# Seminars & Events for 2009-2010

##### Adding a list of numbers (and other determinantal processes)

Adding a list of digits produces 'carries along the way. The distribution theory of carries involves the emerging theory of determinantal point processes (explanations provided). Thinking of carries as cocycles, the story extends to central extensions. There are also nice connections to Koszul algebras. This is joint work with Alexei Borodin and Jason Fulman.

##### Magma

We give a quick tour of some features of the Magma computer algebra system. These will include: modular forms, algebraic geometry (sheaf cohomology and Groebner bases), computing with L-functions, machinery for function fields, lattices, and some group/representation theory. No experience with Magma will be assumed.

##### Algorithmic metatheorems for sparse classes of combinatorial structures

A classic result of Courcelle asserts that every monadic second order logic (MSOL) formula can be decided in linear time for graphs with bounded tree-width. This result unified most of algorithmic results for graphs with bounded tree-width and was followed by many results of the same favor.

##### On the limit curlicue process for theta sums

I shall discuss a random process achieved as the limit for the ensemble of curves generated by interpolating the values of theta sums. The existence and the properties of this process are established by means of purely dynamical tools and rely on generalizations of a result by Marklof and Jurkat and van Horne. (joint work with Jens Marklof).

##### Cosmetic Surgery Conjecture on S3

It has been known over 40 years that every closed orientable 3-manifold is obtained by surgery on a link in $S3$. However, a complete classification has remained elusive due to the lack of uniqueness of this surgery description. In this talk, we discuss the following uniqueness theorem for Dehn surgey on a nontrivial knot in $S3$.

##### Geometric Overconvergent Modular Forms

We will give a geometric definition of the notion of overconvergent modular form of any p-adic weight. As a consequence, we re-obtain Coleman's theory of p-adic families of eigenforms and the eigencurve of Coleman and Mazur without using the Eisenstein family. Similar results have just been obtained independantly by Andreatta, Iovita and Stevens.

##### The dimension of self-affine sets: past, present and future.

Calculating the dimension of sets invariant under non-conformal dynamics is a formidable problem. My talk will be a survey on what is known and expected for self-affine sets, i.e. sets invariant under piece-wise affine expanding maps on Euclidean space. Some emphasis will be given to my joint work with A.

##### The new Intrinsic flat distance between oriented Riemannian manifolds

We define a new distance between oriented Riemannian manifolds that we call the "intrinsic flat distance" based upon Ambrosio-Kirchheim's theory of integral currents on metric spaces. Limits of sequence of manifolds with a uniform upper bound on their volume and diameter are countably H^m rectifiable metric spaces with an orientation and multiplicity that we call "integral current spaces".

##### Fixed Point Theorems and Applications in PDEs

I'll begin with Banach's fixed point theorem in which strict contraction is required and then give examples to show how this simple theorem implies the local and global existence to vary kinds of evolution equations. I'll also introduce Schauder and Schaefer's fixed point theorem which would be of importance in elliptic theory and verify this by several examples if time permitted.

##### Complex variables are not dead

Our lecture will focus on two problems in pde which are solvable by ideas in holomorphic functions of complex variables. The first problem is called the strip theorem. Let $f$ be a function defined in the strip in the complex plane $|Im z| \leq 1$.

##### CAT(0) cube complexes in geometric group theory

CAT(0) cube complexes are high-dimensional generalizations of trees that have emerged as increasingly central objects in combinatorial and geometric group theory. I will describe their prominent geometric properties and explain how cube complexes often arise from infinite groups - the latest examples being the fundamental groups of hyperbolic 3-manifolds.

##### Random polygons in plane convex sets

Consider picking $N$ random points in a convex set $K$ and forming their convex hull $K_N$. Recently, there have been a number of results concerning the asymptotic behavior of random variables such as the area and number of vertices of $K_N$. These are, however, all limited to two special cases: 1) $K$ is a polygon and 2) $K$ is "smooth".

##### On the equivariant K-theory of toric varieties

A recent computation of Bahri, Franz and Ray identified the equivariant cohomology of weighted projective space with a ring of piecewise polynomials. This talk will report on joint work of Ray and the speaker, on recent developments regarding the equivariant topological K-theory of toric varieties.

##### Tree packing conjectures; Graceful tree labeling conjecture

A family of graphs $H_1,...,H_k$ packs into a graph $G$ if there exist pairwise edge-disjoint copies of $H_1,...,H_k$ in $G$. Gyarfas and Lehel conjectured that any family $T_1, ..., T_n$ of trees of respective orders $1, ..., n$ packs into $K_n$. A similar conjecture of Ringel asserts that $2n$ copies of any trees $T$ on $n+1$ vertices pack into $K_{2n+1}$.

##### The structure of groups with a quasiconvex hierarchy

We prove that hyperbolic groups with a quasiconvex hierarchy are virtually subgroups of graph groups. Our focus is on "special cube complexes" which are nonpositively curved cube complexes that behave like "high dimensional graphs" and are closely related to graph groups.

##### The average rank of elliptic curves

A *rational elliptic curve* may be viewed as the set of solutions to an equation of the form $y2=x3+Ax+B$, where $A$ and $B$ are rational numbers. It is known that the rational points on this curve possess a natural abelian group structure, and the Mordell-Weil theorem states that this group is always finitely generated.

##### A Codazzi-like equation and the singular set for surfaces in the Heisenberg group

##### Introduction to DB singularities

##### Radiative decay of bubble oscillations

We consider the dynamics of a gas bubble in an unbounded, inviscid and compressible fluid with surface tension. Kinematic and dynamic (Young-Laplace) boundary conditions couple the dynamics of bubble surface deformations to the dynamics of waves in the fluid. We study the linear decay estimates for the fluid and deforming bubble near the spherical equilibrium.