L3

Let $Q_n$ denote the graph of the $n$-dimensional cube with vertex set $\{0, 1\}^n$

in which two vertices are adjacent if they differ in exactly one coordinate.

Suppose $G$ is a subgraph of $Q_n$ with average degree at least $d$. How long a

path can we guarantee to find in $G$?

My aim in this talk is to show that $G$ must contain an exponentially long

path. In fact, if $G$ has minimum degree at least $d$ then $G$ must contain a path

of length $2^d − 1$. Note that this bound is tight, as shown by a $d$-dimensional

subcube of $Q^n$. I hope to give an overview of the proof of this result and to

discuss some generalisations.

Razborov's flag algebras provide a formal system

for operating with asymptotic inequalities between subgraph densities,

allowing to do extensive "book-keeping" by a computer. This novel use

of computers led to progress on many old problems of extremal

combinatorics. In some cases, finer structural information can be

derived from a flag algebra proof by by using the Removal Lemma or

graph limits. This talk will overview this approach.

We prove that if $\frac{\log^{117} n}{n} \leq p \leq 1 -

n^{-1/8}$, then asymptotically almost surely the edges of $G(n,p)$ can

be covered by $\lceil \Delta(G(n,p))/2 \rceil$ Hamilton cycles. This

is clearly best possible and improves an approximate result of Glebov,

Krivelevich and Szab\'o, which holds for $p \geq n^{-1 + \varepsilon}$.

Based on joint work with Daniela Kuhn, John Lapinskas and Deryk Osthus.

Results of Bourgain and Kindler-Safra state that if $f$ is a Boolean function on $\{0,1\}^n$, and

the Fourier transform of $f$ is highly concentrated on low frequencies, then $f$ must be close

to a ‘junta’ (a function depending upon a small number of coordinates). This phenomenon is

known as ‘Fourier stability’, and has several interesting consequences in combinatorics,

theoretical computer science and social choice theory. We will describe some of these,

before turning to the analogous question for Boolean functions on the symmetric group. Here,

genuine stability does not occur; it is replaced by a weaker phenomenon, which we call

‘quasi-stability’. We use our 'quasi-stability' result to prove an isoperimetric inequality

for $S_n$ which is sharp for sets of size $(n-t)!$, when $n$ is large. Several open questions

remain. Joint work with Yuval Filmus (University of Toronto) and Ehud Friedgut (Weizmann

Institute).

A self-avoiding walk on a lattice is a walk that never visits the same vertex twice.  Self-avoiding walks (SAW) have attracted interest for decades, first in statistical physics, where they are considered as polymer models, and then in combinatorics and in probability theory (the first mathematical contributions are probably due to John Hammersley, from Oxford, in the early sixties). However, their properties remain poorly understood in low dimension, despite the existence of remarkable conjectures.

About two years ago, Duminil-Copin and Smirnov proved an "old" and remarkable conjecture of Nienhuis (1982), according to which the number of SAWs of length n on the honeycomb (hexagonal) lattice grows like mu^n, with mu=sqrt(2 +sqrt(2)).

This beautiful result has woken up the hope to prove other simple looking conjectures involving these objects. I will thus present the proof of a younger conjecture (1995) by Batchelor and Yung, which deals with SAWs confined to a half-plane and interacting with its boundary.

(joint work with N. Beaton, J. de Gier, H. Duminil-Copin and A. Guttmann)

Several recent works by D. Tamarkin, D. Nadler, E. Zaslow make use of the microlocal theory of sheaves of M. Kashiwara and P. Schapira to obtain results in symplectic geometry. The link between sheaves on a manifold $M$ and the symplectic geometry of the cotangent bundle of $M$ is given by the microsupport of a sheaf, which is a conic co-isotropic subset of the cotangent bundle. In the above mentioned works properties of a given Lagrangian submanifold $\Lambda$ are deduced from the existence of a sheaf with microsupport $\Lambda$, which we call a quantization of $\Lambda$.

In the first talk we will see that the graph of a Hamiltonian isotopy admits a canonical quantization and we deduce a new proof of Arnold's non-displaceability conjecture.

A Lagrangian submanifold of a Calabi-Yau manifold is called positive if the real part of the holomorphic volume form restricted to it is positive. A Hamiltonian isotopy class of positive Lagrangian submanifolds admits a Riemannian metric with non-positive curvature. Its universal cover

admits a functional, with critical points special Lagrangians, that is strictly convex with respect to the metric. If time permits, I'll explain

how mirror symmetry relates the metric and functional to the infinite dimensional symplectic reduction picture of Atiyah, Bott, and Donaldson in

the context of the Kobayashi-Hitchin correspondence.