## Schedule

Wednesday 15th

14h Welcome coffee

14h30 - 15h15 Edouard How environmental randomness can reverse the trend

15h30 - 16h15 Stefano On stability of blow up solutions for nonlinear wave equations

16h30 - 17h15 Adrien A new tree formalism for the numerical study of the overdamped Langevin equation

*Thursday 16th*

09h15-10h Ibrahim Explicit stabilized methods for stochastic differential equations, Introduction and some recent results

10h-10h30 Coffee Break

10h30-11h15 Sebastien Localization/De-localization of a 2-dimentional Potts Model's Interface

11h30-12h15 Julia Singular curves on the plane: Blow Them Up

13h45-14h30 Christian TBA

14h45-15h30 Lucas The Bott-Samelson Theorem for positive contact isotopies

16h15-17h15 Colloquium by Daniel Kressler (EPFL) Fast algorithms from low-rank updates

19h Dinner

*Friday 17th*

09h00-09h45 Matthias The complement problem in the plane

09h45-10h15 Coffee Break

10h15-11h Jhih-Huang Star-triangle transformations applied to statistical mechanics

11h-11h30 Remise du prix Birkhäuser

11h30-12h30 Colloque Anton Alekseev Hermitian matrices and planar networks

## Abstracts

**Edouard Strickler (UNINE)***How environmental randomness can reverse the trend*

**Stefano Francesco Burzo (EPFL)***On stability of blow up solutions for nonlinear wave equations.*

Dispersive partial differential equations describe a variety of phenomena ranging from general relativity and quantum field theory to applied physics, nonlinear optics, and water waves. As a purely mathematical discipline, the field has become a cornerstone of modern PDE theory. In the last 15 years there was tremendous progress in the mathematical understanding of dispersive equations. Questions of high interest concern singularity formation in finite time. In this talk, we address the solvability of the energy critical nonlinear wave equation with a focusing nonlinearity. In this case, blow-up in finte time may occur. We show that a certain family of finite time type II blow up solutions are stable along a co-dimension one Lipschitz manifold of data perturbations in a suitable topology. This result is qualitatively optimal.

**Adrien Laurent (UNIGE)***A new tree formalism for the numerical study of the overdamped Langevin equation*

Abstract: The Langevin equation is a strong tool of statistical physics for modelling the motion of particles in a fluid. This talk aims to study numerically a simplified form of the Langevin equation called the overdamped Langevin equation. We will first present some general notions on this stochastic differential equation (SDE), and then we will introduce a new algebraic formalism, based on Butcher series, for the study of high order numerical integrators.

**Ibrahim Almuslimani (UNIGE)***Explicit stabilized methods for stochastic differential equations, Introduction and some recent results.*

We will explain briefly the idea of explicit stabilized Runge-Kutta Chebyshev methods for deterministic problems. Afterwards, we introduce some general notions on SDEs. Finally, we present a new optimal explicit stabilized integrator for stochastic differential equations with nice stability and convergence properties.

**Sebastien Ott (UNIGE)***Localization/De-localization of a 2-dimentional Potts Model's Interface.*

The q-states Potts Model on a graph G \subset Z^2 is a random colouring of G by q colours, the probability of each colouring depending on a parameter: the temperature. We will start by introducing the Potts model and its phase transition: for large G, a typical colouring will be disordered at high temperature and almost monochromatic at low temperature. Then, we will focus on the latter case when two colours are forced to coexist, creating an interface between them.

**Julia Schneider (UNIBAS)***Singular curves on the plane: Blow Them Up *

We will recall the basic notions of singular algebraic curves on the plane and introduce a tool to study them: The Blow Up. We will focus on a certain type of singularities: Type $A_k $, where k 'measures' the 'wildness' of the curve. The following question arises: If we fix a property of the curve (e.g. the degree), how wild can the curve be? In an example, we will see how to answer this using the blow up. There will be lots of pictures

**Christian Urech (UNIBAS) ***TBA*

**Lucas Dahinden (UNINE) ***The Bott-Samelson Theorem for positive contact isotopies *

The Bott-Samelson Theorem is one of the theorems that exclude simple dynamical behaviours for complicated manifolds. One formulation of the theorem is the following: If there is a point on a Riemannian manifold such that every geodesic returns to this same point at the same time, then the manifold must be the quotient of a sphere. I will state the Theorem with more precision and give an idea how to prove and generalize it.

**Mattias Hemmig (UNIBAS) ***The complement problem in the plane *

We consider algebraic curves in the plane, i.e. solution sets of polynomial equations in two variables. A basic problem is the following: Given an isomorphism between the complements of two irreducible algebraic curves, does it follow that the curves are isomorphic? This question was posed by Hanspeter Kraft in 1995. After explaining the basic notions we discuss some results on the relationship between a curve and its complement and also give a counterexample to the complement problem. This is a joint work with Jérémy Blanc and Jean-Philippe Furter.

**Jhih-Huang-Li (UNIGE) ***Star-triangle transformations applied to statistical mechanics *

We introduce the electrical network, the q-color Potts model and the random-cluster model, which are closely related models in statistical mechanics. They share a point in common: they are all preserved by star-triangle transformations. We explain how to use this fact to show interesting results on these models.

More specifically, the models are defined on (embedded) isoradial graphs, a larger family of graphs containing regular lattices such as the square, triangular and hexagonal lattices. Since star-triangle transformations allow us to transform one isoradial graph to another, properties of these models are also transported from one graph to another, which gives a universality result.

**Prof. Anton Alekseev (UNIGE)***Hermitian matrices and planar networks *