Past

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.

A new portfolio choice model in continuous time is formulated and solved, where the quantile function of the terminal cash flow, instead of the cash flow itself, is taken as the decision variable. This formulation covers and leads to solutions to many existing and new models including expected utility maximisation, mean-variance, goal reaching, VaR and CVaR, Yaari's dual model, Lopes' SP/A model, and behavioural model under prospect theory.