Schedule
(LNA=LECTURE NOTES AVAILABLE)News
About the seminar
This is a working seminar open to anyone interested. Lectures are two hours long, with a 10 minutes break and the first introductory part is always aimed at a wider audience. The aim would be to read research articles in subjects explained below, all related to the " right" compactifications of various types of open manifolds. "Right" means that certain questions of global analysis on the complete, smooth but open manifold could be understood by analysis, in nice cases topology, of the (usually singular) compactification. The main focus will be to understand the asymptotics of the metric at infinity and then deduce the "right" compactification for our manifolds. Among them are hyperkähler manifolds: like quiver varieties, moduli spaces of instantons, monopoles and Higgs bundles. The compactifications which may be treated are: Borel-Satake and various other compactifications of locally symmetric spaces, Uhlenbeck compactifications of moduli of instantons, Tian-Yau compactifcations of Ricci-flat manifolds, Morgan-Shalen compactification of character varieties, Thurston compactification of Teichmüller space, Martin compactifications of Riemannian manifolds, Lerman's symplectic cutting etc. Below you can find a list of a few problems of global analysis on certain open manifolds which may be attacked by finding their "right" compactifications. The problems come from string theory and low dimensional topology.
The problems
1. Calculate the L^2 cohomology of moduli of a: instantons (or more generally Nakajima's quiver varieties); b: monopoles and c: Higgs bundles! These problems are motivated by S-duality arguments in string theory. There are predictions of the L^2 cohomology of these moduli spaces in the literature. a: hep-th/9402032; b: hep-th/9408074; c: math.AG/9805071. In general the Zucker conjecture, proven in 90a:32044 and 91m:14027, may give the righ point of view. It claims that the L^2 cohomology of certain locally symmetric spaces is isomorphic with the intersection cohomology of its Borel-Satake compactification. Our question thus is: Is there a compactification of the above hyperkähler moduli spaces whose intersection cohomology agrees with the L^2 cohomology of the original space? Related recent papers: math.DG/9909002
2. Calculate the SU(n) Casson invariant of a three manifold! More generally: define and calculate the Rozansky-Witten invariants of complete but open hyperkähler manifolds! Even more generally calculate intersection numbers on these hyperkähler manifolds! According to a conjecture in hep-th/9612216 the SU(n) Casson invariants of a three manifold should be given by the Rozansky-Witten invariants of the moduli space of SU(2) magnetic monopoles on R^3. However these invariants are given in terms of integrals of various expressions involving the curvature tensor of these manifolds. In order to calculate these integrals on these open manifolds one way of attack is to try to compactify them and somehow calculate the integrals on the compactification. In general there is an indication in hep-th/9712241 how to make sense and calculate integrals on non-compact complete hyperkähler manifolds like the moduli of instantons, monopoles and Higgs bundles. Related recent papers: math.DG/0112210
3. Calculate the SL(2,C) Casson invariant of a three manifold! Immitating the construction of the SU(2) Casson invariant, one is hoping to get SL(2,C) Casson invariants by intersecting Lagrangian subvarieties in the SL(2,C) representation space of a closed Riemann surface. To do the intersection theory in the SL(2,C) case a new problem arises: this space (which is analytically just the hyperkähler moduli of SU(2) Higgs bundles in another complex structure) is non-compact. To solve this problem one may want to compactify the space and do the intersection theory in the compactification. There are various compactifications one can consider like alg-geom/9611008, math.AG/9804083 and math.DG/9810034 . Related recent papers: 1851563
If you are interested and/or have questions or suggestions please contact one of the organizers: