L3

A dimer model on a surface with punctures is an embedded quiver such that its vertices correspond to the punctures and the arrows circle round the faces they cut out. To any dimer model Q we can associate two categories: A wrapped Fukaya category F(Q), and a category of matrix factorizations M(Q). In both categories the objects are arrows, which are interpreted as Lagrangian subvarieties in F(Q) and will give us certain matrix factorizations of a potential on the Jacobi algebra of the dimer in M(Q).

We show that there is a duality D on the set of all dimers such that for consistent dimers the category of matrix factorizations M(Q) is isomorphic to the Fukaya category of its dual,  F((DQ)). We also discuss the connection with classical mirror symmetry.

We shall discuss how the algebra norm can be used to identify structure in groups. No prior familiarity with the area will be assumed.

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.

An embedding of a graph H in a graph G is an injective mapping of the vertices of H to the vertices of G such that edges of H are mapped to edges of G. Embedding problems have been extensively studied. A very powerful tool in this area is Szemeredi's Regularity Temma. It approximates the host graph G by a quasirandom graph which inherits many of the properties of G. Unfortunately the direct use of Szemeredi's Regularity Lemma is useless if the host graph G is sparse.

During the talk I shall expose a technique to deal with embedding trees in sparse graphs. This technique has been developed by Ajtai, Komlos,Simonovits and Szemeredi to solve the Erdos-Sos conjecture. Presently the author together with Hladky, Komlos, Simonovits, Stein and Szemeredi apply this method to solve the related conjecture of Loebl, Komlos and Sos (approximate version).

Symplectic implosion is a construction in symplectic geometry due to Guillemin, Jeffrey and Sjamaar, which is related to geometric invariant theory for non-reductive group actions in algebraic geometry. This talk (based on joint work in progress with Andrew Dancer and Andrew Swann) is concerned with an analogous construction in hyperkahler geometry.

Automorphisms of right-angled Artin groups interpolate between $Out(F_n)$ and $GL_n(\mathbb{Z})$. An active area of current research is to extend properties that hold for both the above groups to $Out(A_\Gamma)$ for a general RAAG. After a short survey on the state of the art, we will describe our recent contribution to this program: a study of how higher-rank lattices can act on RAAGs that builds on the work of Margulis in the free abelian case, and of Bridson and the author in the free group case.

In this talk I'll explain how to build CAT(0) cube complexes and construct Lipschitz maps between them. The existence of suitable Lipschitz maps is used to prove that the asymptotic dimension of a

CAT(0) cube complex is no more than its dimension.

Hex was discovered independently by Piet Hein in Copenhagen in 1942 and byJohn Nash in Princeton in 1948.  The game is interesting because its rules are very simple, yet it is not known how to play best possible.  For example, a winning first move for the first player (who does have  a winning strategy) is still unknown. The talk will tell the history of the game and give simple proofs for basic results about it. Also the reverse game of HEX,sometimes called REX, will be discussed. New results about REX are under publication in Discrete Mathematics in a paper:  How to play Reverse Hex (joint work with Ryan Hayward and Phillip Henderson).