K3 surfaces have been extensively studied over the past decades for several reasons. For once, they have a rich and yet tractable geometry and they are the playground for several open arithmetic questions. Moreover, they form the only class which might admit more than one elliptic fibration with section. A natural question is to ask if one can classify such fibrations, and indeed that has been done by several authors, among them Nishiyama, Garbagnati and Salgado. The particular setting that we were interested in studying is when a K3 surface arises as a double cover of an extremal rational elliptic surface with a unique reducible fiber. This K3 surface will have a non-symplectic involution τ fixing two smooth Galois-conjugate genus 1 curves. In this joint project we determine the fields of definition of the different fibrations on this K3 surface, depending on τ. Moreover, we determine as well the field of definition of the Mordell–Weil group of each fibration. This is a joint work with A. Garbagnati, C. Salgado, A. Trbovi ́c and R. Winter.
Abstract: We will discuss basic properties of normal numbers for theCantor series expansions and a recent result of D. Airey and B. Mance.
Using ideas introduced in this paper, it maybe possible to show that information about algebraic varieties is encoded inthe structure of sets of normal numbers. We will outline this idea and thebarriers one may encounter in finishing it.
Stabilization by boundary noise: a Chafee-Infante equation with dynamical boundary conditions
The stabilization of parabolic PDEs by multiplicative noise is a well know phenomenon that has been studied extensively over the past decades. However, the stabilizing effect of a noise that acts only on the boundary of a domain had not been investigated so far.
As a first model case we consider the Chafee-Infante equation with dynamical boundary conditions and analyze whether a linear multiplicative Itô noise on the boundary can stabilize the equation. In particular, we show that there exists a finite range of noise intensities that imply the exponential stability of the zero steady state. Our results differ from previous works on stabilization, where the noise acts inside the domain, and stabilization typically occurs for an infinite range of noise intensities.
This is joint work with Klemens Fellner, Bao Q. Tang (University of Graz) and Do D. Thuan (Hanoi University of Science and Technology).
Coarse-graining is the procedure of approximating a complex system by a simpler or lower-dimensional one, often in some limiting regime. Coarse-graining limits are by nature singular limits. Therefore rigorous proofs of such limits typically hinge on exploiting certain structural features of the equations such as variational-evolution structures, which for instance are present in gradient flows.
In the first part of the talk we introduce and discuss such a variational structure arising from the theory of large deviations for stochastic processes. We show how in systems, which are characterized by a large deviation rate functional, passing to a limit is facilitated by the dual formulation of the rate functional, in a way that interacts particularly well with coarse-graining. Being closely related to classical variational methods for gradient flows, our approach is also applicable to systems with non-dissipative effects. As an example we apply the technique to derive the large friction (overdamped) limit of the Vlasov-Fokker-Planck equation.
In the second part of the talk we show how the large deviation rate functional can be used to quantify the approximation error.
The talk is based on a joint work with M. Hong Duong, Mark A. Peletier, Andre Schlichting and Upanshu Sharma.
In 1949, L. Onsager proposed a statistical theory for a system of elongated molecules interacting via repulsive short-range forces. Onsager's theory predicted the existence at intermediate densities of a nematic liquid crystal phase, in which the distribution of orientations of the particles is anisotropic,
while the distribution of the particles in space is homogeneous and does not exhibit the periodic variation of densities that characterizes solid crystals (periodicity in all space dimensions).
I will introduce a simplified model for this problem consisting of long rods (in two dimensions) and anisotropic plates (in three dimensions), characterized by purely hard core interactions and a finite number of allowed orientations.
For this model I will review some results/conjectures. This is a joint work with A. Giuliani and I. Jauslin