Past

I shall discuss a number of problems to do with approximating the value function of an American Put option in the Black-Scholes model. This is essentially a variant of the oxygen-consumption problem, a parabolic free boundary (obstacle) problem which is closely related to the Stefan problem. Having reviewed the short-time behaviour from the perspective of ray theory, I shall focus on constructing approximations in the case when there is a discretely paid dividend yield.

I will give a survey-type introduction to manifolds equipped with $G_2$ structures, emphasizing the similarities and differences with Riemannian manifolds equipped with almost complex structures, and with oriented Riemannian 3-manifolds. Along the way I may discuss the Berger classification of Riemannian holonomy, the Calabi-Yau theorem, exceptional geometric structures arising from the algebra of the Octonions, and calibrated submanifolds. This talk is the second of two parts.

'A two-point set is a subset of the plane which meets every line in exactly two points. The existence of two-point sets was shown by Mazurkiewicz in 1914, and the main open problem concerning these objects is to determine if there exist Borel two-point sets. If this question has a positive answer, then we most likely need to be able to construct a two-point set without making use of a well-ordering of the real line, as is currently the usual technique.

We discuss recent work by Robin Knight, Rolf Suabedissen and the speaker, and (independently) by Arnold Miller, which show that it is consistent with ZF that the real line cannot be well-ordered and also that two-point sets exist.'

The question of local existence of a deformation of a simply connected body whose Left Cauchy Green Tensor matches a prescribed, symmetric, positive definite tensor field is considered. A sufficient condition is deduced after formulation as a problem in Riemannian Geometry. The compatibility condition ends up being surprisingly different from that of compatibility of a Right Cauchy Green Tensor field, a fact that becomes evident after the geometric formulation. The question involves determining conditions for the local existence of solutions to an overdetermined system of Pfaffian PDEs with algebraic constraints that is typically not completely integrable.

We modify the usual Erdos-Renyi random graph evolution by letting connected clusters 'burn down' (i.e. fall apart to disconnected single sites) due to a Poisson flow of lightnings. In a range of the intensity of rate of lightnings, the system sticks to a permanent critical state (i.e. exhibits so-called self-organised critical behaviour). The talk will be based on joint work with Balint Toth.

We study non-commutative analogues of Serre-ï¬~Abrations in topology. We shall present several examples of such ï¬~Abrations and give applications for the computation of the K-theory of certain C*-algebras. (Joint work with Ryszard Nest and Herve Oyono-Oyono.)

Let M be a closed spin manifold.

Gromov and Lawson have shown that the presence of certain "enlargeable"

submanifolds of codimension 2 is an obstruction to the existence of a Riemannian metric with positive scalar curvature on M.

In joint work with Hanke, we refine the geoemtric condition of

"enlargeability": it suffices that a K-theoretic index obstruction of the submanifold doesn't vanish.

A "folk conjecture" asserts that all index type obstructions to positive scalar curvature should be read off from the corresponding index for the ambient manifold M (this this is equivalent to a small part of the strong Novikov conjecture). We address this question for the obstruction above and discuss partial results.

The uniform spanning forest (USF) in a graph

is a random spanning forest obtained as the limit of uniformly chosen spanning

trees on finite subgraphs. The USF is known to have stochastic dimension 4 on

graphs that are "at least 4 dimensional" in a certain sense. In this

talk I will look at more detailed estimates on the geometry of a fixed

component of the USF in the special case of the d-dimensional integer lattice,

d > 4. This is motivated in part by the study of random walk restricted to a

fixed component of the USF.

A random environment (in Z^d) is a collection of (random) transition probabilities, indexed by sites. Perform now a random walk using these transitions. This model is easy to describe, yet presents significant challenges to analysis. In particular, even elementary questions concerning long term behavior, such as the existence of a law of large numbers, are open. I will review in this talk the model, its history, and recent advance, focusing on examples of unexpected behavior.

Stress levels embedded in S&P 500 options are constructed and re-ported. The stress function used is MINMAXV AR: Seven joint laws for the top 50 stocks in the index are considered. The first time changes a Gaussian one factor copula. The remaining six employ correlated Brownian motion independently time changed in each coordinate. Four models use daily returns, either run as Lévy processes or scaled, to the option maturity. The last two employ risk neutral marginals from the V GSSD and CGMY SSD Sato processes. The smallest stress function uses CGMY SSD risk neutral marginals and Lévy correlation. Running the Lévy process yields a lower stress surface than scaling to the option maturity. Static hedging of basket options to a particular level of accept- ability is shown to substantially lower the price at which the basket option may be o¤ered.

"Stickelberger's famous theorem (from 1890) gives an explicit ideal which annihilates the imaginary part of the class group of an abelian field as a module for the group-ring of the Galois group. In the 1980s Tate and Brumer proposed a generalisation of Stickelberger's Theorem (and his ideal) to other abelian extensions of number fields, the so-called `Brumer-Stark conjecture'.

I shall discuss some of the many unresolved issues connected with the annihilation of class groups of number fields. For instance, should the (generalised) Stickelberger ideal be the full annihilator, the Fitting ideal or what? And what can we say in the plus part (where Stickelberger's Theorem is trivial)?"

In this talk we present a new construction of a Wedderburn basis for

the generic q-Schur algebra using the Du-Kazhdan-Lusztig basis. We show

that this gives rise to a new view on the Du-Lusztig homomorphism to the

asymptotic algebra. At the end we explain a potential plan for an attack

on James' conjecture using a reformulation by Meinolf Geck.

The talk starts with a gentle recollection of facts about

Iwahori-Hecke-Algebras of type A and q-Schur algebras and aims to be

accessible to people who are not (yet) experts in the representation

theory of q-Schur algebras.

All this is joint work with Olivier Brunat (Bochum).

The butterfly-shaped Lorenz attractor is a fractal set made up of infinitely many periodic orbits. Ever since Lorenz (1963) introduced a system of three simple ordinary differential equations, much of the discussion of his system and its strange attractor has adopted a dynamical point of view. In contrast, we allow time to be a complex variable and look upon such solutions of the Lorenz system as analytic functions. Formal analysis gives the form and coefficients of the complex singularities of the Lorenz system. Very precise (> 500 digits) numerical computations show that the periodic orbits of the Lorenz system have singularities which obey that form exactly or very nearly so. Both formal analysis and numerical computation suggest that the mathematical analysis of the Lorenz system is a problem in analytic function theory. (Joint work with S. Sahutoglu).

We present some recent results on singular solutions of the problem of travelling gravity water waves on flows with vorticity. We show that, for a certain class of vorticity functions, a sequence of regular waves converges to an extreme wave with stagnation points at its crests. We also show that, for any vorticity function, the profile of an extreme wave must have either a symmetric corner of 120 degrees or a horizontal tangent at any isolated stagnation point. Moreover, the profile necessarily has a symmetric corner of 120 degrees if the vorticity is nonnegative near the free surface.

I will give a survey-type introduction to manifolds equipped with $G_2$ structures, emphasizing the similarities and differences with Riemannian manifolds equipped with almost complex structures, and with oriented Riemannian 3-manifolds. Along the way I may discuss the Berger classification of Riemannian holonomy, the Calabi-Yau theorem, exceptional geometric structures arising from the algebra of the Octonions, and calibrated submanifolds. This talk will be in two parts.

This talk will review recent progress in understanding the effective

behavior of free boundaries in heterogeneous media.  Though motivated

by the pinning of martensitic phase boundaries, we shall explain

connections to other problems.  This talk is based on joint work with

Patrick Dondl.

Let X be a Gorenstein orbifold and Y a crepant resolution of

X. Suppose that the quantum cohomology algebra of Y is semisimple. We describe joint work with Iritani which shows that in this situation the genus-zero crepant resolution conjecture implies a higher-genus version of the crepant resolution conjecture. We expect that the higher-genus version in fact holds without the semisimplicity hypothesis.

In space dimension 2, it is well-known that the Smoluchowski-Poisson

system (also called the simplified or parabolic-elliptic Keller-Segel

chemotaxis model) exhibits the following phenomenon: there is a critical

mass above which all solutions blow up in finite time while all solutions

are global below that critical mass. We will investigate the case of the

critical mass along with the stability of self-similar solutions with

lower masses. We next consider a generalization to several space

dimensions which involves a nonlinear diffusion and show that a similar

phenomenon takes place but with some different features.

$PD$-complexes model the homotopy theory of manifolds.

In dimension 3, the unique factorization theorem holds to the extent that a $PD_3$-complex is a connected sum if and only if its fundamental group is a free product, and the indecomposables are aspherical or have virtually free fundamental group [Tura'ev,Crisp]. However in contrast to the 3-manifold case the group of an indecomposable may have infinitely many ends (i.e., not be virtually cyclic). We shall sketch the construction of one such example, and outline some recent work using only group theory that imposes strong restrictions on any other such examples.

The system

u_t = Delta u + buv - cu + u^{1/2} dW

v_t = - uv

models the evolution of a branching population and its usage of a non-renewable resource.

A phase diagram in the parameters (b,c) describes its long time evolution.

We describe this, including some results on asymptotics in the phase diagram for small and large values of the parameters.

Abstract: Recent developments in quantum field theory and twistor-string theory have thrown up surprising structures in the perturbative approach to gravity that cry out for a non-perturbative explanation. Firstly the MHV scattering amplitudes, those involving just two left handed and n-2 right handed outgoing gravitons are particularly simple, and a formalism has been proposed that constructs general graviton scattering amplitudes from these MHV amplitudes as building blocks. This formalism is chiral and suggestive of deep links with Ashtekar variables and twistor theory. In this talk, the MHV amplitudes are calculated ab initio by considering scattering of linear gravitons on a fully nonlinear anti-self-dual background using twistor theory, and a twistor action formulation is provided that produces the MHV formalism as its Feynman rules.

Starting from the problem of perfect hedging under market illiquidity, as introduced by Cetin, Jarrow and Protter, we introduce a class of second order target problems. A dual formulation in the general non-Markov case is obtained by formulating the problem under a convenient reference measure. In contrast with previous works, the controls lie in the classical H2 spaces associated to the reference measure. A dual formulation of the problem in terms of a standard stochastic control problem is derived, and involves control of the diffusion component.

I will present a universal definition of the integers in the field of rational numbers, building on work discussed by Bjorn Poonen in his seminar last term. I will also give, via model theory, a geometric criterion for the non-diophantineness of Z in Q.

It is well-known that there is a strong link between the representation

theories of the general linear group and the symmetric group over the

complex numbers. J.A.Green has shown that this in also true over infinite

fields of positive characteristic. For this he used the Schur functor as

introduced by I.Schur in his PhD thesis.

In this talk I will show that one can do the same thing for the symplectic

group and the Brauer algebra. This is joint work with S.Donkin. As a

consequence we obtain that (under certain conditions) the Brauer algebra and

the symplectic Schur algebra in characteristic p have the same block

relation. Furthermore we obtain a new proof of the description of the blocks

of the Brauer algebra in characteristic zero as obtained by Cox, De Visscher

and Martin.

Operator decomposition methods are an attractive solution strategy for computing complex phenomena involving multiple physical processes, multiple scales or multiple domains. The general strategy is to decompose the problem into components involving simpler physics over a relatively limited range of scales, and then to seek the solution of the entire system through an iterative procedure involving solutions of the individual components. We analyze the accuracy of an operator decomposition finite element method for a conjugate heat transfer problem consisting of a fluid and a solid coupled through a common boundary. We derive accurate a posteriori error estimates that account for both local discretization errors and the transfer of error between fluid and solid domains. We use these estimates to guide adaptive mesh refinement. In addition, we show that the order of convergence of the operator decomposition method is limited by the accuracy of the transferred gradient information, and how a simple boundary flux recovery method can be used to regain the optimal order of accuracy in an efficient manner. This is joint work with Don Estep and Tim Wildey, Department of Mathematics, Colorado State University.