In this talk I will discuss well-posedness of hyperbolic Cauchy problems with multiplicities and the role played by the lower order terms (Levi conditions). I will present results obtained in collaboration with Christian Jäh (Göttingen) and Michael Ruzhansky (QMUL/Ghent) on higher order equations and non-diagonalisable systems.
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.
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).