Bristol

I'll discuss questions about the structure of long sums of

Dirichlet characters --- that is, sums of length comparable to the modulus.

For example: How often do character sums get large? Where do character sums

get large? What do character sums "look like" when then get large? This will

include some combination of theorems and experimental data.

We consider a random geometric graph model relevant to wireless mesh networks. Nodes are placed uniformly in a domain, and pairwise connections

are made independently with probability a specified function of the distance between the pair of nodes, and in a more general anisotropic model, their orientations. The probability that the network is (k-)connected is estimated as a function of density using a cluster expansion approach. This leads to an understanding of the crucial roles of

local boundary effects and of the tail of the pairwise connection function, in contrast to lower density percolation phenomena.

Solutions to translation invariant linear forms in dense sets  (for example: k-term arithmetic progressions), have been studied extensively in additive combinatorics and number theory. Finding solutions to translation invariant quadratic forms is a natural generalization and at the same time a simple instance of the hard general problem of solving diophantine equations in unstructured sets. In this talk I will explain how to modify the  classical circle method approach to obtain quantitative results  for quadratic forms with at least 17 variables.

We discuss the problem to what extend a group action determines geometry of the space. 
More precisely, we show that for a large class of actions measurable isomorphisms must preserve 
the geometric structure as well. This is a joint work with Bader, Furman, and Weiss.

While usual percolation concerns the study of the connected components of

random subgraphs of an infinite graph, bootstrap percolation is a type of

cellular automaton, acting on the vertices of a graph which are in one of

two states: `healthy' or `infected'. For any positive integer $r$, the

$r$-neighbour bootstrap process is the following update rule for the

states of vertices: infected vertices remain infected forever and each

healthy vertex with at least $r$ infected neighbours becomes itself

infected. These updates occur simultaneously and are repeated at discrete

time intervals. Percolation is said to occur if all vertices are

eventually infected.

As it is often difficult to determine precisely which configurations of

initially infected vertices percolate, one often considers a random case,

with each vertex infected independently with a fixed probability $p$. For

an infinite graph, of interest are the values of $p$ for which the

probability of percolation is positive. I will give some of the history

of this problem for regular trees and present some new results for

bootstrap percolation on certain classes of randomly generated trees:

Galton--Watson trees.

Counter-examples to the Hasse principle are known for many families of geometrically rational varieties. We discuss how often such failures arise for Chatelet surfaces and certain higher-dimensional hypersurfaces. This is joint work with Regis de la Breteche.

The Outer Model Programme investigates L-like forcing  extensions of the universe, where we say that a model of Set Theory  is L-like if it satisfies properties of Goedel's constructible universe of sets L. I will introduce the Outer Model Programme, talk  about its history, motivations, recent results and applications. I  will be presenting joint work with Sy Friedman and Philipp Luecke.

We will discuss the Littlewood conjecture from Diophantine approximation, and recent variants of the conjecture in which one of the real components is replaced by a p-adic absolute value (or more generally a "pseudo-absolute value''). The Littlewood conjecture has a dynamical formulation in terms of orbits of the action of the diagonal subgroup on SL_3(R)/SL_3(Z). It turns out that the mixed version of the conjecture has a similar formulation in terms of homogeneous dynamics, as well as meaningful connections to several other dynamical systems. This allows us to apply tools from combinatorics and ergodic theory, as well as estimates for linear forms in logarithms, to obtain new results.