# School of Mathematics

## Planar Convexity, Infinite Perfect Graphs and Lipschitz Continuity

Infinite continuous graphs emerge naturally in the geometric analysis of closed planar sets which cannot be presented as countable union of convex sets. The classification of such graphs leads in turn to properties of large classes of real functions - e.g. the class of Lipschitz continuous functions - and to meta-mathematical properties of sub-ideals of the meager ideal (the sigma-ideal generated by nowhere dense sets over a Polish space) which reduce to finite Ramsey-type relations between random graphs and perfect graphs.

## Configuration Spaces of Hard Discs in a Box

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

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.

## A Rigorous Renormalization Group Study of a p-Adic Quantum Field Theory

ANALYSIS/MATHEMATICAL PHYSICS SEMINAR

## Algebraic Cycles on Picarad Moduli Spaces of Abelian Varieties

Picard moduli spaces parametrize principally polarized abelian varieties with complex multiplication by the ring of integers in an imaginary-quadratic field. The loci where the abelian varieties split off an elliptic curve in a controlled way are divisors on this moduli space. We study the intersection behaviour of these divisors and prove in the non-degenerate case a relation between their intersection numbers and Fourier coefficients of the derivative at s=0 of a certain incoherent Eisenstein series for the unitary group. This is joint work with Kudla.

## Lecture 5

## On the Realization of Some Degenerate Automorphic Forms on Certain Griffiths-Schmid Varieties

GALOIS REPRESENTATIONS AND AUTOMORPHIC FORMS SEMINAR

Some automorphic forms, despite the fact they are algebraic, do not have any interpretation as cohomology classes on a Shimura variety: therefore nothing is known at present on their expected arithmetic properties. I shall explain how such forms appear to be related to more general objects (Griffiths-Schmid varieties) and discuss some related rationality questions.

## An Elementary Proof of the Restricted Invertibility Theorem

We give an elementary proof of a generalization of Bourgain and Tzafriri's Restricted Invertibility Theorem, which says roughly that any matrix with columns of unit length and bounded operator norm has a large coordinate subspace on which it is well-invertible. Our proof gives the tightest known form of this result, is constructive, and provides a deterministic polynomial time algorithm for finding the desired subspace.

## The Graph Removal Lemma

Let H be a fixed graph with h vertices. The graph removal lemma states that every graph on n vertices with o(n^{h}) copies of H can be made H-free by removing o(n^{2}) edges. We give a new proof which avoids Szemeredi's regularity lemma and gives a better bound. This approach also works to give improved bounds for the directed and multicolored analogues of the graph removal lemma. This answers questions of Alon and Gowers.