# Why is math so efficient in physics

## Courant Research Center ´´Higher Order Structures in Mathematics´´

### Research in the Courant Center

Mathematics as a science and cultural achievement has many facets.

In the past hundred years, the identification and application of mathematical structures have been of particular importance. In this way, the field of algebra in Göttingen was revolutionized at the beginning of the twentieth century.

Today we live in a time in which many different areas of mathematics have moved closer together, and methods and ideas are exchanged. New problems arise within this process, for example when the “flexible” world of topology and geometry is applied in the “rigid” world of number theory.

At the moment there is still a lack of understanding of the comprehensive structures that make this possible efficiently.

As another concrete example we can name the connection between mathematics and physics; in history this has always been a real symbiosis: physics has also been the most important impetus for internal mathematical developments; in return, mathematics has always served as the “language of physics”. One of the most important acute questions in physics today is the connection between the general theory of gravitation (Einstein's theory of relativity) and quantum physics. It is very likely that completely new mathematical structures will be necessary to make this possible.

These are two examples of how math struggles to reach new limits. We are convinced that the development and study of new "higher order structures in mathematics" are the necessary solutions to the problems that arise.

The Courant Research Center "Higher Order Structures in Mathematics" consists of about a dozen professors and scientists. The cornerstones are the junior research groups that jointly advance research.

Until January 2011, Prof. Dr. Hannah Markwig Head of the group “Tropical Algebraic Geometry”, which also includes apl. Prof. Hans-Christian Graf von Bothmer. This group explores how to apply “flexible” geometry to “rigid” areas such as number theory. One of the far-reaching goals is to understand more about mirror symmetry (inspired by the string theory of physics).

The junior research group "Higher Differential Geometry" is headed by Prof. Dr. Chenchang Zhu headed. She researches "distributed symmetries" in geometry, especially their analytical aspects. The aim is to classify and apply these symmetries in and later also outside of mathematics.

The group "Mathematical Physics" of Prof. Dr. Dorothea Bahns uses new methods, such as so-called non-commutative geometry, to develop mathematical models that should help in the long term to unite quantum physics and Einstein's general theory of relativity.

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The Courant Center "Higher Order Structures

in Mathematics "conducts research in mathematics and mathematical physics

in and between areas

- Non-commutative 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, Brothers)

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