Shaping solutions

Text: Trio MedienService Bonn

Only a handful of scientists around the globe are conversant with the methods that the celebrated mathematician Marc Levine is developing. In Germany he is the only one. The unconventional, cross-disciplinary thinker from the USA has been researching together with other leading scientists at the mathematics think-tank in Essen since July 2009. There he is helping to build up a new research centre and decisively boost networking.

When Marc Levine talks about his work as a mathematician, there is far more to it than just bare figures. Instead, the researcher enthuses about geometrical configurations: circles, ellipses and spheres. The discipline in which Levine is pushing back the boundaries is called “algebraic geometry”.“I look for solutions to mathematical equations by moving in the world of geometric concepts,” says Levine, as he describes his work. He investigates the structural properties of equations by using methods from another area of mathematics: topology. Topology is the study of the properties of spatial configurations. Levine is endeavouring to use the existing, but not manifest connection between algebra and topology, in order to solve algebraic problems. For instance, a spatial configuration in the shape of trousers is created from three circles and an outer sheath. The corresponding drawing is somewhat reminiscent of a sewing pattern. It is connections between geometrical configurations such as these, as well as in higher dimensions than height, width and depth, that fascinate the unconventional mathematician. Levine says geometry is so flexible, that sometimes it helps in finding solutions more easily than by algebraic means.

Between formulas and forms

Mathematicians associate the name of Marc Levine above all with algebraic cobordism. He developed this theory, which combines geometry, topology and algebra, together with the French mathematician Fabien Morel. “We took the geometric-topological descriptions of the mathematician Daniel Quillen, which cannot be directly set down as algebraic equations, and transformed the descriptions so that they could then actually be applied in algebra,” explains Levine. In this way they succeeded in transferring concepts from topology into algebra – for instance, cobordism.

A cobordism is a so-called equivalence relationship, as in the previously mentioned case of the trousers where one circle is combined with two additional circles. With the aid of their theory Levine and Morel were able to derive several mathematical axioms. As yet, there are no foreseeable applications outside of mathematics, according to the Humboldt Professor. But that can change very quickly, as the scientist explains: the so-called K-Theory, an abstract form of linear algebra on which he used to work, now has applications in quantum physics and string theory.

Marc Levine will be one of the heads of the Essen Institute for Algebraic Geometry and Algebra which is being planned as part of the mathematics department at the University of Duisburg-Essen. Currently, he and his colleagues are working together with more than 50 scientists in eight working groups in this central, yet interdisciplinary field of mathematics. This versatile structure is designed to facilitate joint projects within the department as well as with colleagues at other universities. Marc Levine is really enthusiastic about his new job: “At the campus in Essen we’re working on similar questions. We communicate and cooperate openly with one another in numerous personal exchanges. This creates the potential for groundbreaking results.”

The more interesting colleagues are in Germany

Before he came to Essen the American mathematician taught and researched for 26 years at Northeastern University in Boston, Massachusetts. He now plans to turn these contacts into collaborative projects. He greatly enjoyed his first visit to Germany when he came to Bonn, in 1983. “I received a great deal of encouragement for my research,” he recalls. He explains that although there are a lot more mathematicians and jobs in the USA, in his case he has met the more interesting colleagues in Germany and France.

Levine’s ties with his Essen colleague Hélène Esnault are based on many years of friendship and collaboration. Their mutual esteem led to Levine’s many guest visits to Essen during the 1990s. Hélène Esnault is visibly pleased: “In future we’ll be researching together with an internationally acclaimed mathematician who will strengthen our group and promote our work at the same time. The Alexander von Humboldt Professorship not only brought us a leading-edge mathematician but meant that a talented junior researcher decided to come here, too, even though he had many other possibilities in Germany.”

As soon as he arrived in Essen the Humboldt Professor set about networking: the International Conference on Algebraic Geometry and Arithmetic took place there in mid February 2010. It was dedicated to the memory of Professor Eckart Viehweg who died shortly before the event. He had been instrumental in bringing Marc Levine to Essen as a Humboldt Professor in the first place. Together with his colleagues Levine welcomed 150 participants from around the world, and 20 of the most distinguished mathematicians presented their la test research findings.