Some well-known German mathematicians in combinatorics

I. Current

1. Reinhard Diestel (Universitat Hamburg)

      

      Reinhard Diestel was born in 1959. He obtained a PhD from the University of Cambridge in 1986 and his doctoral advisor was Béla Bollobás. He is the author of the textbook "Graph Theory", which has more than 400 pages and was published by Springer. In this book, as he wrote in the Preface, "I have broken with the tradition of attempting to cover both theory and applications: this book offers an introduction to the theory of graphs as part of (pure) mathematics; it contains neither explicit algorithms nor 'real world' applications.".  

      Currently, he holds the chair of discrete mathematics at the University of Hamburg. He is also the editor of Journal of Combinatorial Theory, Series B, which is a Q1 journal as shown here. Some of his doctoral students, including Daniela Kuhn and Maya Stein, are now well-known mathematicians in the same area. The Diestel-Leader graph was named after him and was the concern of some papers, such as this one. 

2. Mathias Schacht (Universitat Hamburg)

      Mathias Schacht was born in 1977 in Berlin, Germany. He obtained a PhD from Emory University, Atlanta (USA) in 2004 and his doctoral advisor was Vojtěch Rödl. His fields of interest, as shown here, are: 

      - Graph theory

      - Ramsey theory and extremal combinatorics

      - Random discrete structures and probabilistic methods

      - Theoretical computer science

      In 2022, Mathias Schacht was an invited speaker at Combinatorics section, International Congress of Mathematicians (ICM). His talk was titled "Restricted problems in extremal combinatorics". He is also the editor of "Discrete Mathematics". 

3. Daniela Kühn (University of Birmingham, England)

      Daniela Kühn was born in 1973. She obtained a PhD from the University of Hamburg, Germany in 2001, under the supervision of Reinhard Diestel (mentioned in 1.). She was awarded European Prize in Combinatorics in 2003, when this prize was first awarded. In 2014, Daniela Kühn was the recipient of Whitehead Prize and in the same year, she was an invited speaker at Combinatorics section, International Congress of Mathematicians (ICM). The talk that she and Deryk Osthus (who also obtained a PhD in Germany) was titled "Hamilton cycles in graphs and hypergraphs: an extremal perspective". 

4. Julia Böttcher (London School of Economics, England)

      Julia Böttcher completed a PhD in mathematics at the Technical University of Munich in 2009 and her supervisor was Anusch Taraz. In 2018, she was awarded Fulkerson Prize. In 2022, Julia Böttcher was an invited speaker at Combinatorics section, International Congress of Mathematicians (ICM) and her talk was titled "Graph and hypergraph packing". 

5. Johannes Carmesin (TU Freiberg)

      Johannes Carmesin obtained a PhD in mathematics at Universitat Hamburg under the supervision of Reinhard Diestel (mentioned in 1.). He was awarded European Prize in Combinatorics in 2023. As shown here, the research topics of his group include: 

      - Combinatorics in 3 dimensions

      - Graph minors and connectivity

      - Matroids

      - Infinite graphs

6. Felix Joos (Universitat Heidelberg)

      He obtained a PhD from Universitat Ulm in 2015, under the supervision of Dieter Rautenbach. He was awarded European Prize in Combinatorics in 2023. His research, as shown here, deals with graphs and hypergraphs in various aspects including algorithmic, extremal and structural questions as well as problems involving probability theory. 

7. Lisa Sauermann (University of Bonn)

      Lisa Sauermann was first known for her performance in the International Mathematical Olympiad (IMO). She participated in IMO five times and won a silver medal (2007), four gold medals (2008, 2009, 2010, 2011). At IMO 2011, Lisa Sauermann was the only contestant that got a perfect score (42/42). Then, she began studying mathematics at the University of Bonn. In 2014, she completed her bachelor thesis on algebraic geometry under the supervision of Michael Rapoport. If you haven't known Michael Rapoport before, there is a fact that might interest you: Michael Rapoport was the doctoral advisor of Peter Scholze (Fields medalist 2018). 

      After obtaining a BS degree, Lisa Sauermann became a graduate student at Stanford University. Her doctoral advisor was Jacob Fox, who was also the advisor of Pham Tuan Huy at both undergraduate level and graduate level. In 2021, Lisa Sauermann was awarded European Prize in Combinatorics. She came back to the University of Bonn as a professor in 2023. As indicated in Bonn's website, her research area is Combinatorics. 

8. Angelika Steger (ETH Zurich, Switzerland)

 

      Angelika Steger was born in 1962 in Munchen, Germany. She completed a doctorate from the University of Bonn in 1990, under the supervision of Hans Jurgen Promel. She used to work at University of Kiel, University of Duisburg and Technical University of Munich before coming to ETH Zurich in 2003. In 2007, Steger was elected to the Academy of Sciences Leopoldina. In 2014, Angelika Steger was an invited speaker (section: Combinatorics) at International Congress of Mathematicians. 

9. Gunter Ziegler (Free University of Berlin)

   Gunter Ziegler was born in 1963 in Germany. He studied at LMU Munchen from 1981 to 1984. In 1987, he obtained his PhD at Massachusetts Institute of Technology (USA) under the supervision of the Swedish mathematician Anders Bjorner. After postdoctoral positions at University of Augsburg (Germany) and Mittag-Leffler Institute (Sweden), he joined the faculty of the Free University of Berlin.  He was an invited speaker at International Congress of Mathematicians 2002 (section: Combinatorics) and International Congress of Mathematicians 2014 (section: Mathematics Education). He is also one of the two authors of "Proofs from THE BOOK". 

II. Past

1. Gerhard Ringel (1919 - 2008)

   
   Gerhard Ringel obtained a PhD from University of Bonn in 1951, under the supervision of Emanuel Sperner and Ernst Peschl. He was one of the pioneers in graph theory and contributed significantly to
the proof of the Heawood conjecture (known as "the map color theorem" or "Ringel - Youngs theorem"), which is closely related to the well-known four colour theorem. This theorem was also the topic of the book "Map Color Theorem" written by Gerhard Ringel. He also coauthored another book named "Pearls in Graph Theory", in which various topics such as the Oberwolfach problem (which was proposed by Gerhard Ringel in 1967), magic graph, conservative graph, etc. were discussed. 

   
   To understand the (strong) relationship between the map color theorem and the four color theorem, we might think of our usual map as a surface S_0 of genus 0. While the four color theorem considers all maps that can be embedded into this surface, the map color theorem hopes to consider all maps that can be embedded into the surface S_p of genus p (where p >= 1 is a given integer).
    In 1890, Heawood proved that any such map can be colored by at most f(p) = [0.5*(1 + sqrt(48p + 1)] colors (here [x] denotes the largest integer not exceeding x). The Heawood conjecture, or the map color theorem, stated that the number f(p) can't be replaced by any smaller one. Since f(0) = 4, the map color theorem is compatible with the well-known four color theorem. In other words, the combination of the map color theorem (or the Heawood conjecture) and the four color theorem yields the following claims: Let p be a non-negative integer and consider a map that can be embedded into S_p. Then, this map can be colored with no more than f(p) colors, and f(p) can't be replaced by any smaller number.
   The following is an example of a map on the torus which must be colored by at least 7 = f(1) colors. Note that since opposite edges of the rectangle are glued, we have exactly a (connected) region labelled "2" instead of two separated ones as shown in the figure. The case is similar for the regions labelled "3", "4", "5" and "6". Our task is only to prove that for any 0 <= i < j <= 6, region i is adjacent to region j, which can be checked easily by looking at this rectangular figure. 

   From 1954 to 1968, Gerhard Ringel and some other mathematicians tried to prove the Heawood conjecture by finding out examples. They divided the problem into 12 cases based on 12 residue classes modulo 12 and solved these cases step by step. The proof was finalized by Gerhard Ringel and John William Theodore Youngs in 1968, thus the Heawood conjecture might be called "Ringel - Youngs theorem", although there might be a few other mathematicians who also made certain contributions. 
  

2. Emanuel Sperner (1905 - 1980)

   Emanuel Sperner obtained a PhD from Universitat Hamburg in 1928 under the joint supervision of Otto Schreier and Wilhelm Blaschke. Sperner's theorem in extremal set theory and Sperner's lemma - a combinatorial result on colorings of triangulations, were named after him. Both theorems have various generalizations. He was also one of the two doctoral advisors of Gerhard Ringel. 

3. Rudolf Halin (1934 - 2014)

   Rudolf Halin was a German graph theorist. He obtained a PhD from the University of Cologne in 1962 under the supervision of Klaus Wagner and Karl Dorge. After that, he joined the faculty of the University of Hamburg. He is known for defining the ends of infinite graph, for Halin's grid theorem and for extending Menger's theorem to infinite graphs. The Halin's graph constructed from trees by adding a cycle through the leaves of the given tree was also named after him. 

   Regard the definition of ends of infinite graph, its topological version was defined by the German mathematician Hans Freudenthal in the 1930s, while Halin's graph-theoretic version was given about 30 years later. In fact, these two versions are not compatible except for locally finite graphs (i.e. graphs whose vertices are of finite degrees). A comparison between these two definitions was indicated in the paper "Graph-theoretical versus topological ends of graphs" (Journal of Combinatorial Theory, Series B, 2003), which was written by Reinhard Diestel and Daniela Kühn. This concept is considered not only in graph theory/combinatorics but also in some other related areas like geometric group theory. 

4. Klaus Wagner (1910 - 2000)

   Klaus Wagner studied topology at the University of Cologne under the supervision of Karl Dörge, a student of the well-known mathematician Issai Schur and obtained his PhD in 1937. Meanwhile, Klaus Wagner is known for his contributions to graph theory and particularly the theory of graph minors. Wagner's theorem, which successfully characterizes planar graphs via the "absence" of certain representative structures (i.e. K_{3, 3} and K_5), was named after him. The strong perfect graph conjecture/theorem (proved by Maria Chudnovsky, Neil Robertson, Paul Seymour and Robin Thomas in the 2000s) has the similar motif: it characterizes perfect graphs via the absence of cycles whose length are odd number greater than 3. 

                               Klaus Wagner (right) and Frank Harary at Oberwolfach in 1972

   

   In fact, Wagner's theorem is not the only characterization of planar graph. As written in Graph Theory (Reinhard Diestel), in 1937, Saunders Mac Lane, an American mathematician who obtained a PhD from University of Göttingen (Germany) in 1934, provided the following characterization of planar graphs. First, the set of all subsets of the edge set of a graph might be consider as a vector space over the field F_2 where addition corresponds to symmetric difference of two sets. The subspace of this space spanned by all cycles of the graph is called the cycle space. Saunders Mac Lane proved that a graph is planar if and only if the cycle space has a basis such that every edge belongs to at most two subsets in that basis. From my own perspective and experience, Wagner's theorem might be more popular than MacLane's planarity criterion thanks to its elegance. Beginners might start with proving the non-planarity of K_5 and K_{3, 3} as an exercise and they will be probably impressed by the fact that these two seemingly simple structures can be used to characterize planar graphs. Meanwhile, it might be hard to understand and be impressed by MacLane's planarity criterion without an excellent understanding of linear algebra. 

5. Wolfgang Haken (1928 - 2022)

   In fact, Wolfgang Haken was a mathematician specializing in topology. He obtained a PhD in topology in 1951 at University of Kiel under the supervision of Karl-Heinrich Weise. Meanwhile, he was known for proving the four color theorem in the late 1970s. The proof was given by Wolfgang Haken and Kenneth Appel, who were then professors at University of Illinois at Urbana - Champaign (America) and it required certain helps from computers. Thanks to this proof, Wolfgang Haken was invited as a speaker of the section "Discrete Math & math Aspects of Computer Science" at International Congress of Mathematicians 1978 in Helsinki (Finland). He was also known for his contributions to topology, such as the definition of a Haken manifold.