Institute
General information about the institute, such as our mission statement, organizational structure, staff directory, history, directions, etc.

See More
Research
Scientific profile with all research groups, topics, collaborations, as well as columns on research at the institute.

See More
News
News and press releases about the institute, as well as a media archive.

See More
- News Overview
- Press Releases
Events
Overview of all events around the institute, such as talks, seminars, lectures, workshops, conferences and public events.

See More
Publications
Overview of all scientific publications of the institute, as well as our preprint and software repositories.

See More
Career
Information on open positions at the institute, benefits of working with us, graduate school, and postdoctoral supervision.

See More

Events
Discrete Mathematics and Biology: the legacy of Andreas Dress

MPI für Mathematik in den Naturwissenschaften Leipzig

E1 05 (Leibniz-Saal)

conference

26/09/2024 27/09/2024

Discrete Mathematics and Biology: the legacy of Andreas Dress

This two-day workshop celebrates the scientific work of Andreas Dress who passed away in February 2024.

Andreas Dress made fundamental contributions to algebra, geometry and their applications in the life sciences. A long time Professor at the University of Bielefeld, he also served as the founding director of the Max Planck Partner Institute for Computational Biology in Shanghai. Following his early work in group theory and tilings, Dress became a leader at the interface of mathematics and evolutionary biology. He made groundbreaking contributions to the combinatorics of phylogenetic trees. This work also led to his discovery of valuated matroids, which are parametrized by a polyhedral space now known as the Dressian. A generation of scholars in matroid theory and tropical geometry was inspired by his research.

An attractive mix of speakers will showcase Dress' accomplishments, present exciting current research, and share their vision for the future in discrete mathematics and its interface with biology.

Schedule

09:45 - 10:00	Jürgen Jost (Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany) Welcome
10:00 - 10:45	Martin Vingron (MPI-MG Berlin) to be announced
10:45 - 11:30	Robert Simon (London School of Economics and Political Science) Paradoxical Decompositions and Colouring Rules A colouring rule is a way to colour the points x of a probability space according to the colours of finitely many measures preserving tranformations of x. The rule is paradoxical if the rule can be satisfied a.e. by some colourings, but by none whose inverse images are measurable with respect to any finitely additive extension for which the transformations remain measure preserving. We demonstrate paradoxical colouring rules defined via u.s.c. convex valued correspondences (if the colours b and c are acceptable by the rule than so are all convex combinations of b and c). This connects measure theoretic paradoxes to problems of optimization and shows that there is a continuous mapping from bounded group-invariant measurable functions to itself that doesn't have a fixed point (but does has a fixed point in non-measurable functions).
11:30 - 11:45	Coffee Break
11:45 - 12:30	Daniel Huson (Universität Tübingen, Germany) A galaxy of periodic tilings This talk has four parts. In the first part - topology -, we recall the classification of two dimensions manifolds and orbifolds, and introduce their Conway notation. In the second part - geometry -, we discuss two-dimensional periodic tilings and their classification. In the third part - combinatorics -, we introduce Delaney-Dress symbols and indicate how they can be used to systematically enumerate periodic tilings. Finally, in the fourth part - algorithms and software -, we present Tegula, a computer program that can be used to explore the “galaxy” of all 2.4 billion periodic tilings of the euclidean plane, sphere and hyperbolic plane, of low Dress complexity. This is joint work with Olaf Delgado and Rüdiger Zeller. TEGULA - software and information Paper about TEGULA
12:30 - 13:30	Lunch
13:30 - 14:15	Marta Panizzut (University of Tromsø) to be announced
14:15 - 15:00	Mario Kummer (TU Dresden, Germany) Spaces of Lorentzian polynomials We characterize the topology of the space of Lorentzian polynomials with a given support in terms of the local Dressian. We prove that this space can be compactified to a closed Euclidean ball whose dimension is the rank of the Tutte group, a concept introduced by Dress and Wenzel. Finally, we show that a compactification proposed by Brändén is in general not homeomorphic to a manifold with boundary. This is part of a joint work with Matt Baker, June Huh and Oliver Lorscheid.
15:00 - 15:30	Coffee Break
15:30 - 16:15	Tropical Fermat-Weber Polytropes: Ruriko Yoshida (Naval Postgraduate School, Monterey) Tropical Fermat-Weber Polytropes In this talk we focus on the geometry of tropical Fermat-Weber points in terms of the symmetric tropical metric over the tropical projective torus. It is well-known that a tropical Fermat-Weber point of a given sample is not unique and, in this talk, we show that the set of all possible Fermat-Weber points forms a polytrope, which is a classical polytope and a tropical polytope. Then, we introduce the tropical Fermat-Weber gradient and, using them, we show that the tropical Fermat-Weber polytrope is a bounded cell of a tropical hyperplane arrangement given by both min- and max-tropical hyperplanes with apices which are observations in the input data. If time permits, we discuss its application to robust neural networks against adversarial attacks on image data. This is a joint with J. Sabol, D. Barnhill, and K. Miura.
16:15 - 17:00	Kristina Wicke (New Jersey Institute of Technology, USA) From clusters and splits to biodiversity conservation Phylogenetic diversity (PD) is a prominent quantitative measure of the biodiversity of a collection of present-day species. If T is an edge-weighted phylogenetic tree with leaf set X, the PD score of a subset S of X is the sum of the weights of the edges of the minimal subtree of T connecting the species in S. PD underlies various popular evolutionary isolation metrics such as the Fair Proportion or Equal Splits index used to rank the elements in a collection of species for conservation. In the first part of this talk, I will give a general introduction to PD and PD indices as well as an overview of mathematical questions that arise in conservation phylogenetics. In the second part of the talk, I will then discuss two recent developments. First, I will focus on non-treelike evolution and present some natural approaches towards defining and optimizing variants of PD on phylogenetic networks. Second, I will discuss a framework for analyzing and comparing different PD indices, leveraging the interplay between collections of clusters and collections of splits, an approach pioneered in combinatorial phylogenetics by Andreas Dress. I will conclude by highlighting some open questions and directions for future research. Parts of this talk are based on joint work with Magnus Bordewich, Vincent Moulton, Charles Semple, and Andreas Spillner.
18:00 - 21:00	Dinner

09:00 - 09:45	Henning Krause (Universität Bielefeld) to be announced
09:45 - 10:30	Parvaneh Joharinad (MPI MiS, Leipzig) to be announced
10:30 - 11:00	Coffee Break
11:00 - 11:45	Kay Nieselt (University of Tübingen) to be announced
11:45 - 12:30	Marc Hellmuth (Stockholm University) to be announced
12:30 - 13:30	Lunch
13:30 - 14:15	Benjamin Schröter (KTH Stockholm) to be announced
14:15 - 15:00	Matt Baker (Georgia Tech, USA) Lorentzian polynomials and matroids over triangular hyperfields Oleg Viro introduced a family $\{ {\mathbb T}_p \}_{p \geq 0}$ of hyperfields which underlies the process of Maslov dequantization, with ${\mathbb T}_0$ being the tropical hyperfield. There is also a notion of "matroids with coefficients" due to Dress and Wenzel, which allows one to define the notion of ${\mathbb T}_p$-matroid for all $p$ $\geq 0$, with ${\mathbb T}_0$-matroids being the same thing as valuated matroids. We establish an intriguing connection between Lorentzian polynomials, as defined by Brändén and Huh, and ${\mathbb T}_p$-matroids. In particular, we will sketch a proof of the following theorem: For every integer $n$ $\geq 1$ and every matroid $M$ of rank $r$ on $\{ 1,\ldots n \}$, there is a $p>0$ such that ${\mathbb T}_p(M) \subseteq L_M \subseteq {\mathbb T}_2(M)$ inside ${\mathbb R \mathbb P}^{N-1}$, where $N$ = ${n \choose r}$, ${\mathbb T}_p(M)$ denotes the space of ${\mathbb T}_p$-matroids with support $M$, and $L_M$ denotes the space of Lorentzian polynomials with support $M$. Moreover, the dimensions of both $L_M$ and ${\mathbb T}_p(M)$ for $p>0$ are equal to the dimension of the Tutte group of $M$, an algebraic invariant introduced by Dress and Wenzel. We also present examples showing that the value of p in the theorem necessarily depends on $n$. A key ingredient in the proof is the connection between valuated matroids with support $U_{2,n}$ and phylogenetic trees with n leaves, which Dress made use of in both his earlier work on matroid theory and his later work on mathematical biology. This is joint work with June Huh, Mario Kummer, and Oliver Lorscheid.

Participants

Matt Baker

Georgia Tech

Marc Hellmuth

Stockholm University

Daniel Huson

Universität Tübingen

Parvaneh Joharinad

MPI MiS, Leipzig

Jürgen Jost

Max Planck Institute for Mathematics in the Sciences

Michael Joswig

Technical University Berlin

Henning Krause

Universität Bielefeld

Mario Kummer

TU Dresden

Kay Nieselt

University of Tübingen

Marta Panizzut

University of Tromsø

Benjamin Schröter

KTH Stockholm

Robert Simon

London School of Economics and Political Science

Peter Stadler

Leipzig University

Bernd Sturmfels

Max Planck Institute for Mathematics in the Sciences

Martin Vingron

MPI-MG Berlin

Kristina Wicke

New Jersey Institute of Technology

Ruriko Yoshida

Naval Postgraduate School

Scientific Organizers

Jürgen Jost

Max-Planck-Institut für Mathematik in den Naturwissenschaften

Michael Joswig

Technical University Berlin

Peter Stadler

Leipzig University

Bernd Sturmfels

Max Planck Institute for Mathematics in the Sciences

Administrative Contact

Saskia Gutzschebauch

Max Planck Institute for Mathematics in the Sciences Contact via Mail

Mirke Olschewski

Max Planck Institute for Mathematics in the Sciences Contact via Mail

Discrete Mathematics and Biology: the legacy of Andreas Dress

Links

Schedule

Participants

Matt Baker

Marc Hellmuth

Daniel Huson

Parvaneh Joharinad

Jürgen Jost

Michael Joswig

Henning Krause

Mario Kummer

Kay Nieselt

Marta Panizzut

Benjamin Schröter

Robert Simon

Peter Stadler

Bernd Sturmfels

Martin Vingron

Kristina Wicke

Ruriko Yoshida

Scientific Organizers

Jürgen Jost

Michael Joswig

Peter Stadler

Bernd Sturmfels

Administrative Contact

Saskia Gutzschebauch

Mirke Olschewski