Relativity

We present a generalization of the celebrated results by D. Christodoulou and S. Klainerman for solutions of the Einstein vacuum equations in General Relativity.