Past

The transportation problem (TP) is a classic problem in operations research. The problem was posed for the first time by Hitchcock in 1941 [Hitchcock, 1941] and independently by Koopmans in 1947 [Koopmans, 1948], and appears in any standard introductory course on operations research.

The mxn TP has m supply points and n demand points. Each supply Point i holds a quantity r_i, and each demand point j wants a quantity c_j, with the sum of femands equal to the sum of supplies. A solution to the problem can be written as a mxn matrix X with entries decision x_{ij} having value equal to the amount transported from supply point i to demand point j. The objective is to minimize total transportation costs when unit transporation costs between each supply and each demand point are given.

The set of feasible solutions of TP, is called the transportation polytope.

The 1-skeleton (edge graph) of this polytope is defined as the graph with vertices the vertices of the polytope and edges its 1-dimensional faces.

In 1957 W.M. Hirsch stated his famous conjecture cf. [Dantzig, 1963]) saying that any d-dimensional polytope with n facets has diameter at most n-d. So far the best bound for any polytope is O(n^{\log d+1}) [Kalai and Kleitman, 1992]. Any strongly polynomial bound is still lacking. Such bounds have been proved for some special classes of polytopes (for examples, see [Schrijver, 1995]). Among those are some special classes of transportation polytopes [Balinski, 1974],[Bolker, 1972] and the polytope of the dual of TP [Balinski, 1974].

The first strongly polynomial bound on the diameter of the transportation polytope was given by Dyer and Frieze [DyerFrieze, 1994]. Actually, they prove a bound on the diameter of any polytope {x|Ax=b} where A is a totally unimodular matrix. The proof is complicated and indirect, using the probabilistic method. Moreover, the bound is huge O(m^{16}n^3ln(mn))3) assuming m less than or equal to n.

We will give a simple proof that the diameter of the transportation polytope is less than 8(m+n-2). The proof is constructive: it gives an algorithm that describes how to go from any vertex to any other vertex on the transportation polytope in less than 8(m+n-2) steps along the edges.

According to the Hirsch Conjecture the bound on the TP polytope should be

m+n-1. Thus we are within a multiplicative factor 8 of the Hirsch bound.

Recently C. Hurkens refined our analysis and diminished the bound by a factor 2, arriving at 4(m+n-2). I will indicate the way he achieved this as well.

Static vacuum data and their conformal classes play an important role in the discussion of the smoothness of gravitational fields at null infinity. We study the question under which conditions such data admit non-trivial conformal rescalings which lead again to such data. Some of the restrictions implied by this requirement are discussed and it is shown that there exists a 3-parameter family of static vacuum data which are not conformally flat and which admit non-trivial rescalings.

Many biological time series appear nonlinear or chaotic, and from biomechanical principles we can explain these empirical observations. For this reason, methods from nonlinear time series analysis have become important tools to characterise these systems. Nonetheless, a very large proportion of these signals appear to contain significant noise. This randomness cannot be explained within the assumptions of pure deterministic nonlinearity, and, as such, is often treated as a nuisance to be ignored or otherwise mitigated. However, recent work points to this noise component containing valuable information. Random dynamical systems offer a unified framework within which to understand the interplay between deterministic and stochastic dynamical sources. This talk will discuss recent attempts to exploit this synthesis of stochastic and deterministic dynamics in biological signals. It will include a case study from speech science.

I will review the construction of algebraic de Rham cohomology, relative de Rham cohomology, and the Gauss-Manin connection. I will then show how we can find a basis for the cohomology and the matrix for the connection with respect to this basis for certain families of curves sitting in weighted projective spaces.

Joint work with Martin Dyer (Leeds) and Leslie Goldberg (Liverpool).

A spin system may be modelled as a graph, in which edges (bonds) indicate interactions between adjacent vertices (sites). A configuration of the system is an assignment of colours (spins) to the vertices of the graph. The interactions between adjacent spins define a certain distribution, the Boltzmann distribution, on configurations. To sample from this distribution it is usually necessary to simulate one of a number of Markov chains on the space of all configurations. Theoretical analyses of the mixing time of these Markov chains usually assume that spins are updated at single vertices chosen uniformly at random. Actual simulations, in contrast, may make (random) updates according to a deterministic, usually highly structured pattern. We'll explore the relationships between systematic scan and random single-site updates, and also between classical uniqueness conditions from statistical physics and more recent techniques in mixing time analysis.

Both Kazhdan and Haagerup properties turn out to be related to actions

of

groups on median spaces and on spaces with measured walls.

These relationships allows to study the connection between Kazhdan

property (T) and the fixed point property

for affine actions on $L^p$ spaces, on one hand.

On the other hand, they allow to discuss conjugacy classes of subgroups

with property (T) in Mapping Class Groups. The latter result

is due to the existence of a natural structure of measured walls

on the asymptotic cone of a Mapping Class Group.

The talk is on joint work with I. Chatterji and F. Haglund

(first part), and J. Behrstock and M. Sapir (second part).

In the past the theory of rough paths has proven to be an elegant tool for deriving support theorems and large deviation principles. In this talk I will explain how this approach can be used in the analysis of stochastic flows generated by Kunita SDE's. As driving processes I will consider general Banach space valued Wiener processes

I will push Schanuel's conjecture in four directions: defining a dimension

theory (pregeometry), blurred exponential functions, exponential maps of

more general groups, and converses. The goal is to explain how Zilber's

conjecture on complex exponentiation is true at least in a "geometric"

sense, and how this can be proved without solving the difficult number

theoretic conjectures. If time permits, I will explain some connections

with diophantine geometry.

The evaluation of oscillatory integrals is often considered to be a computationally challenging problem. However, in many cases, the exact opposite is true. Oscillatory integrals are cheaper to evaluate than non-oscillatory ones, even more so in higher dimensions. The simplest strategy that illustrates the general approach is to truncate an asymptotic expansion, where available. We show that an optimal strategy leads to the construction of certain unconventional Gaussian quadrature rules, that converge at twice the rate of asymptotic expansions. We examine a range of one-dimensional and higher dimensional, singular and highly oscillatory integrals.