Courant Research Centres ´´Higher Order Structures in Mathematics´´

Research at the Courant Center

Mathematics as science and cultural achievement has many facets.

In the last one hundert years of particular importance was the identification and use of mathematical structure. In this way, at the beginning of the twentieth century the area of algebra was revolutionized in Göttingen.

Today we live in a period during which very different areas of mathematics come closer together and exchange techniques and ideas. In this process, new problems occur, e.g. if the "flexible" world of topology and geometry is used in the "rigid" world of number theory. For the time being, we are still lacking a good understanding of the overarching structure which makes this efficiently possible.

As a further concrete example let us mention the relation between mathematics and physics; in principle this has always been symbiotic: physics is the most important trigger also for inner-mathematical developments, and mathematics always has provided the "language of physics". One of the most urgend fundamental questions of todays theoretical physics is a common theory of gravitation (Einsteins relativity) and quantum physics. It is very likely that completely new mathematical structures are necessary to make this possible.

These are two examples how mathematics s fighting at new frontiers. We are convinced that the development and study of new "higher order structures in mathematics" are necessary to the solution of the problems occuring along the way.

The Courant Research Center "Higher order structures in mathematics" consists of about a dozen professors and further associated researchers. Key pillar are the junior research group, which -focused on detail questions- together push forward the research.

Until January 2011, Prof Hannah Markwig was leader of the group "tropical algebraic geometry", which also includes apl. Prof. Hans-Christian Graf von Bothmer. This group studies how to apply the "flexible" geometry in "rigid" areas like number theory. One of the far reaching goals is to understand more about mirror symmetry (motivated by string theory in physics).

The group "Higher differential geometry" is lead by Prof. Chenchang Zhu. She studies "distributed symmetries" in geometry, in particular their analytic aspects. Goal is the classification and applicaiton of these symmetries in and later also outside of mathematics.

The group "mathematical physics" of Prof. Dorothea Bahns uses new methods, e.g. the so-called non-commutative geometry, to develop mathematical models which in the long run should help to unity quantum physics and Einstein general relativity.

• Noncommutative Geometry (Zhu, Bahns, Meyer, Schick)
  
• Quantum Field Theory (Bahns, Buchholz, Rehren)
  
• Mathematical Physics (Zhu, Buchholz, Rehren, Witt, Bahns)
  
• Category Theory (Zhu, Meyer, Bartholdi)
  
• Algebraic Geometry (Markwig, von Bothmer)
  
• Differential Geometry (Zhu, Bahns, Schick)
  
• Number Theory (Mihailescu, Patterson, Blomer, Brüdern)