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* October 12, 1956

Carl-Friedrich Bödigheimer

Mathematical Institute

Academic career

1975-1979	Studies of mathematics, Heidelberg University
1979	Diplom, Heidelberg University
1980	Master-of-Science, Oxford University
1982-1984	Graduate studies, Heidelberg University
1984	PhD, Heidelberg University
1984-1985	Postdoc, Göttingen University
1985-1991	C1 assistant, Göttingen University
1990	Habilitation, Göttingen University
1991	C2 assistant, Göttingen University
1991-1992	Visiting assistant professor, Johns-Hopkins-University Baltimore
1992-1993	Heisenberg grant
1992-1993	Guest at Institute for Advanced Studies, Princeton
1993--	C3 professor, Bonn University

Research projects and activities

1984-1993 member of the Collaborative Research Center SFB 170 (Göttingen), 1994-2000 coordinator of the Research Training Group GRK 4 (Bonn), 1994-1999 member of the Collaborative Research Center SFB 256 ``Nonlinear Partial Differential Equations'', since 2001 coordinator of the Bonn International Graduate School in Mathematics, Physics and Astronomy, since 2005 coordinator of the Research Training Group GRK 1150 ``Homotopy and Cohomology''.

Research profile

Topology is the study of geometric objects like polyhedra or manifolds and their qualitative properties, for which the linking of two curves in three-dimensional space is an illustrative example. These deformation invariant properties are studied by associating to the geometric object or situation a number (like the Euler characteristic of a polyhedron) or a vector space (like the vector space of all closed differential forms on a manifold) or a group (like the fundamental group or cohomology groups of a space). To understand the homotopy or cohomology groups of mapping or classifying spaces is the key to many problems ranging from differential geometry to theoretical physics.

Contribution to research area "Moduli spaces"

Moduli spaces of Riemann surfaces, historically the first examples of moduli spaces, are of great importance for differential and algebraic geometry as well as for theoretical physics. If the surfaces have boundary, then the moduli space is a smooth manifold and the classifying space of the torsionfree mapping class group. In [5], [6], [9], [10] we have described finite cell complexes homotopy equivalent to these moduli spaces. With these complexes the integral homology of the genus 2 moduli space was computed in [1]. Few structural properties of the homology of moduli spaces and mapping class groups have been known: the homological stability (Harer), the homological diminension (Harer), the Euler characteristic (Harer-Zagier), and a few stable homology groups. Tillmann's theorem on the inifinte loop space structure and the solution of the Mumford conjecture on the rational stable homology type by Madsen-Weiss have spurred new interest in the homotopy and homology type of these moduli spaces. We have started to use the complex mentioned above for structural results on the homology of moduli spaces.

Contribution to research area "Geometric structures in quantum physics"

Segal and Atiyah gave a definition of a conformal field theory which started the interest of mathematicians in new geometric and algebraic structures like vertex operator algebras, Frobenius algebras, quantum cohomology or operads. Many constructions in this field depend continuously on the moduli space of surfaces with in-coming and out-going boundary curves, a model for the world line in string theory; see [9]. Another newly discovered structure is the Chas-Sullivan product in the homology of the loop space of a manifold.

General research topics for PhD theses

Configuration spaces, moduli spaces of Riemann surfaces, mapping class groups, operads.

Recent and planned postgraduate teaching

lecture course : Cohomology of groups, Winter 2003/2004
lecture course: Configuration spaces, Summer 2004
seminar : Deligne-Mumford compactification of moduli spaces, Summer 2005
lecture course : Rational homotopy theory, Winter 2005/2006

Supervised theses

Diplom theses: 38, currently 1
PhD theses: 11, currently 4

Selected PhD Students

Birgit Richter, 2000, Taylor towers and cubical constructions of Gamma modules,
now: professor, Hamburg University
Michael Eisermann, 1999, The number of knot group representations is not a Vassiliev invariant, now : Maitre de Conference, Grenoble

Habilitations

Ulrike Tillmann, 1996, now: professor, Oxford University

Publications

[1]	ABHAU, J., BÖDIGHEIMER, C.-F., AND EHRENFRIED, R. Homology of the moduli space for genus surfaces. preprint (2005).
[2]	BÖDIGHEIMER, C.-F. Splitting the Künneth sequence in K-theory. Math. Annalen 242 (1979), 159-171.
[3]	BÖDIGHEIMER, C.-F. Splitting the Künneth sequence in K-theory, II. Math. Annalen 251 (1980), 249-252.
[4]	BÖDIGHEIMER, C.-F. Stable splittings of mapping spaces. Proceedings Seattle 1985, Springer Lecture Notes in Mathematics 1286 (1987), 174-187.
[5]	BÖDIGHEIMER, C.-F. On the topology of moduli spaces of Riemann surfaces, Part I : Hilbert uniformization. Math. Gottingensis 7+8 (1990).
[6]	BÖDIGHEIMER, C.-F. On the topology of moduli spaces of Riemann surfaces, Part II : homology operations. Math. Gottingensis 23 (1990).
[7]	BÖDIGHEIMER, C.-F. Interval exchange spaces and moduli spaces. Contemporary Math. 150 (1993), 33-50.
[8]	BÖDIGHEIMER, C.-F. Cyclic homology and moduli spaces of Riemann surfaces. Asterisque 226 (1994), 43-55.
[9]	BÖDIGHEIMER, C.-F. Moduli spaces of Riemann surfaces with boundary. preprint (2003).
[10]	BÖDIGHEIMER, C.-F. Hilbert uniformization of Riemann surfaces. preprint (2005).
[11]	BÖDIGHEIMER, C.-F., AND COHEN, F. Rational cohomology of configuration spaces of surfaces. Proceedings Göttingen, Springer Lecture Notes in Mathematics 1361 (1987), 7-13.
[12]	BÖDIGHEIMER, C.-F., COHEN, F., AND MILGRAM, J. Truncated symmetric products and configuration spaces. Math. Zeitschrift 214 (1993), 179-216.
[13]	BÖDIGHEIMER, C.-F., COHEN, F., AND PEIM, M. Mapping class groups and function spaces. Contemporary Math. 271 (2001), 17-39.
[14]	BÖDIGHEIMER, C.-F., COHEN, F., AND TAYLOR, L. On the homology of configuration spaces. Topology 28 (1989), 111-123.
[15]	BÖDIGHEIMER, C.-F., AND MADSEN, I. Homotopy quotients of mapping spaces and their stable splitting. Quart. J. Math. Oxford (2) 39 (1988), 401-409.
[16]	BÖDIGHEIMER, C.-F., AND TILLMANN, U. Stripping and splitting decorated mapping class groups. Birkhäuser Progress in mathematics 196 (2001), 47-57.

projects

lectures

seminars

phd students

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