A Proof of the Cameron-Ku conjecture

A family of permutations A \subset S_n is said to be intersecting if any two permutations in A agree at some point, i.e. for any \sigma, \pi \in A, there is some i such that \sigma(i)=\pi(i). Deza and Frankl showed that for such a family, |A| <= (n-1)!. Cameron and Ku showed that if equality holds then A = {\sigma \in S_{n}: \sigma(i)=j} for some i and j. They conjectured a `stability' version of this result, namely that there exists a constant c < 1 such that if A \subset S_{n} is an intersecting family of size at least c(n-1)!, then there exist i and j such that every permutation in A maps i to j (we call such a family `centred'). They also made the stronger `Hilton-Milner' type conjecture that for n \geq 6, if A \subset S_{n} is a non-centred intersecting family, then A cannot be larger than the family C = {\sigma \in S_{n}: \sigma(1)=1, \sigma(i)=i \textrm{for some} i > 2} \cup {(12)}, which has size (1-1/e+o(1))(n-1)!. We prove the stability conjecture, and also the Hilton-Milner type conjecture for n sufficiently large. Our proof makes use of the classical representation theory of S_{n}. One of our key tools will be an extremal result on cross-intersecting families of permutations, namely that for n \geq 4, if A,B \subset S_{n} are cross-intersecting, then |A||B| \leq ((n-1)!)^{2}. This was a conjecture of Leader; it was recently proved for n sufficiently large by Friedgut, Pilpel and the author.Comment: Updated version with an expanded open problems sectio

Irredundant Families of Subcubes

We consider the problem of finding the maximum possible size of a family of k-dimensional subcubes of the n-cube {0,1}^{n}, none of which is contained in the union of the others. (We call such a family `irredundant'). Aharoni and Holzman conjectured that for k > n/2, the answer is {n choose k} (which is attained by the family of all k-subcubes containing a fixed point). We give a new proof of a general upper bound of Meshulam, and we prove that for k >= n/2, any irredundant family in which all the subcubes go through either (0,0,...,0) or (1,1,...,1) has size at most {n choose k}. We then give a general lower bound, showing that Meshulam's upper bound is always tight up to a factor of at most e.Comment: 24 page

On the structure of random graphs with constant rr-balls

We continue the study of the properties of graphs in which the ball of radius $r$ around each vertex induces a graph isomorphic to the ball of radius $r$ in some fixed vertex-transitive graph $F$, for various choices of $F$ and $r$. This is a natural extension of the study of regular graphs. More precisely, if $F$ is a vertex-transitive graph and $r \in \mathbb{N}$, we say a graph $G$ is {\em $r$-locally $F$} if the ball of radius $r$ around each vertex of $G$ induces a graph isomorphic to the graph induced by the ball of radius $r$ around any vertex of $F$. We consider the following random graph model: for each $n \in \mathbb{N}$, we let $G_n = G_n(F,r)$ be a graph chosen uniformly at random from the set of all unlabelled, $n$-vertex graphs that are $r$-locally $F$. We investigate the properties possessed by the random graph $G_n$ with high probability, for various natural choices of $F$ and $r$. We prove that if $F$ is a Cayley graph of a torsion-free group of polynomial growth, and $r$ is sufficiently large depending on $F$, then the random graph $G_n = G_n(F,r)$ has largest component of order at most $n^{5/6}$ with high probability, and has at least $\exp(n^{\delta})$ automorphisms with high probability, where $\delta>0$ depends upon $F$ alone. Both properties are in stark contrast to random $d$-regular graphs, which correspond to the case where $F$ is the infinite $d$-regular tree. We also show that, under the same hypotheses, the number of unlabelled, $n$-vertex graphs that are $r$-locally $F$ grows like a stretched exponential in $n$, again in contrast with $d$-regular graphs. In the case where $F$ is the standard Cayley graph of $\mathbb{Z}^d$, we obtain a much more precise enumeration result, and more precise results on the properties of the random graph $G_n(F,r)$. Our proofs use a mixture of results and techniques from geometry, group theory and combinatorics.Comment: Minor changes. 57 page

Could a nearby supernova explosion have caused a mass extinction?

We examine the possibility that a nearby supernova explosion could have caused one or more of the mass extinctions identified by palaeontologists. We discuss the likely rate of such events in the light of the recent identification of Geminga as a supernova remnant less than 100 pc away and the discovery of a millisecond pulsar about 150 pc away, and observations of SN 1987A. The fluxes of $\gamma$ radiation and charged cosmic rays on the Earth are estimated, and their effects on the Earth's ozone layer discussed. A supernova explosion of the order of 10 pc away could be expected every few hundred million years, and could destroy the ozone layer for hundreds of years, letting in potentially lethal solar ultraviolet radiation. In addition to effects on land ecology, this could entail mass destruction of plankton and reef communities, with disastrous consequences for marine life as well. A supernova extinction should be distinguishable from a meteorite impact such as the one that presumably killed the dinosaurs.Comment: 10 pages, CERN-TH.6805/9

Geometric mechanics and Lagrangian reduction

The purpose of this thesis is two-fold: Firstly, to contribute to the tools available to geometric mechanics; secondly, to apply the geometric perspective to two particular problems. The thesis falls into three parts. The first part deals with the dynamics of charged molecular strands (CMS). The second part contributes general tools for use in geometric mechanics. The third part develops a new geometric modelling technique and applies it to image dynamics. Part I develops equations of motion for the dynamical folding of CMS (such as DNA). The CMS are modelled as flexible continuous filamentary distributions of interacting rigid charge conformations, and their dynamics are derived via a modified Hamilton-Pontryagin variational formulation. The new feature is the inclusion of nonlocal screened Coulomb interactions, or Lennard-Jones potentials between pairs of charges. The CMS equations are shown to arise from a form of Lagrangian reduction initially developed for complex fluids. Subsequently, the equations are also shown to arise from Lagrange-Poincaré reduction of a field theory. This dual interpretation of the CMS equations motivates the undertakings of Part II. In Part II, a general treatment of Lagrange-Poincaré (LP) reduction theory is undertaken. The LP equations are cast into a field theoretic context together with their associated constrained variational principle. An integrability/reconstruction condition is established that relates solutions of the original problem with those of the reduced problem. The new contribution of the LP framework is to unify the Lagrange-Poincaré field reduction with the canonical theory, which involves a single independent variable, and to extend LP field reduction to the general fibre bundle setting. The Kelvin-Noether theorem is generalised in two new ways; from the Euler- Poincaré to the LP setting, and from the canonical to the field setting. The importance of the extended Kelvin-Noether theorem is elucidated by an application to the CMS problem, yielding new qualitative insight into molecular strand dynamics. Finally, Part III gives a full geometric development of a new technique called un-reduction, that uses the canonical LP reduction back-to-front. Application of un-reduction leads to new developments in image dynamics

Thermal conductivity and thermal expansion of graphite fiber/copper matrix composites

The high specific conductivity of graphite fiber/copper matrix (Gr/Cu) composites offers great potential for high heat flux structures operating at elevated temperatures. To determine the feasibility of applying Gr/Cu composites to high heat flux structures, composite plates were fabricated using unidirectional and cross-plied pitch-based P100 graphite fibers in a pure copper matrix. Thermal conductivity of the composites was measured from room temperature to 1073 K, and thermal expansion was measured from room temperature to 1050 K. The longitudinal thermal conductivity, parallel to the fiber direction, was comparable to pure copper. The transverse thermal conductivity, normal to the fiber direction, was less than that of pure copper and decreased with increasing fiber content. The longitudinal thermal expansion decreased with increasing fiber content. The transverse thermal expansion was greater than pure copper and nearly independent of fiber content