Hans U. Boden

MODULI SPACES OF PARABOLIC HIGGS BUNDLES AND PARABOLIC K(D) PAIRS OVER SMOOTH CURVES: I

This paper concerns the moduli spaces of rank-two parabolic Higgs bundles and parabolic K(D) pairs over a smooth curve. Precisely which parabolic bundles occur in stable K(D) pairs and stable Higgs bundles is determined. Using Morse theory, the moduli space of parabolic Higgs bundles is shown to be a non-compact, connected, simply connected manifold, and a computation of its Poincare polynomial is given.

Representations of orbifold groups and parabolic bundles

6 Applications 32 6.1 Equivariant cohomology . . . . . . . . . . . . . . . . . . . . . . . . . 32 6.2 The filtration on C . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 6.3 The gauge groups G and P . . . . . . . . . . . . . . . . . . . . . . . 34 6.4 The equivariant cohomology of Css . . . . . . . . . . . . . . . . . . . 35 6.5 The cohomology of S in the case Css = Cs . . . . . . . . . . . . . . . 37 6.6 Results for genus 0 and 1 . . . . . . . . . . . . . . . . . . . . . . . . . 39 6.7 Relationship between S and R(Σ). . . . . . . . . . . . . . . . . . . . 42 6.8 Explicit computations . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Rationality of Moduli Spaces of Parabolic Bundles

The moduli space of parabolic bundles with fixed determinant over a smooth curve of genus greater than one is proved to be rational whenever one of the multiplicities associated to the quasi-parabolic structure is equal to one. It follows that if rank and degree are coprime, the moduli space of vector bundles is stably rational, and the bound obtained on the level is strong enough to conclude rationality in many cases.

Gauge theoretic invariants of Dehn surgeries on knots

New methods for computing a variety of gauge theoretic invariants for homology 3{spheres are developed. These invariants include the Chern{Simons invariants, the spectral flow of the odd signature operator, and the rho invariants of irreducible SU(2) representations. These quantities are calculated for flat SU(2) connections on homology 3{spheres obtained by 1=k Dehn surgery on (2;q) torus knots. The methods are then applied to compute the SU(3) gauge theoretic Casson invariant (introduced in [5]) for Dehn surgeries on (2;q ) torus knots for q =3 ; 5 ; 7a nd 9.

Unitary representations of Brieskorn spheres

In this article, we commence an investigation of the SU(N) representation space of Seifert fibered homology spheres Σ(a1, . . . , an). Under mild assumptions (e.g. if N is prime), then Theorem 3.1 implies that any closed connected component of irreducible SU(N) representations of Σ(a1, . . . , an) is homeomorphic to a component of SU(N) representations of an associated genus zero Fuchsian group. The latter representation spaces can be studied using the general correspondence between representations of Fuchsian groups and the moduli of parabolic bundles given by Mehta and Seshadri. For example, the inductive procedure of Atiyah-Bott-Nitsure determines the cohomology of this moduli space and it follows that the odd dimensional cohomology groups of any component of irreducible SU(N) representations of Σ(a1, . . . , an) vanish. In particular, any irreducible component of the SU(3) representation space of a Brieskorn spheres Σ(p, q, r) is either a point or a two sphere. By repeated application of the inductive procedure, the precise number of points and two spheres in this representation space is determined. Specific results for the Brieskorn spheres with p = 2 are given, where the representation space is a collection of points. In the last section, the SU(N) spectral flow of irreducible representations of Seifert fibered homology spheres is shown to be even. This gives a calculation of the leading term in a gauge-theoretic definition of the generalized Casson invariants.

Chern-Simons gauge theory : 20 years after

In 1989, Edward Witten discovered a deep relationship between quantum field theory and knot theory, and this beautiful discovery created a new field of research called Chern-Simons theory. This field has the remarkable feature of intertwining a large number of diverse branches of research in mathematics and physics, among them low-dimensional topology, differential geometry, quantum algebra, functional and stochastic analysis, quantum gravity, and string theory. The 20-year anniversary of Wittens discovery provided an opportunity to bring together researchers working in Chern-Simons theory for a meeting, and the resulting conference, which took place during the summer of 2009 at the Max Planck Institute for Mathematics in Bonn, included many of the leading experts in the field. This volume documents the activities of the conference and presents several original research articles, including another monumental paper by Witten that is sure to stimulate further activity in this and related fields. This collection will provide an excellent overview of the current research directions and recent progress in Chern-Simons gauge theory.

Alexander invariants for virtual knots

Given a virtual knot

METABELIAN SL(N, ℂ) REPRESENTATIONS OF KNOT GROUPS, III: DEFORMATIONS

K

The SL(2,C) Casson invariant for Dehn surgeries on two-bridge knots

, we construct a group

Virtual knot groups and almost classical knots

VG_K