Andreas Dress

Andreas Dress: The Mathematician Who Kept Changing Fields

There is a variety in tropical geometry called the Dressian. There is a combinatorial object used to classify periodic tilings called the Delaney–Dress symbol. There is a body of work on reconstructing evolutionary trees from distance data known as T-theory. All three came from the same man, and they belong to three different branches of mathematics that most researchers would consider separate careers. Andreas Dress, who died on 23 February 2024 at eighty-five, spent six decades doing something unusual even among first-rate mathematicians: he kept starting over.

Kiel, 1962

Dress was born on 26 August 1938. He studied at the Freie Universität Berlin, at Tübingen and at Kiel, taking his doctorate at Kiel in 1962 under Friedrich Bachmann and Karl-Heinrich Weise. The dissertation was titled *Konstruktion metrischer Ebenen* — the construction of metric planes — which places his origins squarely in the German tradition of foundations of geometry. It is worth noting how young this makes him: a doctorate at twenty-four, in a subject that had been one of the great German specialities since Hilbert.

Bielefeld from Scratch

In 1969 the state of North Rhine-Westphalia opened a new university at Bielefeld, conceived as a research-first institution with an interdisciplinary charter. Dress was one of its founding professors of mathematics, and he stayed until his retirement in 2003 — thirty-four years in the same department, which for a mathematician of his restlessness is a curious statistic until one notices that he changed subject roughly every decade without ever changing address. Bielefeld's mathematics faculty became one of the strongest in Germany in that period, and Dress was a substantial part of the reason.

Functors, Tilings, Matroids

His first major body of work was in representation theory, where he developed Mackey and Green functors — formal machinery that supplies a unified framework for the restriction and induction theorems that pervade the representation theory of finite groups. This is abstract algebra of the purest kind, and it remains standard equipment in the field.

In the early 1980s he moved to discrete geometry and introduced, with Andrew Delaney, the combinatorial encoding now called the Delaney–Dress symbol: a finite structure that captures the symmetry type of a periodic tiling completely enough to permit systematic classification. Crystallographers and mathematicians both took it up, because it converts a geometric classification problem into a combinatorial one that a computer can enumerate.

He then turned to matroid theory, building comprehensive algebraic approaches to a subject that had been largely combinatorial, including the construction of the Tutte group of a matroid. From there the work fed forward into tropical geometry: Dress established connections between valuated matroids and tropical linear spaces, and the parametrising object that emerged bears his name — the Dressian.

Trees from Distances

The final and most publicly visible chapter took him out of pure mathematics altogether, into biology. The problem is easily stated and hard to solve: given a matrix of measured dissimilarities between species, sequences or populations, reconstruct the evolutionary tree that generated them — or determine that no tree fits, and represent what does. Dress attacked this with the tools of metric geometry, developing T-theory and the theory of tight spans, which supply a canonical way of embedding a finite metric space and of reading off its tree-like and non-tree-like structure. The subject he helped create came to be called phylogenetic combinatorics, and in 2012 he set it out in book form with Katharina Huber, Jacobus Koolen, Vincent Moulton and Andreas Spillner as *Basic Phylogenetic Combinatorics*, published by Cambridge University Press.

Shanghai

After retiring from Bielefeld in 2003 Dress went to the Max Planck Institute for Mathematics in the Sciences in Leipzig, where he remained a long-standing external scientific member. In 2005 he became a founding director of the Partner Institute for Computational Biology in Shanghai, a joint venture between the Max Planck Society and the Chinese Academy of Sciences, serving until 2010. He was an Invited Speaker at the International Congress of Mathematicians in Berlin in 1998.

Why Andreas Is Called a Genius

The quality that distinguishes Dress is structural transfer — the recognition that a piece of machinery developed for one problem is secretly about something else entirely. This is visible at every stage of his career. The metric geometry he learned for his 1962 dissertation on metric planes became, forty years later, the apparatus for deciding whether a matrix of genetic distances is tree-like. The algebraic instincts he developed in representation theory became the algebraisation of matroid theory, which in turn became a foundation for tropical geometry. The combinatorial eye that produced the Delaney–Dress symbol for tilings is recognisably the same eye that produced split decomposition for biological data. Most mathematicians deepen; Dress translated. Doing it once is luck. Doing it four times, across representation theory, discrete geometry, matroids and mathematical biology, is a distinctive cast of mind, and the fact that three separate objects carry his name in three separate fields is the objective evidence for it.

The counter-case deserves an airing. Dress solved no famous open problem, and none of his contributions carries the kind of singular fame that attaches to a Fields Medal result. Several of his best-known constructions were collaborative — the Delaney–Dress symbol, the phylogenetic work with Huber, Koolen, Moulton and Spillner, the connections in tropical geometry that others substantially developed — and his gift was often for setting a framework that colleagues then built out. He was an Invited rather than a Plenary speaker at the ICM, and his formal honours are modest relative to his influence. A fair verdict is that he was an exceptional generator of frameworks and a superb identifier of hidden analogies, rather than a solver of celebrated problems. Within mathematics those are different reputations, and the first is easier to underrate.

Legacy

Phylogenetic combinatorics is now an established field with a textbook, a research community and direct application in evolutionary biology and sequence analysis — the closest thing Dress has to a founded discipline. The Delaney–Dress symbol remains the working tool for the systematic classification of periodic tilings. The Dressian sits in the standard vocabulary of tropical geometry. And in Shanghai, the Partner Institute he helped found trained a generation of Chinese computational biologists at a moment when that country was building the field from almost nothing. Bielefeld, Leipzig and Shanghai each mourned him as their own, which is roughly the right description of a career that never stayed in one place long enough to be claimed by any single subject.

Achievements

• Held posts at Bielefeld University, Freie Universität Berlin and Institute for Advanced Study