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

26.09.24 27.09.24

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 Welcome
10:00 - 10:45	Martin Vingron Association Plots and joint clustering and embedding of genes and cells Single-cell transcriptome analysis poses many problems in visualizing and analyzing data. Based on the geometric interpretation of correspondence analysis biplots we define "Association Plots" to depict genes that characterize ("are associated with") a cluster of cells. We further develop this geometric intuition into a method to build a nearest-neighbor graph that encompasses both genes and cells. Using this data structure one can apply clustering and embedding methods which then naturally show clusters of cells together with their associated genes. The principles of this analyses are not restricted to single-cell transcriptomics, but should apply to any data in the form of a contingency table.
10:45 - 11:30	Robert Simon 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 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 The Dressian, the tropical Grassmannian and their positive parts The Dressian and the tropical Grassmannian parametrize abstract and realizable tropical linear spaces, which are defined using the notion of valuated matroid introduced by Dress and Wenzel. In the last twenty years there has been an extensive study of their beautiful geometric and combinatorial structures, with a recent focus on their positive parts motivated by problems in mathematical physics. In this talk we will introduce these spaces and their main properties.
14:15 - 15:00	Mario Kummer 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	Ruriko Yoshida Tropical Geometry Tools for Machine Learning and Phylogenomics Linear algebra provides key tools in statistical and machine learning models. Tropical geometry can be described as the piecewise-linear analogue of classical geometry and in this talk we discuss recent exciting developments in “tropical geometric machine learning tools” in the rapidly growing area of phylogenomics, which can provide insight into relationships and evolutionary patterns of a diversity of organisms, from humans, plants and animals, to microbes and viruses. This is joint work with a subset of the following wonderful researchers: David Barnhill, George Aliatimis, John Sabol, Chris Teska, Keiji Miura, Kurt Pasque, Jefferson Huang.
16:15 - 17:00	Kristina Wicke 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 -	Dinner

09:00 - 09:45	Henning Krause Fom representations of finite groups to biology - the early work of Andreas Dress The early contributions of Andreas Dress centred around groups and their representations. For example a fundamental induction theorem is named after him, generalising well known results of Artin and Brauer, and opening new directions in modular representation theory through the use of of algebraic K-theory. His interest in applications of mathematics manifested when he moved in 1969 to Bielefeld as one of the first professors of the newly founded faculty of mathematics. My talk will present some personal impressions from the perspective of a former colleague (and a representation theorist).
09:45 - 10:30	Parvaneh Joharinad Hyperconvexity as a generalization of Negative curvature In this talk, we introduce a novel approach to understanding negative sectional curvature in general metric spaces, drawing inspiration from the concept of the hyperconvex envelope. Hyperconvexity, introduced by Aronszajn and Panitchpakdi in 1956, represents a key concept in metric space theory, characterizing nonexpansive absolute retracts. An important theory developed by Isbell (1965) and later in another context by Dress (1984) associates to each metric space a hyperconvex envelope via Kuratowski embedding. Dress uses this envelope to define a combinatorial dimension for finite metric spaces. An intriguing aspect of the hyperconvex hull, as demonstrated by Dress, is its capability to reveal tree-like structures within metric spaces. Specifically, a metric space exhibits tree-like characteristics if its hyperconvex hull exhibits similar properties. Inspired by this characteristic, we explore how hyperconvexity can be interpreted as a generalized form of negative curvature.
10:30 - 11:00	Coffee Break
11:00 - 11:45	Kay Nieselt to be announced
11:45 - 12:30	Marc Hellmuth Symbolic Ultrametrics and Beyond In this talk, we provide a brief introduction to symbolic ultrametrics, a concept pioneered by Andreas Dress and Sebastian Böcker in the late 1990s. Symbolic ultrametrics extend the idea of traditional metrics by using symbols instead of real-valued numbers. Similar to ultrametrics, one may ask whether there exists a dated or labeled tree (T,t) such that the label t(lca(xy)) of the least common ancestor of two leaves x and y coincides with the symbol d(x,y) determined by the underlying symbolic map d. We will discuss how this work laid the foundation for a deeper understanding of orthologous genes. Additionally, we will explore characterizations and generalizations of symbolic ultrametrics.
12:30 - 13:30	Lunch
13:30 - 14:15	Benjamin Schröter Matroids and splits Andreas Dress studied several discrete structures, among them matroids and (finite) metric spaces with a focus on their interrelation. His famous work includes the introduction of valuated matroids, and the split decomposition theorem for metric spaces, both of which are highly relevant in phylogenetics. From a polyhedral point of view, valuated matroids are lifting functions on matroids (base) polytopes that subdivide the polytope into matroidal cells. Furthermore, splits in this language are subdivisions into exactly two maximal cells. It follows that split decomposable subdivisions are dual to trees and the matroids occurring in the case of a valuated matroid are well structured. These matroids, called split matroids, have been intensively studied in the last years and have remarkable features, in particular with respect to matroid invariants and specially the Tutte polynomial a important matroid invariant. In this talk I will report on the latest developments around split matroids.
14:15 - 15:00	Matt Baker Matroids with coefficients and Lorentzian polynomials In the first half of the talk, I will briefly survey the theory of matroids with coefficients, which was introduced by Andreas Dress and Walter Wenzel in the 1980s and refined by the speaker and Nathan Bowler in 2016. This theory provides a unification of vector subspaces, matroids, valuated matroids, and oriented matroids. Then, in the second half, I will outline an intriguing connection between Lorentzian polynomials, as defined by Petter Brändén and June Huh, and matroids with coefficients. The connection hinges on the relationship between valuated matroids and phylogenetic trees, 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.

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