Scientific Context

Understanding the geometry and topology of the moduli space of Riemann surfaces has been a goal of central importance in mathematics for a century. Later the moduli space of holomorphic bundles over Riemann Surfaces has emerged as an important mathematical theme as well.

The term moduli space of structures of a given kind refers to the set of all such structures, suitably topologized. Moduli spaces of geometric structures, such as Riemann surfaces and bundles over Riemann Surfaces, also carry beautiful geometric structures themselves.

Primarily, these moduli spaces were studied using techniques of algebraic geometry and Teichmüller theory, but during the last 2 decades also by using techniques from quantum field theory from mathematical physics. Most recently, algebraic topology and the theory of Toeplitz-operators have been applied to moduli spaces, and that with great success.

There has been three major breakthroughs within the last couple of years:

• The Madsen-Weiss solution of the Mumford conjecture.
• The Chas-Sullivan discovery of new structures on loop spaces of manifolds.
• The Andersen proof of Turaev's asymptotic faithfulness conjecture.

A more in-depth description of the new research opportunities that is represented by these 3 developments as well as their interrelation can be found here.

The center activities are focused on the boarder areas between algebraic geometry, algebraic topology, Riemann surface theory, gauge theory, topological quantum field theory and Berezin-Toeplitz quantization, and will therefore strive to contribute to the integration of these areas. The purpose is to foster a collaboration between senior mathematicians who are specialists in these different areas to explore specific questions in geometry and topology that now seem ready for attack. It is an equally important goal of the center to train graduate students and postdocs in the foundation and methods of these areas with the intent of working as senior mathematicians in this exciting developing field of mathematics.

It is the long-term vision of the researchers in moduli problems to fuse and apply the emerging mathematical frameworks of the above three developments to the mathematical study of quantum field theory and string theory. The center also endeavours to assist in this process in particular in collaboration and interaction with the Center for Theory in the Natural Sciences at the Faculty of Science, the University of Aarhus.

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