# Hausdorff-Kolloquium 2014/15

Date: October 29, 2014 - February 4, 2015

Venue: Mathematik-Zentrum, Lipschitz Lecture Hall, Endenicher Allee 60, Bonn

## Wednesday, February 4

 15:15 Hugo Duminil-Copin (Universitéde Genève) 16:45 Alan Rendall (Gutenberg Universität, Mainz): The MAPK cascade and the mathematics of signalling networks in cell biology

### Abstracts:

#### S. Jon Chapman: Asymptotics beyond all orders: the devil's invention?

"Divergent series are the invention of the devil, and it is shameful to base on them any demonstration whatsoever." --- N. H. Abel.
The lecture will introduce the concept of an asymptotic series, showing how useful divergent series can be, despite Abel's reservations. We will then discuss Stokes' phenomenon, whereby the coefficients in the series appear to change discontinuously. We will show how understanding Stokes phenomenon is the key which allows us to determine the qualitative and quantitative behaviour of the solution in many practical problems.
Examples will be drawn from the areas of surface waves on fluids, crystal growth, dislocation dynamics, localised pattern formation, and Hele-Shaw flow.

#### Hugo Duminil-Copin: The self-avoiding walk on the hexagonal lattice: from combinatorics to Conformal Field Theory

We will discuss the self-avoiding walk model on the hexagonal lattice. Starting with the combinatorial aspects of the model, and in particular the proof of a conjecture made by B. Nienhuis regarding the so-called connective constant of the hexagonal lattice, we will then explain how the scaling limit of the model is (conjecturally) described by conformally invariant objcets. More precisely, we will show that on the hexagonal lattice, the number an of self-avoiding walks of length n (starting at the origin) satisfies:

$\lim_{n^ \rightarrow ∞} a_n^\frac{1}{n} = \sqrt{2+\sqrt{2}}$

The proof uses a parafermionic observable for the self-avoiding walk, which satisfies a half of the so-called discrete Cauchy-Riemann relations. Establishing the other half of the relations (which conjecturally holds in the scaling limit) would also imply convergence of the self-avoiding walk to SLE(8=3), an object which also appears in different aspects of Conformal Field Theory.
The talk will be elementary and will not require any background. This is a joint work with S. Smirnov.

#### Yaron Ostrover: From symplectic measurements to Mahler conjecture via billiard dynamics

We shall discuss certain interrelations between symplectic geometry and the theory of mathematical billiards. In particular, we will show how certain symplectic invariants in the classical phase space can be used to obtain information on the length of the shortest periodic billiard trajectory in a smooth convex domain, and how a symplectic isoperimetric conjecture by Viterbo is related to a 70-years old open conjecture by Mahler regarding the volume product of convex sets. Based on a joint work with Shiri Artstein-Avidan and Roman Karasev.

#### Mario Pulvirenti: Scaling limits in Kinetic Theory and derivation of effective equations

In this talk I will review the various scaling limits yielding the Vlasov, Boltzmann and Landau equations for classical and quantum systems. I also discuss the present status regarding the rigorous derivation of such kinetic equations.

#### Alan Rendall: The MAPK cascade and the mathematics of signalling networks in cell biology

In order to react to their environment living cells have to transmit information between different parts of the molecular machinery they contain. This transfer takes place by means of sets of chemical reactions known as signalling networks. Often these can be modelled mathematically by systems of ordinary differential equations for the concentrations of the substances involved. It is important to know when these equations exhibit phenomena such as multiple stable steady states and periodic solutions. The lecture will illustrate the mathematical questions which arise in this context and the techniques which can be used to answer them (such as bifurcation theory and geometric singular perturbation theory) by a discussion of one rich example, the MAPK (mitogen-activated protein kinase) cascade. This system is of interest not only as a source of challenging mathematical problems but also because of the relevance of its dynamical properties to innovative cancer therapies.