Cambridge

We say that a hypergraph is \emph{stable} if each sufficiently large subset of its vertices either spans many hyperedges or is very structured. Hypergraphs that arise naturally in many classical settings posses the above property. For example, the famous stability theorem of Erdos and Simonovits and the triangle removal lemma of Ruzsa and Szemeredi imply that the hypergraph on the vertex set $E(K_n)$ whose hyperedges are the edge sets of all triangles in $K_n$ is stable. In the talk, we will present the following general theorem: If $(H_n)_n$ is a sequence of stable hypergraphs satisfying certain technical conditions, then a typical (i.e., uniform random) $m$-element independent set of $H_n$ is very structured, provided that $m$ is sufficiently large. The above abstract theorem has many interesting corollaries, some of which we will discuss. Among other things, it implies sharp bounds on the number of sum-free sets in a large class of finite Abelian groups and gives an alternate proof of Szemeredi’s theorem on arithmetic progressions in random subsets of integers.

Joint work with Noga Alon, Jozsef Balogh, and Robert Morris.

There exist two constructions of families of exotic monotone Lagrangian tori in complex projective spaces and products of spheres, namely the one by Chekanov and Schlenk, and the one via the Lagrangian circle bundle construction of Biran. It was conjectured that these constructions give Hamiltonian isotopic tori. I will explain why this conjecture is true in the complex projective plane and the product of two two-dimensional spheres.

I will discuss new rigidity and rationality phenomena

(related to the phenomenon of Arnold tongues) in the theory of

nonabelian group actions on the circle. I will introduce tools that

can translate questions about the existence of actions with prescribed

dynamics, into finite combinatorial questions that can be answered

effectively. There are connections with the theory of Diophantine

approximation, and with the bounded cohomology of free groups. A

special case of this theory gives a very short new proof of Naimi’s

theorem (i.e. the conjecture of Jankins-Neumann) which was the last

step in the classification of taut foliations of Seifert fibered

spaces. This is joint work with Alden Walker.

Khovanov homology is an invariant of knots in S^3 which categorifies the Jones polynomial. Let C be a singular plane curve. I'll describe some conjectures relating the geometry of the Hilbert scheme of points on C to a variant of Khovanov homology which categorifies the HOMFLY-PT polynomial. These conjectures suggest a relation between HOMFLY-PT homology of torus knots and the representation theory of the rational Cherednik algebra. As a consequence, we get some easily testable predictions about the Khovanov homology of torus knots.

The Market Selection Hypothesis is a principle which (informally) proposes that `less knowledgeable' agents are eventually eliminated from the market. This elimination may take the form of starvation (the proportion of output consumed drops to zero), or may take the form of going broke (the proportion of asset held drops to zero), and these are not the same thing. Starvation may result from several causes, diverse beliefs being only one.We firstly identify and exclude these other possible causes, and then

prove that starvation is equivalent to inferior belief, under suitable technical conditions. On the other hand, going broke cannot be characterized solely in terms of beliefs, as we show. We next present a remarkable example with two agents with different beliefs, in which one agent starves yet amasses all the capital, and the other goes broke yet consumes all the output -- the hungry miser and the happy bankrupt.

This example also serves to show that although an agent may starve, he may have long-term impact on the prices. This relates to the notion of price impact introduced by Kogan et al (2009), which we correct in the final section, and then use to characterize situations where asymptotically equivalent

pricing holds.

In joint work with Tim Perutz, we give a complete characterization of the Fukaya category of the punctured torus, denoted by $Fuk(T_0)$. This, in particular, means that one can write down an explicit minimal model for $Fuk(T_0)$ in the form of an A-infinity algebra, denoted by A, and classify A-infinity structures on the relevant algebra. A result that we will discuss is that no associative algebra is quasi-equivalent to the model A of the Fukaya category of the punctured torus, i.e., A is non-formal. $Fuk(T_0)$ will be connected to many topics of interest: 1) It is the boundary category that we associate to a 3-manifold with torus boundary in our extension of Heegaard Floer theory to manifolds with boundary, 2) It is quasi-equivalent to the category of perfect complexes on an irreducible rational curve with a double point, an instance of homological mirror symmetry.

I will introduce the notion of a nilsequence, which is a kind of

"higher" analogue of the exponentials used in classical Fourier analysis. I

will summarise the current state of our understanding of these objects. Then

I will discuss a variety of applications: to solving linear equations in

primes (joint with T. Tao), to a version of Waring's problem for so-called

generalised polynomials (joint with V. Neale and Trevor Wooley) and to

solving certain pairs of diagonal quadratic equations in eight variables

(joint work with L. Matthiesen). Some of the work to be described is a

little preliminary!