# School of Mathematics

## Configuration Spaces of Hard Discs in a Box

Matthew Kahle
November 15, 2010

The "hard discs" model of matter has been studied intensely in statistical mechanics and theoretical chemistry for decades. From computer simulations it appears that there is a solid--liquid phase transition once the relative area of the discs is about 0.71, but little seems known mathematically. Indeed, Gian-Carlo Rota suggested that if we knew the total measure of the underlying configuration space, "we would know, for example, why water boils at 100 degrees on the basis of purely atomic calculations."

## Fractional Perfect Matchings in Hypergraphs

Andrzej Rucinski
Adam Mickiewicz University in Polznan, Poland; Emory University
November 15, 2010

A perfect matching in a k-uniform hypergraph H = (V, E) on n vertices
is a set of n/k disjoint edges of H, while a fractional perfect matching
in H is a function w : E → [0, 1] such that for each v ∈ V we have
e∋v w(e) = 1. Given n ≥ 3 and 3 ≤ k ≤ n, let m be the smallest
integer such that whenever the minimum vertex degree in H satisfies
δ(H) ≥ m then H contains a perfect matching, and let m∗ be defined
analogously with respect to fractional perfect matchings. Clearly, m∗ ≤
m.

## Lecture 5

Pierre Colmez
National Center for Scientific Research; Member, School of Mathematics
November 11, 2010