Jørgen Ellegaard Andersen (Aarhus) and Ib Madsen (Aarhus).
Contents
Understanding the geometry and topology of the moduli space of Riemann surfaces has been a goal of central importance in mathematics for a century. A little later the moduli space of holomorphic bundles over Riemann Surfaces has emerged as an important mathematical theme. - 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 them self. - These moduli spaces were primarily studied using techniques of algebraic geometry and Teichmuller 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 has been applied to the moduli spaces, and that with great success. There have 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.
The course will seek to establish a collective overview of these developments with the view to further understanding there interrelations. The course will be organized in a number of relative independent shorter lecture series.
