L3

Phylogenetics is the reconstruction and analysis of 'evolutionary'

trees and graphs in biology (and related areas of classification, such as linguistics). Discrete mathematics plays an important role in the underlying theory. We will describe some of the ways in which concepts from combinatorics (e.g. poset theory, greedoids, cyclic permutations, Menger's theorem, closure operators, chordal graphs) play a central role. As well as providing an overview, we also describe some recent and new results, and outline some open problems.

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.

The Rado Graph R is a graph on a countably infinite number of vertices that can be characterized by the following property: for every pair of finite, disjoint subsets of vertices X and Y, there exists a vertex that is adjacent to every vertex in X and none in Y. It is often called the Random Graph for the following reason: for any 0

One of the main tools in studying sparse random graphs with independence between different edges is local comparison with branching processes. Recently, this method has been used to determine the asymptotic behaviour of the diameter (largest graph distance between two points that are in the same component) of various sparse random graph models, giving results for $G(n,c/n)$ as special cases. Nick Wormald and I have applied this method to $G(n,c/n)$ itself, obtaining a much stronger result, with a best-possible error term. We also obtain results as $c$ varies with $n$, including results almost all the way down to the phase transition.

Let $K \subset {\mathbb R}^d$ be a convex set. Choose $n$ random points in $K$, and denote by $P_n$ their convex hull. We call $P_n$ a random polytope. Investigations concerning the expected value of functionals of $P_n$, like volume, surface area, and number of vertices, started in 1864 with a problem raised by Sylvester and now are a classical part of stochastic and convex geometry. The last years have seen several new developments about distributional aspects of functionals of random polytopes. In this talk we concentrate on these recent results such as central limit theorems and tail inequalities, as the number of random points tends to infinity.

A central task in conservation biology is measuring, predicting, and preserving biological diversity as species face extinction. Dating back to 1992, phylogenetic diversity is a prominent notion for measuring the biodiversity of a collection of species. This talk gives a flavour of some the combinatorial and algorithmic problems and recent solutions associated with computing this measure. This is joint work with Magnus Bordewich (Durham University, UK) and Andreas Spillner (University of East Anglia, UK).

We show that the diameter of G(n,p) is concentrated on one of three values provided the average degree p(n-1) goes to infity with n. This is joint work with N. Fountoulakis even though he refuses to admit it.

On parabolic and elliptic equations with VMO coefficients.

Abstract: An L_p-theory of divergence and non-divergence form elliptic and parabolic equations is presented.

The main coefficients are supposed to belong to the class VMO_x, which, in particular, contains all functions independent of x.

Weak uniqueness of the martingale problem associated with such equations is obtained