## Interview with MATH+ Distinguished Fellow Michael Joswig

**Michael Joswig was honored with the MATH+ Distinguished Fellow award by the Excellence Cluster MATH+ at the end of 2023. In the interview, he discusses his journey into mathematics, the specific topics that fascinate him, and how, following a MATH+ research project, NASA even reached out and expressed interest in the results.**

**Short Bio**

Michael Joswig studied mathematics and computer science at the Universität Tübingen, where he earned his doctorate in 1994. He obtained his habilitation in 2000 at TU Berlin on the topic of “Contributions to polytope theory and incidence geometry.” After research stays in the USA, including at MSRI in Berkeley, he was appointed as a professor for “Algorithmic Discrete Mathematics” at TU Darmstadt from 2004 to 2013. Since 2013, he has been the Einstein Professor for “Discrete Mathematics/Geometry” at TU Berlin. In 2019, he became a Max Planck Fellow at the Max Planck Institute for Mathematics in the Sciences (MPI MiS). In late 2023, MATH+ honored him with the “MATH+ Distinguished Fellow” award.

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**MATH+ Distinguished Fellowship Program **

The MATH+ Distinguished Fellowship is bestowed upon individuals in recognition of outstanding contributions to the mathematical sciences in general. The program’s goal is to support the work of world-leading mathematicians who hold permanent professorships at one of the MATH+ partner institutions, i.e., FU, HU, TU Berlin, WIAS, or ZIB. MATH+ provides Distinguished Fellows with a fellowship allowance for fostering innovative ideas and research activities, initiating exciting, vibrant research collaborations, and expanding the Fellow’s international research network and activities. There are currently five MATH+ Distinguished Fellows: __Gavril Farkas__, __Peter K. Friz__, __Michael Joswig__, __Bruno Klingler____,__ und __Alexander Mielke__.

For international colleagues, MATH+ offers funding as (Distinguished) Visiting Scholars.

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**What does the MATH+ Distinguished Fellow award mean to you?**

I am very pleased about the recognition I have received through the MATH+ award. It relates to my commitment to the field of tropical geometry, where the simultaneity of optimization and geometry takes place. I will continue my work in this area. Further, I will initiate innovative, interdisciplinary projects with scientists in the field of “Topological Data Analysis,“ i.e., data analysis using topological methods.

**Why did you choose to study mathematics? **

At school, I also seriously considered studying ancient languages, Latin and Greek. However, at the age of 14, I began programming and became interested in understanding how software works, including its mathematical aspects. Self-study and courses in computer science further increased my interest in mathematics. I then studied mathematics with a minor in computer science at the Universität Tübingen to better comprehend computers and algorithms.

**What fascinates you about mathematics? What is the beauty of mathematics?**

There is this clarity in mathematics. Unlike other areas of science, you can clarify definitively whether a fact is valid or not, and you can provide precise answers to questions. Regarding beauty, clear thought processes and visual aspects are significant. Everything I engage in falls under the umbrella of geometry, whereby geometry can be seen as a visual representation of a part of mathematics. I am heavily involved in photography in my private life, and the visual aspect is important to me. I enjoy contemplating mathematical issues visually.

**What are the most important and influential milestones in your professional career?**

During my mathematics studies in Tübingen, I spent the third academic year (1988/89) at ETH Zurich because the computer science program in Tübingen was still in its early stages, and ETH was considered the best institution for computer science. To this day, I maintain contact with colleagues in mathematics and computer science at ETH.

A highly influential milestone for my current research was my postdoctoral period starting in 1996 in the research group led by Günter M. Ziegler at TU Berlin. Prior to that, I had focused on different mathematical topics, and in Berlin, I completely changed my research area.

Another significant milestone was my time at MSRI in Berkeley, where I stayed during the fall semester of 2003, focusing on the theme “Discrete and topological combinatorics.” During this period, the new topic of tropical geometry came up, which has strongly shaped and influenced my research to this day. This topic originated from questions within algebraic geometry, leading to the connection of the geometry of polynomial equations with polyhedral methods, a concept that was fascinating. Here, algorithmic questions and the relationship with computers emerged again, allowing us to make valuable contributions with the software we had developed.

However, the perspectives and theoretical considerations from tropical geometry also prove useful for applications. For instance, this approach enabled us to study and understand certain biological issues, as presented in our recent PNAS publication „__Master regulators of biological systems in higher dimension____“.__

**What is currently the most important aspect of your research activities? Where is your focus?**

In my research group, several topics are always being addressed simultaneously, mutually enriching each other. The topics of computers and software are a crucial driving force for me in two directions. On the one hand, when facing mathematical questions, I want to understand how to calculate certain things, leading me to algorithms and their complexity. On the other hand, the computer provides results that would not be accessible otherwise. This, in turn, leads to theoretical work, creating a dynamic interplay between theory and computer-based research.

On the geometric side, polytopes always play a significant role. They represent feasible regions of linear programs, as discussed in our __2021 SIAM Reviews publication__ on Stephen Smale’s ninth problem. This was about the complexity of linear optimization, i.e., how quickly linear programs can be solved. These are fundamental optimization problems that are crucial, as they often appear as building blocks in more complex, natural optimization problems. As we conduct this interview right now, millions of linear programs are being solved worldwide.

Polytopes are the geometric counterpart to this, representing the feasible regions on which linear programs operate. In the collaborative __research project on auctions (AA3-5)__ with __MATH+ colleague Max Klimm__ (TU Berlin), polytopes are used, and their geometric properties describe certain auction mechanisms. The goal is to set prices for certain objects in a given scenario, maximizing the expected profit for the seller. This is expressed through the volumes of polytopes that need to be calculated for problem-solving, which is a highly classical problem that becomes particularly interesting when parameterized.

**Is there a favorite project in your research?**

Throughout my research career, there has always been a larger guiding theme that I have oriented myself towards. The fundamental question of the “Complexity of Linear Optimization,” the ninth Smale problem, has various facets, many of which have led to my own research questions. While some aspects have been clarified, others remain open. This can keep you busy your whole mathematical life.

When I started working with polytopes in the 1990s, there was a combinatorial variant of it. The so-called “Hirsch Conjecture” was still unresolved at that time and was only clarified in 2000 by my colleague Francisco Santos (Universidad de Cantabria) through a constructed counterexample.

**You have been involved in four MATH+ projects so far. Which aspects have caught your interest?**

Each funded project has a certain direction. MATH+ aims to bring mathematics into application, which I always find very interesting, and that’s why I feel perfectly aligned with MATH+. The structuring into the __Application Areas and Emerging Fields__ primarily places “Discrete Mathematics” in the __AA 3 Group „Next Generation Networks.“__ The main topics of combinatorial optimization and graph algorithms provide natural connections to tropical geometry. Many problems in tropical geometry are related to classical graph algorithms.

In **all MATH+ projects**, there were exciting results, and I would like to highlight two in particular:

Firstly__, Project AA 3-5 with Max Klimm__ from TU Berlin as I needed to familiarize myself with the economic issues of the auction mechanisms. It turned out that the available methods were well suited to working on this together. I learned a lot, and good results were achieved, as we could demonstrate the optimality of certain auction mechanisms, going beyond what was previously known. To calculate the prices for the goods to be auctioned concretely, computer algebra is important, which complements the work very well.

Another __interesting project (AA3-10)__ was positioned exactly at the interface of graph algorithms and tropical geometry. With my former doctoral student Benjamin Schröter, we aimed to classify a certain type of convex polytopes motivated by tropical geometry and implement it algorithmically. This is related to the shortest path problem in optimization, providing the basis for navigation systems, for example. The classical problem is initially straightforward: we have a transportation network, such as the road network in Berlin, and the shortest path problem provides the shortest route from, for example, the TU Berlin to Berlin-Kreuzberg. However, our question was, what happens if I don’t know exactly how many minutes of travel time I need for each segment, specifying an interval, such as 3-7 minutes for a specific segment. Then, you have a parametric shortest-path problem, for which we developed algorithms.

One of the **highlights of my MATH+ projects** was that shortly after its publication in 2019, a research group at NASA approached us with a request to use our work. They needed to ensure communication between distant spacecraft in space missions, such as a human-crewed mission to Mars and were considering building an interstellar internet. On Earth, the concern is routing data from one antenna to the next, involving the shortest path problem. However, in interstellar space, the challenge arises because objects can no longer be assumed to be stationary; it involves satellites moving in orbits, rendering classical routing algorithms ineffective. This NASA working group found precisely what they needed in our work.

**What advantages do you see in a large research center like MATH+, which comprises over 500 members from the three major universities in Berlin and two research institutes?**

It is fundamentally beneficial that many researchers come together in such a large-scale project, providing numerous stimuli, even from unexpected directions. Additionally, excellent structures have been developed over the years, such as the __Graduate School “Berlin Mathematical School” (BMS)__, which is a genuine achievement, internationally visible, and has become a brand. Such an Excellence Cluster functions both internationally to represent Berlin externally and internally, for instance, through regular MATH+/BMS Fridays with presentations by colleagues from around the world, offering a meeting place and exchange for all colleagues in Berlin.

**What opportunities do you see for your field in MATH+? What do you plan to achieve with the MATH+ Distinguished Fellow funding?**

There are numerous possibilities and points of entry for my mathematics, as my research is highly versatile and can be applied and combined in various ways. This Distinguished Fellow award relates explicitly to my involvement in the field of tropical geometry, where the simultaneity of optimization and geometry takes place. In addition to my work in Berlin, I have strong connections with the __Max Planck Institute for Mathematics in the Sciences__ in Leipzig, where much research on tropical geometry is conducted.

Furthermore, the __Max Planck Institute of Molecular Cell Biology and Genetics__ in Dresden has a new colleague: __Heather Harrington__, a mathematician and director of the institute. She is engaged in „Topological Data Analysis,“ collaborating with colleagues from biology and chemistry, a prospect I find very inspiring.

The polyhedral methods also have a topological component, an area I have worked on extensively in the past. Therefore, along with Heather Harrington and a new postdoctoral researcher, I intend to work on projects related to topological data analysis. This aligns with the research of the Collaborative Research Center (SFB/TRR 109) at TU Berlin, led by __Frank Lutz__ and __Myfanwy Evans__ (U Potsdam), focusing on __“The Structure within Disordered Cellular Materials” (AA3-14__). The field of data science is also highly interesting for MATH+. While it is often equated with artificial intelligence, this perspective is overly simplistic though. Various other topics also play a role, including topological data analysis, to which I plan to dedicate more of my time.

**Thank you very much for the interview and the interesting insights into your research!**

(The interview was conducted in January 2024 by Beate Rogler, MATH+.)

**LINKS**

- Michael Joswig was involved in the following MATH+ projects:
- Michael Joswig
__at Technische Universität Berlin__and at__Max Planck Institute for Mathematics in the Sciences (MPI MiS__) - More on the MATH+ Distinguished Fellowship Program and the MATH+ research areas
- The interview in German (PDF)