The new class of Banach spaces, so-called asymptotic lp spaces, is introduced and it is shown that every Banach space with bounded distortions contains a subspace from this class. The proof is based on an investigation of certain functions, called enveloping functions, which are intimately connected with stabilization properties of the norm. 0 Introduction During the last year several problems ...

Dave Benson, in [2], conjectured that for any finite group G and any prime p the Castelnuovo-Mumford regularity of the cohomology ring, H(G,Fp), is zero. He showed that reg(H(G,Fp)) ≥ 0 and succeeded in proving equality when the difference between the dimension and the depth is at most two. The purpose of this paper is to prove Benson’s Regularity Conjecture as a corollary of the following result.

A celebrated result of Gowers states that for every > 0 there is a graph G such that every -regular partition of G (in the sense of Szemerédi’s regularity lemma) has order given by a tower of exponents of height polynomial in 1/ . In this note we give a new proof of this result that uses a construction and proof of correctness that are significantly simpler and shorter.

In this paper, we are interested in the minimization of the second eigenvalue of the Laplacian with Dirichlet boundary conditions amongst convex plane domains with given area. The natural candidate to be the optimum was the “stadium”, convex hull of two identical tangent disks. We refute this conjecture. Nevertheless, we prove existence of a minimzer. We also study some qualitative properties o...

We present alternative proofs of density versions of some combinatorial partition theorems originally obtained by H. Furstenberg and Y. Katznelson. These proofs are based on an extremal hypergraph result which was recently independently obtained by W. T. Gowers and B. Nagle, V. Rödl, M. Schacht, J. Skokan by extending Szemerédi’s regularity lemma to hypergraphs.

We develop a multigraded variant of Castelnuovo-Mumford regularity. Motivated by toric geometry, we work with modules over a polynomial ring graded by a finitely generated abelian group. As in the standard graded case, our definition of multigraded regularity involves the vanishing of graded components of local cohomology. We establish the key properties of regularity: its connection with the m...

We prove regularity criteria for harmonic heat flow and a liquid crystals model. Mathematics Subject Classifications: 35K55, 58E20

We prove the existence of many complete graphs in almost all sufficiently dense partitions obtained by an application of Szemerédi’s Regularity Lemma. More precisely, we consider the number of complete graphs K` on ` vertices in `-partite graphs where each partition class consists of n vertices and there is an ε-regular graph onm edges between any two partition classes. We show that for all β >...