L3

Let $A$ be a finite set in some ambient group. We say that $A$ is a $K$-approximate group if $A$ is symmetric and if the set $A.A$ (the set of all $xy$, where $x$, $y$ lie in $A$) is covered by $K$ translates of $A$. I will illustrate this notion by example, and will go on to discuss progress on the "rough classification" of approximate groups in various settings: abelian groups, nilpotent groups and matrix groups of fixed dimension. Joint work with E. Breuillard.

The Gilbert model of a random geometric graph is the following: place points at random in a (two-dimensional) square box and join two if they are within distance $r$ of each other. For any standard graph property (e.g.  connectedness) we can ask whether the graph is likely to have this property.  If the property is monotone we can view the model as a process where we place our points and then increase $r$ until the property appears.  In this talk we consider the property that the graph has a Hamilton cycle.  It is obvious that a necessary condition for the existence of a Hamilton cycle is that the graph be 2-connected. We prove that, for asymptotically almost all collections of points, this is a sufficient condition: that is, the smallest $r$ for which the graph has a Hamilton cycle is exactly the smallest $r$ for which the graph is 2-connected.  This work is joint work with Jozsef Balogh and B\'ela Bollob\'as

For graphs $L_1,\dots,L_k$, the Ramsey number $R(L_1,\ldots,L_k)$ is the minimum integer $N$ such that for any edge-colouring of the complete graph $K_N$ by $k$ colours there exists a colour $i$ for which the corresponding colour class contains $L_i$ as a subgraph.

In this talk, we shall discuss recent developments in the case when the graphs $L_1,\dots,L_k$ are all cycles and $k\ge2$.

I will discuss recent joint work with A. Ionescu and S.
Klainerman on the black hole uniqueness problem. A classical result of
Hawking (building on earlier work of Carter and Robinson) asserts that any
vacuum, stationary black hole exterior region must be isometric to the
Kerr exterior, under the restrictive assumption that the space-time metric
should be analytic in the entire exterior region.
We prove that Hawking's theorem remains valid without the assumption of
analyticity, for black hole exteriors which are apriori assumed to be "close"
to the Kerr exterior solution in a very precise sense. Our method of proof
relies on certain geometric Carleman-type estimates for the wave operator.

A polynomial $f$ is said to be a Brieskorn-Pham polynomial if

$ f = x_1^{p_1} + ... + x_n^{p_n}$

for positive integers $p_1,\ldots, p_n$. In the talk, I will discuss my joint work with Masahiro Futaki on the equivalence between triangulated category of matrix factorizations of $f$ graded with a certain abelian group $L$ and the Fukaya-Seidel category of an exact symplectic Lefschetz fibration obtained by Morsifying $f$.

We define a chain complex B_*(C, M) (the "blob complex") associated to an n-category C and an n-manifold M. This is in some sense the derived category version of a TQFT. Various special cases of the blob complex are

familiar: (a) if M = S^1, then the blob complex is homotopy equivalent to the Hochschild complex of the 1-category C; (b) for * = 0, H_0 of the blob complex is the Hilbert space of the TQFT based on C; (c) if C is a commutative polynomial ring (viewed as an n-category), then the blob complex is homotopy equivalent to singular chains on the configuration (Dold-Thom) space of M. The blob complex enjoys various nice formal properties, including a higher dimensional generalization of the Deligne conjecture for Hochschild cohomology.

If time allows I will discuss applications to contact structures on 3-manifolds and Khovanov homology for links in the boundaries of 4-manifolds. This is joint work with Scott Morrison.

I shall describe the notion of finite decomposition complexity (FDC), introduced in joint work with Romain Tessera and Guoliang Yu on the Novikov and related conjectures. The talk will focus on the definition of FDC and examples of groups having FDC.