L3

A family of graphs F on a fixed set of n vertices is said to be triangle-intersecting if for any two graphs G,H in F, the intersection of G and H contains a triangle. Simonovits and Sos conjectured that such a family has size at most (1/8)2^{\binom{n}{2}}, and that equality holds only if F
consists of all graphs containing some fixed triangle. Recently, the author, Yuval Filmus and Ehud Friedgut proved a strengthening of this conjecture, namely that if F is an odd-cycle-intersecting family of graphs, then |F| \leq (1/8) 2^{\binom{n}{2}}. Equality holds only if F consists of all graphs containing some fixed triangle. A stability result also holds: an odd-cycle-intersecting family with size close to the maximum must be close to a family of the above form. We will outline proofs of these results, which use Fourier analysis, together with an analysis of the properties of random cuts in graphs, and some results in the theory of Boolean functions. We will then discuss some related open questions.

All will be based on joint work with Yuval Filmus (University of Toronto) and Ehud Friedgut (Hebrew University of Jerusalem).

Recently, two new Calabi-Yau threefolds have been discovered which have small Hodge numbers, and give rise to three chiral generations of fermions via the so-called 'standard embedding' compactification of the heterotic string.
In this talk I will describe how to deform the standard embedding on these manifolds in order to achieve the correct gauge group.  I will also describe how to calculate the resulting spectrum and interactions, which is still work in progress.

The is the second part of the Analysis and Geometry Seminar today.

Communication between observers in a relativistic scenario has proved to be

a setting for a fruitful dialogue between quantum field theory and quantum

information theory. A state that an inertial observer in Minkowski space

perceives to be the vacuum will appear to an accelerating observer to be a

thermal bath of radiation. We study the impact of this Davies-Fulling-Unruh

noise on communication, particularly quantum communication from an inertial

sender to an accelerating observer and private communication between two

inertial observers in the presence of an accelerating eavesdropper. In both

cases, we establish compact, tractable formulas for the associated

communication capacities assuming encodings that allow a single excitation

in one of a fixed number of modes per use of the communications channel.