Founded more than 20 years ago, the Max Planck Institute for Mathematics in the Sciences is an important research center. The mission of the institute is to conduct cutting-edge research in pure and applied mathematics and to promote the interlinking of ideas and concepts between mathematics and the sciences in both directions. However, given our size, topics covered in the institute are limited. Yet, many fascinating developments happen in the mathematical sciences all the time.

This conference is meant to overcome this limitation and to give insights in recent developments. We aim to discuss current trends and gain an overview of today’s mathematical landscape – in particular with respect to the label “Mathematics in the Sciences”. Speakers will motivate their work and embed it into a general context. Talks will be accessible to mathematicians from a wide range of backgrounds. There will be ample time for questions and informal interactions.

Images are a rich source of beautiful mathematical formalism and analysis. Associated mathematical problems arise in functional and non-smooth analysis, the theory and numerical analysis of partial differential equations, harmonic, stochastic and statistical analysis, and optimisation. Starting with a discussion on the intrinsic structure of images and their mathematical representation, in this talk we will learn about some of these mathematical problems, about variational models for image analysis and their connection to partial differential equations and deep learning. The talk is furnished with applications to art restoration, forest conservation and cancer research.

Single particle cryo-EM is becoming an increasingly popular technique for determining 3-D molecular structures at high resolution. We will discuss the mathematical principles for reconstruction using cryo-EM and then focus on computational challenges, in particular, reconstruction of small molecules and heterogeneity analysis.

Many tasks in machine learning and signal processing can be solved by minimizing a convex function of a measure. This includes sparse spikes deconvolution or training a neural network with a single hidden layer. For these problems, we study a simple minimization method: the unknown measure is discretized into a mixture of particles and a continuous-time gradient descent is performed on their weights and positions. This is an idealization of the usual way to train neural networks with a large hidden layer. We show that, when initialized correctly and in the many-particle limit, this gradient flow, although non-convex, converges to global minimizers (Joint work with Lénaïc Chizat)

We wish to understand when a tensor s can be transformed into a tensor t by application of linear maps to its tensor legs (we then say s restricts to t). In the language of restrictions, the rank of a tensor t is given by the minimal size of a diagonal tensor restricting to t. The study of rank and restrictions are motivated by algebraic complexity theory, where the rank corresponds to the computational complexity of a bilinear map (e.g. matrix multiplication) which then is viewed as a tensor with three legs.
Interestingly, some important open problems can be formulated in terms of asymptotic properties of restriction, among them the exponent of matrix multiplication. Following the seminal work of Volker Strassen, we will therefore study whether for large n the (n+o(n))'th tensor power of s can be restricted to the n'th tensor power of t. The information-theoretic flavor of the problem is apparent and was heavily used by Strassen in conjunction with the discovery of algebraic structures (his spectral theorem).
Identifying k-leg-tensors with states of quantum systems of k particles allows us to bring tools and ideas from quantum information theory to the table, among them entanglement polytopes and quantum entropy. I will use these to construct a family of functionals - the quantum functionals - that can serve as obstructions to asymptotic restrictions. The functionals are the first of their kind applicable to all tensors, thereby solving a problem by Strassen from 1986.

In 2010 Nakanishi and the speaker introduced a "one-pass theorem" which precludes almost homoclinic orbits in certain infinite dimensional dynamical systems. More specifically, solutions to nonlinear wave equations starting from a small neighborhood of a ground state soliton and which return to another such neighborhood after some finite time. Such statements are non-perturbative and require truly global arguments based on monotone quantities. In the original setting invariant manifolds played an essential role, especially the one-dimensional unstable manifold associated with the steady-state solution. I will review these results, and discuss some of their ramifications over the past ten years. In particular, I will present a recent theorem by Jendrej and Lawrie on two soliton dynamics for energy critical equivariant wave maps into the 2-sphere (Inventiones 2018). The main tool in this classification result is a subtle one-pass type theorem based on the nonlinear inelastic collision of two dynamically rescaled harmonic maps.

A key result in classical probability and statistics is the Central Limit Theorem, asserting asymptotic normality under conditions on the underlying distribution of observations. The limit is taken as the number of observations tends to infinity, but real data are finite. Hence explicit bounds on the distance between the object of interest and the limiting distribution are required to account for the approximation error. A method which has proven useful for obtaining such explicit bounds in rather general situations, which may include complex dependence, is Stein's method. This talk will give a very brief introduction into Stein's method and then illustrate it with an explicit bounds on an approximation of exponential random graphs.

To mathematicians, the Hilbert transform is a singular integral operator that gives access to harmonic conjugate functions via a convolution of boundary values. To others, it may mean a signal transformer that produces an audible signal, sent to a radio's speaker or, it might model the trajectories of charged particles in a potential. The concept of martingales on the other hand, is often associated with old betting strategies in so-called fair games.
In part of this lecture we focus on connections between the Hilbert transform and transforms on systems of martingales.
We give an idea of various results that were obtained using this connection between analysis and probability. Applications include characterisations of certain Banach spaces, norm estimates in high dimensions and so called weighted estimates with applications to PDE and multivariate stationary stochastic processes.

Statistical physics is concerned with the understanding of the large scale behaviour of microscopically defined models. Simple representative models that have been studied starting 100 years ago, and continue to be intensively studied today, include the Ising and Heisenberg models, Percolation, Interface models, Random Schroedinger Operators, Random Polymers, Random Walks in Random Environments, and many others.
As a function of a temperature parameter (or similar), these models often undergo phase transition that connect qualitatively very different behaviour. Understanding the nature of these transitions, and often related universal behaviour that is independent of precise microscopic definitions, remain fundamental mathematical questions. After giving a short overview of the status, I will highlight the concept of renormalisation that plays a fundamental role in physics in explaining many of these questions at a heuristic level.
In the second part of the talk, I will give a concrete example of the use of renormalisation in understanding stochastic dynamics of such models, an aspect that is mathematically much less understood than its static counterpart. We show that the idea of the renormalisation group in the spirit of Polchinski connects naturally to the classical theory for Log-Sobolev inequalities of Bakry and Emery. As an application, we prove a Log-Sobolev inequality for the continuum Sine-Gordon model.
(The second part is joint work with T. Bodineau.)

Density Functional Theory (DFT) is a method used to approximate solutions to the linear Schrödinger equation which describes the behavior of $N$ quantum electrons in an atom or a molecule. It is employed everywhere in quantum chemistry and physics and it has become the method of choice to simulate matter at the microscopic scale. In this talk I will explain how to formulate DFT as a rigorous mathematical theory and will present some recent results obtained in collaboration with Elliott H. Lieb (Princeton) and Robert Seiringer (IST Austria).

Global manifolds are the backbone of a dynamical system and key to the characterisation of its behaviour. They arise in the classical sense of invariant manifolds associated with saddle-type equilibria or periodic orbits and, more recently, in the form of finite-time invariant manifolds in systems that evolve on multiple time scales. Dynamical systems theory relies heavily on the knowledge of such manifolds, because of the geometric insight that they can offer into how observed behaviour arises. Global invariant manifolds need to be computed and visualised numerically, which is a serious challenge, but an effort that pays off. Our approach is based on the continuation of solutions to a two-point boundary value problem, and has successfully been used for vector fields, as well as invertible and noninvertible maps. We will explain how global invariant manifolds can be computed to study the geometric complexity arising from classical chaotic dynamics. We will also discuss recent research on a new type of chaos that arises robustly in higher-dimensional systems. Specifically, we will present results for a 3D Henon-like map that address the discrepancy between theoretical results and their manifestations in explicit examples.

In 1872 Felix Klein introduced a new approach to view geometry as the study of symmetries. This approach has been very influential both in mathematics as well as in physics. It allows to study different geometric structures in a unified framework, and to investigate the interplay between topological manifolds and their geometric realizations in a new way.
In the past twenty years several new developments at the intersection of geometry and topology led to a revival of these ideas and paradigm shifts in the study of geometric structures.
In this talk I will highlight some of these developments. In the end I will hint at the potential of these ideas for the discovery of structure in data.

Recent progress in genomics makes it possible to perform perturbation experiments at a very large scale. This motivates the development of a causal inference framework that is based on observational and interventional data. We characterize the causal relationships that are identifiable and present the first provably consistent algorithm for learning a causal network from such data. I will then couple gene expression with the 3D genome organization. In particular, we will discuss approaches for integrating different data modalities such as sequencing or imaging via autoencoders. We end by a theoretical analysis of autoencoders linking overparameterization to memorization. In particular, we will show that overparameterized single-layer fully connected autoencoders as well as deep convolutional autoencoders memorize images, i.e., they produce outputs in the span of the training images. Collectively, this talk will highlight the symbiosis between biology and machine learning, showing how biology can lead to new theorems, which in turn can guide biological experiments.

Accurate molecular simulation requires computationally expensive quantum chemistry models that makes simulating complex material phenomena or large molecules intractable. This talk will be an overview of classical as well as "modern" data-driven multi-scale and coarse-grained models intended to overcome this barrier. I will focus in particular on analysis and numerical analysis aspects of modelling material defects:
(1) formulation and analysis of model problems; (2) approximation by multi-scale/hybrid QM/MM models; (3) approximation by data-driven interatomic potentials based on machine learning methodology.Mathematical aspects of these topics include for example the analysis of the potential energy surface (regularity, sparsity, ...), high-dimensional approximation under symmetry constraints, and new inverse problems.

Participants

Zachary Adams

Max Planck Institute for Mathematics in the Sciences

Renan Assimos Martins

Max Planck Institute for Mathematical Sciences

Francis Bach

INRIA

Roland Bauerschmidt

University of Cambridge

Peter Benner

MPI for Dynamics of Complex Technical Systems

Türkü Özlüm Çelik

Max Planck Institute for Mathematics in the Sciences

Matthias Christandl

University of Copenhagen

Peter Dayan

Max Planck Institute for Biological Cybernetics

Eliana Duarte

Max Planck Institute for Mathematics in the Sciences

Paweł Duch

Max Planck Institute for Mathematics in the Sciences

Henrik Eisenmann

Max Planck Institute for Mathematics in the Sciences

Benjamin Gess

MPI MIS Leipzig & University of Bielefeld

Paul Görlach

Max Planck Institute for Mathematics in the Sciences

Wolfgang Hackbusch

Max Planck Institute for Mathematics in the Sciences

Paul Heine

Max Planck Institute for Mathematics in the Sciences

Per Helander

MPI for Plasma Physics

Johannes Henn

Max Planck Institute for Physics

Jürgen Jost

Max Planck Institute for Mathematics in the Sciences

Michael Joswig

Technische Universität Berlin

Gülce Kardeş

Middle East Technical University

Venera Khoromskaia

Max Planck Institute for Mathematics in the Sciences

Boris Khoromskij

Max Planck Institute for Mathematics in the Sciences

Bernd Kirchheim

Leipzig University

Aleksander Klimek

Max Planck Institute for Mathematics in the Sciences

Florian Kunick

Max Planck Institute for Mathematics in the Sciences

Stefan Kunis

University Osnabrueck

Carlotta Langer

Max Planck Institute for Mathematics in the Sciences

Jörg Lehnert

Max Planck Institute for Mathematics in the Sciences

Mathieu Lewin

Université Paris-Dauphine

Stephan Luckhaus

Leipzig University

Duc Luu

Max Planck Institute for Mathematics in the Sciences

Slava Matveev

Max Planck Institute for Mathematics in the Sciences

Richard McElreath

MPI for Evolutionary Anthropology

Hermann Nicolai

MPI for Gravitational Physics

Eckehard Olbrich

Max Planck Institute for Mathematics in the Sciences

Christoph Ortner

University of Warwick

Hinke M. Osinga

University of Auckland

Felix Otto

Max Planck Institute for Mathematics in the Sciences

Joël Ouaknine

MPI for Software Systems

Albert Georg Passegger

Max Planck Institute for Mathematics in the Sciences

Stefanie Petermichl

University of Würzburg

Gesine Reinert

University of Oxford

Tobias Ried

Max Planck Institute for Mathematics in the Sciences

Artem Sapozhnikov

Leipzig University

Wilhelm Schlag

Yale University

Carola-Bibiane Schönlieb

University of Cambridge

Amit Singer

Princeton University

Eric Sonnendrücker

Max Planck Institute for Plasma Physics

Miruna-Stefana Sorea

Max Planck Institute for Mathematics in the Sciences

Peter Stadler

University of Leipzig

Bernd Sturmfels

Max Planck Institute for Mathematics in the Sciences

László Székelyhidi

University of Leipzig

Arleta Szkoła

Leipzig University

Peter Teichner

Max Planck Institute for Mathematics

Simon Telen

KU Leuven

Felix Tellander

Lund University

Andreas Thom

Technische Universität Dresden

Jürgen Tolksdorf

Max Planck Institute for Mathematics in the Sciences

Tat Dat Tran

University of Leipzig

Arne Traulsen

Max Planck Institute for Evolutionary Biology

Caroline Uhler

Massachusetts Institute of Technology

André Uschmajew

Max Planck Institute for Mathematics in the Sciences

Pau Vilimelis Aceituno

Max Planck Institute for Mathematics in the Sciences

Martin Vingron

Max Planck Institute for Molecular Genetics

Max von Renesse

Leipzig University

Anna Wienhard

University of Heidelberg

Marius Yamakou

Max Planck Institute for Mathematics in the Sciences

Lin Zhang

Max Planck Institute for Mathematics in the Sciences

Scientific Organizers

Jürgen Jost

Max Planck Institute for Mathematics in the Sciences

Felix Otto

Max Planck Institute for Mathematics in the Sciences

Bernd Sturmfels

Max Planck Institute for Mathematics in the Sciences

Administrative Contact

Valeria Hünniger

Max Planck Institute for Mathematics in the Sciences
Contact via Mail

Jörg Lehnert

Max Planck Institute for Mathematics in the Sciences
Contact via Mail