Cambridge

Symplectic cohomology is an invariant of symplectic manifolds with contact type boundary. For example, for disc cotangent bundles it recovers the

homology of the free loop space. The aim of this talk is to describe algebraic operations on symplectic cohomology and to deduce applications in

symplectic topology. Applications range from describing the topology of exact Lagrangian submanifolds, to proving existence theorems about closed

Hamiltonian orbits and Reeb chords.

Suppose that $A \subset \mathbb F_2^n$ has density $\Omega(1)$. How

large a subspace is $A+A:=\{a+a’:a,a’ \in A\}$ guaranteed to contain? We

discuss this problem and how the the result changes as the density

approaches $1/2$.

Lagrangian submanifolds are an important class of objects in symplectic geometry. They arise in diverse settings: as vanishing cycles in complex algebraic geometry, as invariant sets in integrable systems, as Heegaard tori in Heegaard-Floer theory and of course as "branes" in the A-model of mirror symmetry. We ask the difficult question: when are two Lagrangian submanifolds isotopic? Restricting to the simplest case of Lagrangian spheres in rational surfaces we will give examples where this question has a complete answer. We will also give some very pictorial examples (due to Seidel) illustrating how two Lagrangians can fail to be isotopic.

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.