# Seminars & Events for Department Colloquium

##### Counting Faces of Randomly-Projected Polytopes, with applications to Compressed Sensing, Error-Correcting Codes, and Statistical Data Mining

##### The geometry and topology of arithmetic hyperbolic 3-manifolds

This talk will discuss recent advances in regards to some of the main open problems about hyperbolic 3-manifolds in the context of arithmetic hyperbolic 3-manifolds.

##### Unitary representations of simple Lie groups

By 1950, work of Gelfand and others had led to a general program for "non-commutative harmonic analysis": understanding very general mathematical problems (particularly of geometry or analysis) in the presence of a (non-commutative) symmetry group G. A first step in that program is classification of unitary representations - that is, the realizations of G as automorphisms of a Hilbert space. Despite tremendous advances from the work of Harish-Chandra, Langlands, and others, completing this first step is still some distance away. Since functional analysis is not as fashionable now as it was in 1950, I'll explain some of the ways that Gelfand's problem can be related to algebraic geometry (particularly to equivariant K-theory).

##### Three conjectures in arithmetic geometry

We discuss the Manin-Mumford conjecture (about the closure of any set of torsion points in an abelian variety), the André-Oort conjecture (about the closure of any set of CM-points in a moduli space) and the Hecke Orbit Conjecture (about the closure of the Hecke orbit of a moduli point). These conjectures, on the borderline of geometry and arithmetic, seem to be (have been) solved. We explain the similarities. We will discuss the motivation for these conjectures, and in some cases we will say something about methods of proofs.

##### Cremona transformations and homeomorphisms of topological surfaces

##### Fundamental lemma and Hitchin fibration

The fundamental lemma is an identity of orbital integrals on p-adic reductive groups which was stated precisely by Langlands and Shelstads as a conjecture in the 80's. We now have a proof due to the efforts of many peoples with many ingredients. I will only explain how a certain particular type of geometry like affine Springer fibers and Hitchin were helpful in this proof.

##### A new proof of Gromov's theorem on groups of polynomial growth

##### 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. Ideally, the moduli spaces giving rise to these invariants should be critical sets of "holomorphic Chern-Simons functions." Currently, such holomorphic Chern-Simons functions exist at best locally (see my seminar talk on Monday), and it is unlikely that they exist globally. I will describe geometric structures on the moduli spaces (some conjectural) that exist globally and reflect the fact that the moduli spaces look as if they were the zero loci of holomorphic maps.

##### Mathematical Questions Arising from Bose-Einstein Condensation

Bose-Einstein condensation was predicted by Einstein in 1925 and was experimentally discovered 70 years later. This discovery was followed by a flurry of activity in the physics community with many new experiments and with attempts to construct a theory of the newly discovered state of matter. In this talk I will review some recent rigorous results in the subject and outline open problems.

##### Large N limit of random matrices, free probability and the graded algebra of a planar algebra

##### Internal aggregation Models: From Diaconis-Fulton addition to a free boundary problem

Start with $n$ particles at each of $k$ points in the $d$-dimensional lattice, and let each particle perform simple random walk until it reaches an unoccupied site. The law of the resulting random set of occupied sites does not depend on the order in which the walks are performed, as shown by Diaconis and Fulton. We prove that if the distances between the starting points are suitably scaled, then the set of occupied sites has a deterministic scaling limit. In two dimensions, the boundary of the limiting shape is an algebraic curve of degree $2k$. (For $k=1$ it is a circle, as proved in 1992 by Lawler, Bramson and Griffeath.) The limiting shape can also be described in terms of a free-boundary problem for the Laplacian and quadrature identities for harmonic functions.

##### Quantum Unique Ergodicity and Number Theory

A fundamental problem in the area of quantum chaos is to understand the distribution of high eigenvalue eigenfunctions of the Laplacian on certain Riemannian manifolds. A particular case which is of interest to number theorists concerns hyperbolic manifolds arising as a quotient of the upper half-plane by a discrete "arithmetic" subgroup of $SL_2(R)$ (for example, $SL_2(Z)$, and in this case the corresponding eigenfunctions are called Maass cusp forms). In this case, Rudnick and Sarnak have conjectured that the high energy eigenfunctions become equi-distributed. I will discuss some recent progress which has led to a resolution of this conjecture, and also on a holomorphic analog for classical modular forms.

##### What goes on in a plasma

##### On a conjecture of De Giorgi

In 1978 De Giorgi made a conjecture about the symmetry of global solutions to a certain semilinear elliptic equation. He stated that monotone, bounded solutions of $$ \triangle u=u^3-u$$ in $\mathbb{R}^n$ are one dimensional (i.e. the level sets of $u$ are hyperplanes) at least in dimension $n \le 8$. This problem is in fact closely related to the theory of minimal surfaces and it is sometimes referred to as "the $\varepsilon$ version of the Bernstein problem for mininimal graphs". In my talk I will explain this relation and I will give an idea about the proof of this conjecture for $n \le 8$. We mention that recently Del Pino, Kowalzyk and Wei provided a counterexample in dimension $n \ge 9$.

##### Invariant distributions and scaling in parabolic dynamics

A smooth dynamical system is often called parabolic if nearby orbits diverge with power-like (polynomial) speed. There is no general theory of parabolic dynamics and a few classes of examples are relatively well-understood: area-preserving flows with saddle singularities on surfaces (or, equivalently, interval exchange transformations) and to a lesser extent 'rational' polygonal billiards; $SL(2,R)$ unipotent subgroups (horocycle flows on surfaces of constant negative curvature) and nilflows. In all the above cases, the typical system is uniquely ergodic, hence ergodic averages of continuous functions converge unformly to the mean. A fundamental question concerns the speed of this convergence for sufficiently smooth functions. In many cases it is possible to prove power-like (polynomial) upper bounds.