Jan Bohr (University of Bonn): Entropy: Geometric optics, scattering and rigidity 

 In geometric optics, light propagation is described by paths that (locally) minimise travel time. In a medium with variable refractive index these optical paths can bend, loop around a point, and create other interesting patterns. This leads to a number of interesting questions (can we build an invisible lens? which scattering patterns can be created by a lens?) that can be brought into a precise mathematical form and are subject to current research. In the talk I'll motivate these questions and discuss how Riemannian geometry, complex analysis and twistor spaces enter the picture. 

 Iulia Cristian (University of Bonn): On coagulation, gel formation, and rain models 

We explore a model for blood coagulation and polymerization. We analyze its properties to reveal phenomena like gel formation. We then show how we can modify the model to describe other fun things such as the onset of rain. 

Alexandros Eskenazis (CNRS, Sorbonne Université): Learning low-degree functions on the discrete hypercube

Let f be an unknown function on the n-dimensional discrete hypercube. How many values of f do we need in order to approximately reconstruct the function? In this talk we shall discuss the random query model for this fundamental problem from computational learning theory. We will explain a recently discovered connection with a family of polynomial inequalities going back to Littlewood (1930) which will in turn allow us to derive sharper estimates for the the query complexity of this model, exponentially improving those which follow from the classical Low-Degree Algorithm of Linial, Mansour and Nisan (1989), while maintaining a running time of the same order. Time permitting, we will also show a matching information-theoretic lower bound and extensions beyond the discrete hypercube. Based on joint works with Paata Ivanisvili and Lauritz Streck.

Volker Mehrmann (TU Berlin): Mathematical modeling, simulation and control of open physical systems via port-Hamiltonian systems

Most real world dynamical systems consist of subsystems from different physical domains, modelled by partial-differential equations, ordinary differential equations, and algebraic equations, combined with input and output connections. To simulate and control such complex systems, in recent years the class of dissipative port-Hamiltonian (pH) descriptor systems has emerged as a very powerful mathematical modeling paradigm. The main reasons are that the network based interconnection of pH systems is again pH, Galerkin projection in PDE discretization and model reduction preserve the pH structure. Furthermore, the physical properties are encoded in the geometric properties of the flow as well as the algebraic properties of the equations. Dissipative pH systems form a very robust representation under structured perturbations and directly indicate Lyapunov functions for stability and passivity analysis. Another advantage of energy based modeling via pH systems is that each separate model of a physical system can be a whole model catalog from which models can be chosen in an adaptive way within simulation and optimization methods. We discuss the model class of constrained pH systems and its mathematical properties. We illustrate the results with some real world examples from gas transport and district heating systems and point out emerging mathematical challenges.

Johannes Linn (MPIM): Bounding Exponential Sums 

Exponential Sums or Character Sums appear at many places in mathematics and estimating them is often an essential step in solving related problems. In this talk, we want to give an intuition for the meaning of the size of exponential sums and look at some examples of sums, their origin, and techniques to bound them. 

Ruoyuan Liu (MI): Entropy: Role of invariant Gibbs measures in Hamiltonian PDEs 

Gibbs measures, rooted in statistical mechanics, provide a probabilistic perspective in the study of Hamiltonian partial differential equations (PDEs). As opposed to individual trajectories, invariant Gibbs measures inform us of the typical behavior of solutions to an Hamiltonian PDEs. In this talk, I will introduce invariant Gibbs measures for Hamiltonian systems, from the finite dimensional setting via ordinary differential equations to the more complicated infinite dimensional PDEs. In particular, I will mention Bourgain's invariant measure argument in constructing a solution for the nonlinear Schrödinger equation. 

Jonas Lührmann (Texas A&M University, USA): A Tale of Two Solitons

Solitons are special solutions to dispersive evolution equations that preserve their shape as time goes by. I will begin with a historical introduction to the study of soliton dynamics. In the main part of this talk,I will then discuss some recent developments in the study of the asymptotic stability problem for kinks. These are simple examples of topological solitons that arise in scalar field theories on the line.

Matthew Kwan (Institute for Science and Technology Austria): The Quadratic Littlewood-Offord problem

Consider a quadratic polynomial Q(ξ1, . . . , ξn) of a random binary sequence ξ1, . . . , ξn. To what extent can Q(ξ1, . . . , ξn) concentrate on a single value? This is a quadratic version of the classical Littlewood-Offord problem, popularised by Costello, Tao and Vu in their study of symmetric random matrices. In this talk we will introduce the problem, discuss some new bounds and some connections to analytic number theory. Featuring joint work with Zhihan Jin, Lisa Sauermann and Yiting Wang.

Annika Tarnowsky (MPIM): Computing Differentiable Stack Cohomology 

Higher Differential Geometry is the intersection point between (Higher) Category Theory and Differential Geometry. The field concerns itself with concepts such as that of a Lie groupoid, which is a special type of category where objects and morphisms assemble into manifolds. A Differentiable Stack is a notion that has arisen more recently and is elusive at first glance, but it turns out to be closely related to well-known objects - in particular Lie groupoids - which can be employed for their study. Using this relation, Differentiable Stack Cohomology can be interpreted as a generalisation of Equivariant Cohomology, for which there are models that significantly simplify its computation. This poses the question if a similar method can be applied to Differentiable Stack Cohomology in general. In this talk, I will survey the relevant notions mentioned here and the relations between them as well as the research progress on the given problem including recent advances during my PhD project.