Past

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.

In the lecture, I am going to explain a connection between

local regularity theory for the Navier-Stokes equations

and Liouville type theorems for bounded ancient solutions to

these equations.

I talk about a recent article of mine that aims at giving an alternative proof to a formula by Carne on random walks. Consider a discrete, reversible random walk on a graph (not necessarily the simple walk); then one has a surprisingly simple formula bounding the probability of getting from a vertex x at time 0 to another vertex y at time t, where it appears a universal Gaussian factor essentially depending on the graph distance between x and y. While Carne proved that result in 1985, through‘miraculous’ (though very pretty!) spectral analysis reasoning, I will expose my own ‘natural' probabilistic proof of that fact. Its main interest is philosophical, but it also leads to a generalization of the original formula. The two main tools we shall use will be techniques of forward and backward martingales, and a tricky conditioning argument to prevent a random walk from being `’too transient'.

One of the outstanding successes of mathematical population genetics is Kingman's coalescent. This process provides a simple and elegant description of the genealogical trees relating individuals in a sample of neutral genes from a panmictic population, that is, one in which every individual is equally likely to mate with every other and all individuals experience the same conditions. But real populations are not like this. Spurred on by the recent flood of DNA sequence data, an enormous industry has developed that seeks to extend Kingman's coalescent to incorporate things like variable population size, natural selection and spatial and genetic structure. But a satisfactory approach to populations evolving in a spatial continuum has proved elusive. In this talk we describe the effects of some of these biologically important phenomena on the genealogical trees before describing a new approach (joint work with Nick Barton, IST Austria) to modelling the evolution of populations distributed in a spatial continuum.

Abstract: It is known that many Calabi-Yau manifolds form a connected web. The question of whether all CY manifolds form a single web depends on the degree of singularity that is permitted for the varieties that connect  the distinct families of smooth manifolds. If only conifolds are allowed then, since shrinking two-spheres and three-spheres to points cannot affect the fundamental group, manifolds with different fundamental groups will form disconnected webs. We examine these webs for the tip of the distribution of CY manifolds where the Hodge numbers $(h^{11},h^{21})$ are both small. In the tip of the distribution the quotient manifolds play an important role. We generate via conifold transitions from these quotients a number of new manifolds. These include a manifold with $\chi =-6$, that is an analogue of the $\chi=-6$ manifold found by Yau,  and manifolds with an attractive structure that may prove of interest for string phenomenology.

In 2006, a bad field was constructed (together with Baudisch, Hils and Wagner) collapsing Poizat's green fields. In this talk, we will not concentrate on the general methodology for collapsing specific structures, but more on a specific result in algebraic geometry, a weaker version of the Conjecture on Intersection with Tori (CIT). We will present a model theoretical proof of this result as well as discuss the possible generalizations to positive characteristic. We will try to make the talk  self-contained and aimed for an audience with a basic acquaintance with Model Theory.

Travelling waves are highly symmetric solutions to the Hamiltonian lattice equation and are determined by nonlinear advance-delay differential equations. They provide much insight into the microscopic dynamics and are moreover fundamental building blocks for macroscopic

lattice theories.

In this talk we concentrate on travelling waves in convex FPU chains and study both periodic waves (wave trains) and homoclinic waves (solitons). We present a new existence proof which combines variational and dynamical concepts.

In particular, we improve the known results by showing that the profile functions are unimodal and even.

Finally, we study the complete localization of wave trains and address additional complications that arise for heteroclinic waves (fronts).(joint work with Jens D.M. Rademacher, CWI Amsterdam)