Past

In this talk, we present an extension of the theory of rough paths to partial differential equations. This allows a robust approach to stochastic partial differential equations, and in particular we can replace Brownian motion by more general Gaussian and Markovian noise. Support theorems and large deviation statements all become easy corollaries of the corresponding statements of the driving process. This is joint work with Peter Friz in Cambridge.

It is a classical result that the Alexander polynomial of a fibered knot has to be monic. But in general the converse does not hold, i.e. the Alexander polynomial does not detect fibered knots. We will show that the collection of all twisted Alexander polynomials (which are a natural generalization of the ordinary Alexander polynomial) detect fibered 3-manifolds.

As a corollary it follows that given a 3-manifold N the product S1 x N is symplectic if and only if N is fibered.

A new general approach to optimal stopping problems in L\'evy models, regime switching L\'evy models and L\'evy models with stochastic volatility and stochastic interest rate is developed. For perpetual options, explicit solutions are found, for options with finite time horizon, time discretization is used, and explicit solutions are derived for resulting sequences of perpetual options.

The main building block is the option to abandon a monotone payoff stream. The optimal exercise boundary is found using the operator form of the Wiener-Hopf method, which is standard in analysis, and interpretation of the factors as {\em expected present value operators} (EPV-operators) under supremum and infimum processes.

Other types of options are reduced to the option to abandon a monotone stream. For regime-switching models, an additional ingredient is an efficient iteration procedure.

L\'evy models with stochastic volatility and/or stochastic interest rate are reduced to regime switching models using discretization of the state space for additional factors. The efficiency of the method for 2 factor L\'evy models with jumps and for 3-factor Heston model with stochastic interest rate is demonstrated. The method is much faster than Monte-Carlo methods and can be a viable alternative to Monte Carlo method as a general method for 2-3 factor models.

Joint work of Svetlana Boyarchenko,University of Texas at Austin and Sergei Levendorski\v{i},

University of Leicester

We discuss the valuation problem for a broad spectrum of derivatives, especially in Levy driven models. The key idea in this approach is to separate from the computational point of view the role of the two ingredients which are the payoff function and the driving process for the underlying quantity. Conditions under which valuation formulae based on Fourier and Laplace transforms hold in a general framework are analyzed. An interesting interplay between the properties of the payoff function and the driving process arises. We also derive the analytically extended characteristic function of the supremum and the infimum processes derived from a Levy process. Putting the different pieces together, we can price lookback and one-touch options in Levy driven models, as well as options on the minimum and maximum of several assets.

The large-time behavior of solutions to Burgers equation with

small viscosity is described using invariant manifolds. In particular,

a geometric explanation is provided for a phenomenon known as

metastability,which in the present context means that

solutions spend a very long time near the family of solutions known as

diffusive N-waves before finally converging to a stable self-similar

diffusion wave. More precisely, it is shown that in terms of

similarity, or scaling, variables in an algebraically weighted $L^2$

space, the self-similar diffusion waves correspond to a one-dimensional

global center manifold of stationary solutions. Through each of these

fixed points there exists a one-dimensional, global, attractive,

invariant manifold corresponding to the diffusive N-waves. Thus,

metastability corresponds to a fast transient in which solutions

approach this ``metastable" manifold of diffusive N-waves, followed by

a slow decay along this manifold, and, finally, convergence to the

self-similar diffusion wave.

(joint work with E. Hrushovski and A. Pillay)

If G is a definably compact, connected group definable in an o-minimal structure then, as is known, G/Z(G) is semisimple (no infinite normal abelian subgroup).

We show, that in every o-minimal expansion of an ordered group:

If G is a definably connected central extension of a semisimple group then it is bi-intepretable, over parameters, with the two-sorted structure (G/Z(G), Z(G)). Many corollaries follow for definably connected, definably compact G.
Here are two:

1. (G,.) is elementarily equivalent to a compact, connected real Lie group of the same dimension.

2. G can be written as an almost direct product of Z(G) and [G,G], and this last group is definable as well (note that in general [G,G] is a countable union of definable sets, thus not necessarily definable).

In 1989, Happel raised the following question: if the Hochschild cohomology

groups of a finite dimensional algebra vanish in high degrees, then does the

algebra have finite global dimension? This was answered negatively in a

paper by Buchweitz, Green, Madsen and Solberg. However, the Hochschild

homology version of Happel's question, a conjecture given by Han, is open.

We give a positive answer to this conjecture for local graded algebras,

Koszul algebras and cellular algebras. The proof uses Igusa's formula for

relating the Euler characteristic of relative cyclic homology to the graded

Cartan determinant. This is joint work with Dag Madsen.

This is not intended to be a systematic History, but a selection of highlights, with some digressions, including:

The early days of the Computing Lab;

How the coming of the Computer changed some of the ways we do Computation;

A problem from the Study Groups;

Influence of the computing environment (hardware and software);

Convergence analysis for the heat equation, then and now.

A function $u: \mathbb{R}^{n} \to \mathbb{R}^{m}$ is one-homogeneous if $u(ax)=au(x)$ for any positive real number $a$ and all $x$ in $\R^{n}$. Phillips(2002) showed that in two dimensions such a function cannot solve an elliptic system in divergence form, in contrast to the situation in higher dimensions where various authors have constructed one-homogeneous minimizers of regular variational problems. This talk will discuss an extension of Phillips's 2002 result to $x-$dependent systems. Some specific one-homogeneous solutions will be constructed in order to show that certain of the hypotheses of the extension of the Phillips result can't be dropped. The method used in the construction is related to nonlinear elasticity in that it depends crucially on polyconvex functions $f$ with the property that $f(A) \to \infty$ as $\det A \to 0$.

We identify a natural way of ordering functions, which we call the interval dominance order, and show that this concept is useful in the theory of monotone comparative statics and also in statistical decision theory. This ordering on functions is weaker than the standard one based on the single crossing property (Milgrom and Shannon, 1994) and so our monotone comparative statics results apply in some settings where the single crossing property does not hold. For example, they are useful when examining the comparative statics of optimal stopping time problems. We also show that certain basic results in statistical decision theory which are important in economics - specifically, the complete class theorem of Karlin and Rubin (1956) and the results connected with Lehmann's (1988) concept of informativeness – generalize to payoff functions that obey the interval dominance order.

Geometrically, the problem of descent asks when giving some structure on a space is the same as giving some structure on a cover of the space, plus perhaps some extra data.
In algebraic geometry, faithfully flat descent says that if $X\rightarrow Y$ is a faithfully flat morphism of schemes, then giving a sheaf on $Y$ is the same as giving a collection of sheaves on a certain simplicial resolution constructed from $X$, satisfying certain compatibility conditions. Translated to algebra, it says that if $S\rightarrow R$ is a faithfully flat morphism of rings, then giving an $S$-module is the same as giving a certain simplical module over a simplicial ring constructed from $R$. In topology, given an etale cover $X\rightarrow Y$ one can recover $Y$ (at least up to homotopy equivalence) from a simplical space constructed from $X$.

How can one guarantee the presence of an oriented four-cycle in an oriented graph G? We shall see, that one way in which this can be done, is to demand that G contains no large `biased. subgraphs; where a `biased. subgraph simply means a subgraph whose orientation exhibits a strong bias in one direction.

Furthermore, we discuss the concept of biased subgraphs from another standpoint, asking: how can an oriented graph best avoid containing large biased subgraphs? Do random oriented graphs give the best examples? The talk is partially based on joint work with Omid Amini and Florian Huc.

There is a non-degenerate 2-form on S^6, which is compatible with the almost complex structure that S^6 inherits from its inclusion in the imaginary octonions. Even though this 2-form is not closed, we may still define Lagrangian submanifolds. Surprisingly, they are automatically minimal and are related to calibrated geometry. The focus of this talk will be on the Lagrangian submanifolds of S^6 which are fibered by geodesic circles over a surface. I will describe an explicit classification of these submanifolds using a family of Weierstrass formulae.