Past

On a manifold there is the graded algebra of polyvector fields with its Lie-Schouten bracket, and the module of de Rham differentials with exterior differentiation. This package is called a "calculus". The moduli

space of sheaves (or derived category objects) on a Calabi-Yau threefold has a kind of "virtual calculus" on it, at least conjecturally. In particular, this moduli space has virtual de Rham cohomology groups, which categorify Donaldson-Thomas invariants, at least conjecturally. We describe some attempts at constructing such a virtual calculus. This is work in progress.

How many triangles must a graph of density d contain? This old question due to Erdos was recently answered by Razborov, after many decades of progress by numerous authors.

We will consider the analogous question for tripartite graphs. Given a tripartite graph with prescribed edges densities between each

pair of classes how many triangles must it contain?

On a manifold there is the graded algebra of polyvector fields with its Lie-Schouten bracket, and the module of de Rham differentials with exteriour differentiation. This package is called a "calculus". The moduli space of sheaves (or derived category objects) on a Calabi-Yau threefold has a kind of "virtual calculus" on it, at least conjecturally. In particular, this moduli space has virtual de Rham cohomology groups, which categorify Donaldson-Thomas invariants, at least conjecturally. We describe some attempts at constructing such a virtual calculus. This is work in progress.

We investigate the behaviour of free-surface waves on time-varying potential flow in the limit as the Froude number becomes small. These waves are exponentially small in the Froude number, and are therefore inaccessible to ordinary asymptotic methods. As such, we demonstrate how exponential asymptotic techniques may be applied to the complexified free surface in order to extract information about the wave behaviour on the free surface, using a Lagrangian form of the potential flow equations. We consider the specific case of time-varying flow over a step, and demonstrate that the results are consistent with the steady state case.

Let $u \in W^{1,p}(\Omega,\R^N)$, $\Omega$ a bounded domain in

$\R^n$, be a minimizer of a convex variational integral or a weak solution to

an elliptic system in divergence form. In the vectorial case, various

counterexamples to full regularity have been constructed in dimensions $n

\geq 3$, and it is well known that only a partial regularity result can be

expected, in the sense that the solution (or its gradient) is locally

continuous outside of a negligible set. In this talk, we shall investigate

the role of the space dimension $n$ on regularity: In arbitrary dimensions,

the best known result is partial regularity of the gradient $Du$ (and hence

for $u$) outside of a set of Lebesgue measure zero. Restricting ourselves to

the partial regularity of $u$ and to dimensions $n \leq p+2$, we explain why

the Hausdorff dimension of the singular set cannot exceed $n-p$. Finally, we

address the possible existence of singularities in two dimensions.

We prove that no Riemannian manifold quasiisometric to

complex hyperbolic plane can have a better curvature pinching. The proof

uses cup-products in $L^p$-cohomology.

We study a simple heat-bath type dynamic for a simple model of
polymer interacting with an interface. The polymer is a nearest neighbor path in
Z, and the interaction is modelised by energy penalties/bonuses given when the
path touches 0. This dynamic has been studied by D. Wilson for the case without
interaction, then by Caputo et al. for the more general case. When the interface
is repulsive, the dynamic slows down due to the appearance of a bottleneck in the
state space, moreover, the systems exhibits a metastable behavior, and, after time
rescaling, behaves like a two-state Markov chain.

We consider a directed random polymer interacting with an interface
that carries random charges some of which attract while others repel
the polymer. Such a polymer can be in a localized or delocalized
phase, i.e., it stays near the interface or wanders away respectively.
 The phase it chooses depends on the temperature and the average bias
of the disorder. At a given temperature, there is a critical bias
separating the two phases. A question of particular interest, and
which has been studied extensively in the Physics and Mathematics
literature, is whether the quenched critical bias differs from the
annealed critical bias. When it does, we say that the disorder is
relevant.

Using a large deviations result proved recently by Birkner, Greven,
and den Hollander, we derive a variational formula for the quenched

critical bias. This leads to a necessary and sufficient condition for
disorder relevance that implies easily some known results as well as
new ones.

The talk is based on joint work with Frank den  Hollander.

Anomalous ( non local) diffusion processes appear in many subjects: phase transition, fracture dynamics, game theory I will describe some of the issues involved, and in particular, existence and regularity for some non local versions of the p Laplacian, of non variational nature, that appear in non local tug of war.

The talk will be devoted to criteria of absence of arbitrage opportunities under small transaction costs for a family of multi-asset models of financial market.

• Ian Griffiths - "Taylor Dispersion in Colloidal Systems".
• James Lottes - "Algebraic multigrid for nonsymmetric problems".
• Derek Moulton - "Surface growth kinematics"
• Rob Style - "Ice lens formation in freezing soils"

Please note the earlier than usual start-time!

The Mathematical Institute invites you to attend the Inaugural Lecture of Professor Gui-Qiang G. Chen. Professor in the Analysis of Partial Differential Equations. Examination Schools, 75-81 High Street, Oxford, OX 4BG.

There is no charge to attend but registration is required. Please register your attendance by sending an email to @email specifying the number of people in your party. Admission will only be allowed with prior registration.

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ABSTRACT

While calculus is a mathematical theory concerned with change, differential equations are the mathematician's foremost aid for describing change. In the simplest case, a process depends on one variable alone, for example time. More complex phenomena depend on several variables – perhaps time and, in addition, one, two or three space variables. Such processes require the use of partial differential equations. The behaviour of every material object in nature, with timescales ranging from picoseconds to millennia and length scales ranging from sub-atomic to astronomical, can be modelled by nonlinear partial differential equations or by equations with similar features. The roles of partial differential equations within mathematics and in the other sciences become increasingly significant. The mathematical theory of partial differential equations has a long history. In the recent decades, the subject has experienced a vigorous growth, and research is marching on at a brisk pace.

In this lecture, Professor Gui-Qiang G. Chen will present several examples to illustrate the origins, developments, and roles of partial differential equations in our changing world.

Efficient linear and infinitely smooth approximation of functions from equidistant samples is a fascinating problem, at least since Runge showed in 1901 that it is not delivered by the interpolating polynomial.

In 1988, I suggested to substitute linear rational for polynomial interpolation by replacing the denominator 1 with a polynomial depending on the nodes, though not on the interpolated function. Unfortunately the so-obtained interpolant converges merely as the square of the mesh size. In 2007, Floater and Hormann have given for every integer a denominator that yields convergence of that prescribed order.

In the present talk I shall present the corresponding interpolant as well as some of its applications to differentiation, integration and the solution of boundary value problems. This is joint work with Georges Klein and Michael Floater.

The maximum principle is one of the main tools use to understand the behaviour of solutions to the Ricci flow. It is a very powerful tool that can be used to show that pointwise inequalities on the initial data of parabolic PDE are preserved by the evolution. A particular weak maximum principle for vector bundles will be discussed with references to Hamilton's seminal work [J. Differential Geom. 17 (1982), no. 2, 255–306; MR664497] on 3-manifolds with positive Ricci curvature and his follow up paper [J. Differential Geom. 24 (1986), no. 2, 153–179; MR0862046] that extends to 4-manifolds with various curvature assumptions.

This talk highlights the role of Gaussian Process models in sequential

data analysis. Issues of active data selection, global optimisation,

sensor selection and prediction in the presence of changepoints are

discussed.

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.