Past

Abstract. In this talk, I will present joint work with Uri Onn, Mark Berman, and Pirita Paajanen.

Let G be a linear algebraic group defined over the integers. Let O be a compact discrete valuation ring with a finite residue field of cardinality q and characteristic p. The group

G(O) has a filtration by congruence subgroups

G_m(O) (which is by definition the kernel of reduction map modulo P^m where P is the maximal ideal of O).

Let c_m=c_m(G(O))  denote the number of conjugacy classes in the finite quotient group G(O)/G_m(O) (which is called the mth congruence quotient of G(O)).  The conjugacy class zeta function of

G(O) is defined to be the Dirichlet series Z_{G(O)}(s)=\sum_{m=0,1,...} c_m q^_{-ms}, where s is a complex number with Re(s)>0. This zeta function was defined by du Sautoy when G is a p-adic analytic group and O=Z_p, the ring of p-adic integers, and he proved that in this case it is a rational function in p^{-s}.  We consider the question of dependence of this zeta function on p and more generally on the ring O.

We prove that for certain algebraic groups, for all compact discrete valuation rings with finite residue field of cardinality q and sufficiently large residue characteristic p, the conjugacy class zeta function is a rational function in q^{-s} which depends only on q and not on the structure of the ring. Note that this applies also to positive characteristic local fields.

A key in the proof is a transfer principle. Let \psi(x) and f(x) be resp.

definable sets and functions in Denef-Pas language.

For a local field K, consider the local integral Z(K,s)=\int_\psi(K)

|f(x)|^s dx, where | | is norm on K and dx normalized absolute value

giving the integers O of K volume 1. Then there is some constant

c=c(f,\psi) such that  for all local fields K of residue characteristic larger than c and residue field of cardinality q, the integral Z(K,s) gives the same rational function in q^{-s} and takes the same value as a complex function of s.

This transfer principle is more general than the specialization to local fields of the special case when there is no additive characters of the motivic transfer principle of Cluckers and Loeser since their result is the case when the integral is zero.

The conjugacy class zeta function is related to the representation zeta function which counts number of irreducible complex representations in each degree (provided there are finitely many or finitely many natural classes) as was shown in the work of Lubotzky and Larsen, and gives information on analytic properties of latter zeta function.

We will investigate what one can detect about a discrete group from its profinite completion, with an emphasis on considering geometric properties.

Cryopreservation (using temperatures down to that of liquid nitrogen at

–196 °C) is the only way to preserve viability and function of mammalian cells for research and transplantation and is integral to the quickly evolving field of regenerative medicine. To cryopreserve tissues, cryoprotective agents (CPAs) must be loaded into the tissue. The loading is critical because of the high concentrations required and the toxicity of the CPAs. Our mathematical model of CPA transport in cartilage describes multi-component, multi-directional, non-dilute transport coupled to mechanics of elastic porous media in a shrinking and swelling domain.

Parameters are obtained by fitting experimental data. We show that predictions agree with independent spatially and temporally resolved MRI experimental measurements. This research has contributed significantly to our interdisciplinary group’s ability to cryopreserve human articular cartilage.

A family of graphs F on a fixed set of n vertices is said to be triangle-intersecting if for any two graphs G,H in F, the intersection of G and H contains a triangle. Simonovits and Sos conjectured that such a family has size at most (1/8)2^{\binom{n}{2}}, and that equality holds only if F
consists of all graphs containing some fixed triangle. Recently, the author, Yuval Filmus and Ehud Friedgut proved a strengthening of this conjecture, namely that if F is an odd-cycle-intersecting family of graphs, then |F| \leq (1/8) 2^{\binom{n}{2}}. Equality holds only if F consists of all graphs containing some fixed triangle. A stability result also holds: an odd-cycle-intersecting family with size close to the maximum must be close to a family of the above form. We will outline proofs of these results, which use Fourier analysis, together with an analysis of the properties of random cuts in graphs, and some results in the theory of Boolean functions. We will then discuss some related open questions.

All will be based on joint work with Yuval Filmus (University of Toronto) and Ehud Friedgut (Hebrew University of Jerusalem).

We study the nonhomogeneous boundary value problem for the

Navier--Stokes equations

\[

\left\{ \begin{array}{rcl}

-\nu \Delta{\bf u}+\big({\bf u}\cdot \nabla\big){\bf u} +\nabla p&=&{0}\qquad \hbox{\rm in }\;\;\Omega,\\[4pt]

{\rm div}\,{\bf u}&=&0 \qquad \hbox{\rm in }\;\;\Omega,\\[4pt]

{\bf u}&=&{\bf a} \qquad \hbox{\rm on }\;\;\partial\Omega

\end{array}\right

\eqno(1)

\]

in a bounded multiply connected domain

$\Omega\subset\mathbb{R}^n$ with the boundary $\partial\Omega$,

consisting of $N$ disjoint components $\Gamma_j$.

Starting from the famous J. Leray's paper published in 1933,

problem (1) was a subject of investigation in many papers. The

continuity equation in (1) implies the necessary solvability

condition

$$

\int\limits_{\partial\Omega}{\bf a}\cdot{\bf

n}\,dS=\sum\limits_{j=1}^N\int\limits_{\Gamma_j}{\bf a}\cdot{\bf

n}\,dS=0,\eqno(2)

$$

where ${\bf n}$ is a unit vector of the outward (with respect to

$\Omega$) normal to $\partial\Omega$. However, for a long time

the existence of a weak solution ${\bf u}\in W^{1,2}(\Omega)$ to

problem (1) was proved only under the stronger condition

$$

{\cal F}_j=\int\limits_{\Gamma_j}{\bf a}\cdot{\bf n}\,dS=0,\qquad

j=1,2,\ldots,N. \eqno(3)

$$

During the last 30 years many partial results concerning the

solvability of problem (1) under condition (2) were obtained. A

short overview of these results and the detailed study of problem

(1) in a two--dimensional bounded multiply connected domain

$\Omega=\Omega_1\setminus\Omega_2, \;\overline\Omega_2\subset

\Omega_1$ will be presented in the talk. It will be proved that

this problem has a solution, if the flux ${\cal

F}=\int\limits_{\partial\Omega_2}{\bf a}\cdot{\bf n}\,dS$ of the

boundary datum through $\partial\Omega_2$ is nonnegative (outflow

condition).

In this talk I will introduce some of the basic ideas linking the theory of complex multiplication for elliptic curves and class field theory. Time permitting, I'll mention Shimura and Taniyama's work on the case of abelian varieties.

Let $G $ be a compact Lie group, and consider the variety $\text {Hom} (\bb Z^k,G)$

of representations of the rank $k$ abelian free group $\bb Z^k$ into $G$. We prove

that the fundamental group of $\text {Hom} (\bb Z^k,G) $ is naturally isomorphic to direct

product of $k$ copies of the fundamental group of $G$. This is joint work with

Jose Manuel Gomez and Juan Souto.

In this talk, we will look at the diffusive scaling limit of a class of

one-dimensional random walks in a random space-time environment. In the

scaling limit, this gives rise to a so-called stochastic flow of kernels as

introduced by Le Jan and Raimond and generalized by Howitt and Warren. We will

prove several new results about these stochastic flows of kernels by making

use of the theory of the Brownian web and net. This is joint work with R. Sun

and E. Schertzer.

Abstract:  An instance of the Potts model is defined by a graph of interactions and a number, q, of  different ``spins''.  A configuration in this model is an assignment of spins to vertices. Each configuration has a weight, which in the ferromagnetic case is greater when more pairs of adjacent spins are alike.  The classical Ising model is the special case

of q=2 spins.  We consider the problem of computing an approximation to the partition function, i.e., weighted sum of configurations, of

an instance of the Potts model.  Through the random cluster formulation it is possible to make sense of the partition function also for non-integer q.  Yet another equivalent formulation is as the Tutte polynomial in the positive quadrant.

About twenty years ago, Jerrum and Sinclair gave an efficient (i.e., polynomial-time) algorithm for approximating the partition function of a ferromagnetic Ising system. Attempts to extend this result to q≠2 have been unsuccessful. At the same time, no convincing evidence has been presented to indicate that such an extension is impossible.  An interesting feature of the random cluster model when q>2 is that it exhibits a first-order phase transition, while for 1≤q≤2 only a second-order phase transition is apparent.  The idea I want to convey in this talk is that this first-order phase transition can be exploited in order to encode apparently hard computational problems within the model.  This provides the first evidence that the partition function of the ferromagnetic Potts model may be hard to compute when q>2.

This is joint work with Leslie Ann Goldberg, University of Liverpool.

In this talk a number of identities will be discussed that relate to the event of level crossing of certain types of stochastic processes. Some of these identities are surprisingly simple and have interpretations in surplus modelling of insurance portfolios, the design of taxation schemes, optimal dividend strategies and the pricing of barrier options.

POSTPONED!!!

PLEASE NOTE THAT THIS WORKSHOP IS TO BE HELD IN 21 BANBURY ROAD BEGINNING AT 9AM! \\

We will give three short presentations of current work here on small scale mechanics :

1) micron-scale cantilever testing and nanoindentation - Dave Armstrong

2) micron-scale pillar compression – Ele Grieveson

3) Dislocation loop shapes – Steve Fitzgerald

These should all provide fuel for discussion, and I hope ideas for future collaborative work.\\

The meeting will be in the committee room in 21 Banbury Rd (1st floor, West end).

Vopenka's Principle is a very strong large cardinal axiom which can be used to extend ZFC set theory. It was used quite recently to resolve an important open question in algebraic topology: assuming Vopenka's Principle, localisation functors exist for all generalised cohomology theories. After describing the axiom and sketching this application, I will talk about some recent results showing that Vopenka's Principle is relatively consistent with a wide range of other statements known to be independent of ZFC. The proof is by showing that forcing over a universe satisfying Vopenka's Principle will frequently give an extension universe also satisfying Vopenka's Principle.