Arthur B. Coble Memorial Lectures
37th Annual Lecture Series
March 31 - April 2, 2008

Ib Madsen
University of Aarhus, Denmark

will present
Moduli Spaces and Topology

Moduli spaces are varieties whose points are themselves geometric objects. The classical examples are the moduli space of all Riemann surfaces and the closely related moduli space of differentiable surfaces embedded in Euclidian space of arbitrarily high dimension. Other examples include the Gromov-Witten moduli space of pseudo-holomorphic curves in a symplectic background, the moduli space of flat connections on principal bundles on a surface, and the moduli space of graphs. The algebraic K-theory of a ring may be viewed as the moduli space of its projective modules.

The lecture series aims at explaining the role of topology in the study of moduli problems, exemplified by the Madsen-Weiss solution of Mumford's "standard" conjecture.

The first lecture will evolve around Riemann's moduli space. I will explain its connection to the mapping class group via Teichmüller theory on the one hand, and its connection to smooth surface bundles on the other hand. The focus will be on the role of the tangent and normal bundles, following an old tradition in geometric topology.

The second and third lectures will explain the relationship between categories and topological spaces with special attention to various cobordism categories, the domains of Field Theories. I will then focus on the role of homological stability theorems for relating cobordism categories to moduli problems, and present a number of applications. Mumford's conjecture is a special case.

The first lecture will be held Monday, March 31 at 4 p.m. followed by a reception in 314 Altgeld Hall.
The lectures on April 1 and 2 will be held at 4 p.m. in 245 Altgeld Hall. Cookies and coffee will be served at 3:30 p.m. in 321 Altgeld before the April 1 and 2 Coble lectures.