L3

The famous theorem of Szemerédi says that for any natural number $k$ and any $a>0$ there exists $n$ such that if $N\ge n$ then any subset $A$ of the set $[N] =\{1, 2,\ldots , N\}$ of size $|A| \ge a N$ contains an arithmetic progression of length $k$. We consider the question of when such a theorem holds in a random set. More precisely, we say that a set $X$ is $(a, k)$-Szemerédi if every subset $Y$ of $X$ that contains at least $a|X|$ elements contains an arithmetic progression of length $k$. Let $[N]_p$ be the random set formed by taking each element of $[N]$ independently with probability $p$. We prove that there is a threshold at about $p = N^{-1/(k-1)}$ where the probability that $[N]_p$ is $(a, k)$-Szemerédi changes from being almost surely 0 to almost surely 1.

There are many other similar problems within combinatorics. For example, Turán’s theorem and Ramsey’s theorem may be relativised, but until now the precise probability thresholds were not known. Our method seems to apply to all such questions, in each case giving the correct threshold. This is joint work with Tim Gowers.

A celebrated result in mathematical general relativity is the uniqueness of the Kerr(-Newman) black-holes as regular solutions to the stationary and axially-symmetric Einstein(-Maxwell) equations. The axial symmetry can be removed if one invokes Hawking's rigidity theorem. Hawking's theorem requires, however, real analyticity of the solution. A recent program of A. Ionescu and S. Klainerman seeks to remove the analyticity requirement in the vacuum case. They were able to show that any smooth extension of "Kerr data" prescribed on the horizon, satisfying the Einstein vacuum equations, must be Kerr, using a characterization of Kerr metric due to M. Mars. In this talk I will give a characterization for the Kerr-Newman metric, and extend the rigidity result to cover the electrovacuum case.

I will introduce the theory of cluster categories after Amiot and Plamondon. For a quiver with a potential, the cluster category is defined as the quotient of the category of perfect dg-modules by the category of dg-modules with finite dimensional cohomologies. We can show the existence of the equivalence in the first talk as an application of the cluster category. I will also propose a definition of a counting invariant for each element in the cluster category.

The Green-Griffiths conjecture from 1979 says that every projective algebraic variety $X$ of general type contains a certain proper algebraic subvariety $Y$ such that all nonconstant entire holomorphic curves in $X$ must lie inside $Y$. In this talk we explain that for projective hypersurfaces of degree $d>dim(X)^6$ this is the consequence of a positivity conjecture in global singularity theory.

The subject of this talk is the structure of the space of homomorphisms from a free abelian group to a Lie group G as well as quotients spaces given by the associated space of representations.

These spaces of representations admit the structure of a simplicial space at the heart of the work here.

Features of geometric realizations will be developed.

What is the fundamental group or the first homology group of the associated space in case G is a finite, discrete group ?

This deceptively elementary question as well as more global information given in this talk is based on joint work with A. Adem, E. Torres, and J. Gomez.