—  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.

Markus Tempelmayr — A diagram-free approach to the stochastic estimates in regularity structures (06.05.2022)

We explore the version of Hairer's regularity structures based on a greedier index set than trees, as introduced by Otto, Sauer, Smith and Weber. More precisely, we construct and stochastically estimate the renormalized model, avoiding the use of Feynman diagrams but still in a fully automated, i.e., inductive way. This is carried out for a class of quasi-linear parabolic PDEs driven by noise in the full singular but renormalizable range.
We assume a spectral gap inequality on the (not necessarily Gaussian) noise ensemble. The resulting control on the variance of the model naturally complements its vanishing expectation arising from the BPHZ-choice of renormalization. We capture the gain in regularity on the level of the Malliavin derivative of the model by describing it as a modeled distribution. Symmetry is an important guiding principle and built-in on the level of the renormalization Ansatz. The approach is analytic and top-down rather than combinatorial and bottom-up. This is joint work with Pablo Linares, Felix Otto, and Pavlos Tsatsoulis.

Corresponding paper: "A diagram-free approach to the stochastic estimates in regularity structures"

Claudia Fevola — KP Solitons from Tropical Limits (25.08.2021)

In this talk, we study solutions to the Kadomtsev-Petviashvili equation whose underlying algebraic curves undergo tropical degenerations. Riemann’s theta function becomes a finite exponential sum that is supported on a Delaunay polytope. We introduce the Hirota variety which parametrizes all tau functions arising from such a sum. After introducing solitons solutions, we compute tau functions from points on the Sato Grassmannian that represent Riemann-Roch spaces. This is joint work with Daniele Agostini, Yelena Mandelshtam and Bernd Sturmfels.

Pablo Linares & Markus Tempelmayr — A tree-free construction of the structure group (25.08.2021)

We present a new approach to regularity structures, and in particular to the construction of the structure group, which replaces the tree-based framework of Hairer by a more Lie-geometric setting. We consider the space of pairs (a,p), where a is a placeholder for the nonlinearity and p is a polynomial which locally parameterizes the manifold of solutions to a given (stochastic) PDE, together with natural actions of shift by space-time vectors and tilt by space-time polynomials. Moreover, we provide a coordinate representation for the algebra of functions of (a,p) in terms of multi-indices, and lift the actions of shift and tilt to endomorphisms of this algebra. The infinitesimal generators of these actions form a Lie algebra, from which we trace a completely algebraic path towards the structure group via the universal enveloping algebra. We also connect to the tree-based approach in the driven ODE case (branched rough paths), showing that our multi-indices actually correspond to specific linear combinations of trees.Joint work with Felix Otto.

Corresponding paper: "The structure group for quasi-linear equations via universal enveloping algebras"

Stefan Hollands — General Relativity and Quantum Theory (03.08.2021)

In this short overview talk I outline some of my core research interests, ranging from Mathematical General Relativity, to Quantum Field- and Information Theory. Towards the end of my talk I indicate the mathematical areas that my research is connected to.

Daniele Agostini — Curves and theta functions: algebra, geometry & physics (03.08.2021)

Riemann’s theta function is a central object throughout mathematics, from algebraic geometry to number theory, and from mathematical physics to statistics and cryptography. One of my long term projects is to develop a program to study and connect the various aspects - geometric, computational, tropical and applied - of the theta function. In this talk, I present some of the results obtained in my group about theta functions of algebraic curves, and especially how they connect to the physics of water waves through the KP equation.

Guido Montúfar — Implicit Bias in Wide Neural Networks (19.07.2021)

We investigate gradient descent training of overparametrized neural networks with rectified linear units and the corresponding implicit bias in function space. For 1D mean squared error regression, the solution found by gradient descent is a function which interpolates the training data and has a small spatially weighted two norm of the second derivative relative to the initial function. The curvature penalty function is expressed in terms of the probability distribution that is utilized to initialize the network parameters, and we compute it explicitly for various common parameter initialization procedures. Based on these results, the training trajectories can be described in function space as trajectories of spatially adaptive smoothing splines with decreasing regularization strength. The results generalize to multivariate regression and different activation functions. This is joint work with Hui Jin.

Rostislav Matveev — Tropical Probability and Entropic Inequalities (19.07.2021)

Tropical Probability is a new method to address problems in Information Theory and related fields of research, that is being developed in collaboration with J. Portegies (TU/e). We consider the monoid of commutative diagrams of probability spaces and look at its asymptotic cone. The dual cone roughly correspond to the space of bounded characters on the original monoid. Its elements are entropy-like quantities.

I will briefly explain our ideas and present one of the applications of the theory outside of itself, more specifically, to the entropic (non-Shannon) inequalities.

Wilmer Leal — Exploration of the chemical space (24.06.2021)

Chemical substances and reactions underlie any activity ascribed to chemistry. Empirical information from more than two centuries of chemical exploration has been systematically recorded in Reaxys@, an electronic database that continues the tradition of the Gmelin and Beilstein handbooks. In this talk, we explore large-scale patterns emerging in the last two centuries of chemical history, in particular, on the growth of chemical knowledge, the use of reactants and the synthesis of products, which reveal both conservatism and sharp transitions in the exploration of the chemical space. Furthermore, we show how a network structure arises when sets of reactants and sets of products are connected by arcs. Our program of producing formal tools for probing the local geometry of the resulting directed hypergraph is illustrated by presenting two concepts of curvature introduced by our group, along with the local connectivity patterns that can be identified when they are used together.

Noémie Combe — How many Frobenius manifolds are there? (24.06.2021)

In this talk an overview of my recent results is presented. In a joint work with Yu. Manin (2020) we discovered that an object central to information geometry: statistical manifolds (related to exponential families) have an F-manifold structure. This algebraic structure is a more general version of Dubrovin’s Frobenius manifolds, serving in the axiomatisation of Topological Field Theory. This unexpected result enters the scene of tetralogy of Frobenius manifolds involving some of the deepest domains of the last past decades: quantum cohomology (related to Gromov—Witten invariants), Saito manifold (unfolding of singularities) and solutions to Maurer—Cartan (appearing in Barannikov—Kontsevitch’s theory).

These statistical manifolds turn out to have incredibly rich algebraic and geometrical properties. Moreover, it can be shown that classes of Frobenius manifolds have deep connections. Recently I proved the existence of statistical Gromov--Witten invariants for statistical manifolds.

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.

06.05.2022, 11:05