—  Putting current research center stage

Our institute represents a great variety of research topics that affect current developments both within the natural sciences as well as social and economic life. In our column Research Spotlight MiS scientists are presenting key ideas and giving insights into recent developments of their research in a short video format.

Pavlos Tsatsoulis — Singular SPDEs and Renormalisation (17.06.2021)

In this short video I will discuss topics on Singular Stochastic Partial Differential Equations (SPDEs) with interesting connections to different fields of mathematics. More precisely, I will focus on problems related to the Calculus of Variations, Numerical Analysis, and Algebra, which either arise from the study of specific models or appear in the development of a general solution theory to Singular SPDEs.

Simon Telen — Likelihood Equations and Scattering Amplitudes (17.06.2021)

We identify the scattering equations from particle physics as the likelihood equations for a particular statistical model. The scattering potential plays the role of the log-likelihood function. We employ recent methods from numerical nonlinear algebra to solve challenging instances of the scattering equations. We revisit the theory of stringy canonical forms proposed by Arkani-Hamed, He and Lam, introducing positive statistical models and their amplitudes. This is joint work with Bernd Sturmfels.

Tobias Ried — Optimal Transportation, Monge–Ampère, and the Matching Problem (10.06.2021)

We present a fully variational approach to the regularity theory for the Monge-Ampère equation, or rather of optimal transportation, with interesting applications to the problem of optimally matching a realisation of a Poisson point process to the Lebesgue measure. Following De Giorgi’s strategy for the regularity theory of minimal surfaces, it is based on the approximation of the displacement by a harmonic gradient, and leads to a quantitative linearisation result for the Monge-Ampère equation. One of the benefits of our approach is that it also works for irregular data, in particular in situations where Caffarelli’s celebrated regularity theory is not expected to work.

Mima Stanojkovski — Groups from determinantal curves (10.06.2021)

Groups are fundamental entities in mathematics and in the sciences, which, when viewed as symmetries of objects, can help understand better or tell the objects in question apart. As most groups are p-groups, we are motivated to understand structure and symmetries of p-groups, even though a complete classification seems far out of reach (unlike for the case of simple groups). I will present joint work with Christopher Voll, in which we study p-groups coming from determinantal representations of curves.

I will explain how to construct, analyze, and distinguish these groups using tools coming from (algebraic and arithmetic) geometry, number theory, and algebra.

Michael Joswig — What is Mathematical Software (03.06.2021)

What Is Mathematical Software?

A short answer to this question is: Mathematical Software is what mathematics receives as a benefit from the digital age. This is relevant because Mathematical Software is useful in many ways.

For instance, Mathematical Software serves as a tool to support traditional methods of mathematical research; this includes routine computations and the construction of explicit examples. Secondly, via massive computations, Mathematical Software can liberate mathematics from being restricted to proofs that are very short. Finally, Mathematical Software represents a large body of mathematical knowledge; in this way mathematical results become accessible to non-experts in other fields of science.

Benjamin Gess — Fluctuations in non-equilibrium and stochastic PDE (03.06.2021)

Macroscopic fluctuation theory provides a general framework for far from equilibrium thermodynamics, based on a fundamental formula for large fluctuations around (local) equilibria. This fundamental postulate can be informally justified from the framework of fluctuating hydrodynamics, linking far from equilibrium behavior to zero-noise large deviations in conservative, stochastic PDE. In this talk, we will give rigorous justification to this relation in the special case of the zero range process. More precisely, we show that the rate function describing its large fluctuations is identical to the rate function appearing in zero noise large deviations to conservative stochastic PDE. The proof is based on the well-posedness of the skeleton equation -- a degenerate parabolic-hyperbolic PDE with irregular coefficients, the proof of which extends DiPerna-Lions' concept of renormalized solutions to nonlinear diffusions.

Raffaella Mulas — Spectral theory of hypergraphs (28.05.2021)

Hypergraphs are a generalization of graphs in which vertices are joined by edges of any size. In this talk, we generalize the graph normalized Laplace operators to the case of hypergraphs, and we discuss some properties of their spectra. We discuss the geometrical meaning of the largest and smallest eigenvalues, and we show how the Cheeger inequalities can be generalized to the case of uniform hypergraphs. We also discuss some relations between the eigenvalues and constants such as the coloring number and the independence number of the hypergraph. We talk about hypergraph symmetries, and we discuss spectral measures and spectral classes.

Matteo Smerlak — Aspects of evolutionary dynamics from viruses to whales (28.05.2021)

Our interdisciplinary group studies ecological and evolutionary dynamics across scales. In this video I present a selection of recent results that illustrate the role of mathematics in furthering our understanding of biological evolution:
(1) a formalization of the concept of “selection” which highlights its analogy with extreme value statistics;
(2) a mapping between positive linear systems and Markov processes on graphs, with applications to the prediction problem in viral evolution;
(3) stability results for Lotka-Volterra type systems rooted in newly discovered macro-ecological scaling relations.

17.06.2021, 14:29