SR1

This talk will be about the systematic simplification of differential equations.

After giving a geometric reformulation of the concept of a differential equation using prolongations, I will show how we can prolong group actions relatively easily at the level of Lie algebras. I will then discuss group-invariant solutions.

The key example will be the heat equation.

Klein's famous lecture proposes that to study geometry we study homogeneous spaces ie study transformation groups acting on a space. E. Cartan found a generalization now known as "Cartan geometries", these are a curved generalization of homogeneous spaces, eg Riemannian manifolds are Cartan geometries modeled on {Euclidean group}/{orthogonal group}.

Topics for my talk will be

Cartan geometries / Cartan connections

Parabolic geometries - a special class of Cartan geometries

Examples - depending on how much time but I will probably explain conformal

geometry as a parabolic geometry

To understand the definable sets of a theory, it is helpful to have some invariants, i.e. maps from the definable sets to somewhere else which are invariant under definable bijections. Denef and Loeser constructed a very strong such invariant for the theory of pseudo-finite fields (of characteristic zero): to each definable set, they associate a virtual motive. In this way one gets all the known cohomological invariants of varieties (like the Euler characteristic or the Hodge polynomial) for arbitrary definable sets.

I will first explain this, and then present a generalization to other fields, namely to perfect, pseudo-algebraically closed fields with pro-cyclic Galois group. To this end, we will construct maps between the set of definable sets of different such theories. (More precisely:

between the Grothendieck rings of these theories.) Moreover, I will show how, using these maps, one can extract additional information about definable sets of pseudo-finite fields (information which the map of Denef-Loeser loses).

Glauber dynamics on $\mathbb{Z}^d$ is a dynamic representation of the zero-temperature Ising model, in which the spin (either $+$ or $-$) of each vertex updates, at random times, to the state of the majority of its neighbours. It has long been conjectured that the critical probability $p_c(\mathbb{Z}^d)$ for fixation (every vertex eventually in the same state) is $1/2$, but it was only recently proved (by Fontes, Schonmann and Sidoravicius) that $p_c(\mathbb{Z}^d)

A graph property is a first order property if it can be written as a logic sentence with variables ranging over the vertices of the graph.

A sequence of random graphs (G_n)_n satisfies the zero-one law if the probability that G_n satisfies P tends to either zero or one for every first order property P. This is for instance the case for G(n,p) if p is fixed. I will survey some of the most important results on the G(n,p)-model and then proceed to discuss some work in progress on other graph models.

The Lovasz Local Lemma is known to have an extension for cases where the dependency graph requirement is relaxed to negative dependency graph (Erdos-Spencer 1991). The difficulty is to find relevant negative dependency graphs that are not dependency graphs. We provide two generic constructions for negative dependency graphs, in the space of random matchings of complete and complete bipartite graphs. As application, we prove existence results for hypergraph packing and Turan type extremal problems. We strengthen the classic probabilistic proof of Erdos for the existence of graphs with large girth and chromatic number by prescribing the degree sequence, which has to satisfy some mild conditions. A more surprising application is that tight asymptotic lower bounds can be obtained for asymptotic enumeration problems using the Lovasz Local Lemma. This is joint work with Lincoln Lu.

Digital trees is a general structure to manipulate sequences of characters. We propose a novel approach to the structure of digital trees.

It shades some new light on the profile of digital trees, and provides a unified explanation of the relationships between different kinds of digital trees. The idea relies on the distinction of nodes based on their type, i.e., the set of their children. Only two types happen to matter when studying the number of nodes lying at a specified level: the nodes with a full set of children which constitutes the core, and the nodes with a single child producing spaghetti-like trees hanging down the core. We will explain the distinction and its applications on a number of examples related to data structures such as the TST of Bentley and Sedgewick.

This is joint work with Luc Devroye.

We will prove an extended Poincaré - Hopf theorem, identifying several invariants of a manifold $M$. These are its Euler characteristic $\chi(M)$, the sum $\sum_{x_i} ind_V(x_i)$ of indices at zeroes of a vector field $V$ on $M$, the self-intersection number $\Delta \cap \Delta$ of the diagonal $\Delta \subset M \times M$ and finally the integral $\int_M e(TM)$ of the Euler class of the tangent bundle.