Brylinski J-L. Brylinski will continue his work on the geometric structures occurring in mathematical physics, particularly in gauge theory, loop spaces, and moduli spaces of vector bundles. Gauge groups are of fundamental importance in modern geometry. The gauge groups on a circle are loop groups, and gauge groups in dimension 2 are closely related to conformal field theory. Brylinski will study certain cohomology classes he has constructed for gauge groups, which are natural generalizations of the degree-two cohomology class giving the central extension of a loop group. These classes are obtained by a transgression process from the Chern-Simons classes and the Chern classes of Beilinson. The first new example is that of a degree-three cohomology class in the gauge group of a closed surface. Brylinski has shown that these classes obey reciprocity laws that generalize those of conformal field theory. He will investigate to what extent they are related to higher-dimensional field theories. He will study the Kaehler geometry of moduli spaces of vector bundles, continuing his work with P. Foth, and intends to study the geometric quantization of these moduli spaces. Brylinski will also continue his work on the Quillen metric on determinant line bundles, which should lead to methods to compute it geometrically, as opposed to analytically, in many situations. Here he will use both the differential geometry of gerbes that he developed and a variant of Deligne cohomology he introduced, which incorporates hermitian metrics. Brylinski will continue his investigation of the differential geometry of the space of knots in a smooth three-manifold, which he previously showed is a Kaehler manifold, more precisely a union of coadjoint orbits of the group of unimodular diffeomorphisms. A particular object of study will be the structure of various Lie algebras of functionals with respect to the Poisson bracket. The research of J.-L. Brylinski is concerned with the interface between geometry and mathematical physics. This interface has been growing steadily in the recent past and has had profound consequences for various branches of geometry: algebraic, differential, and symplectic, among others. For instance, Edward Witten (School of Natural Sciences, Institute for Advanced Study) interpreted the Jones polynomial for knots in terms of field theory in three dimensions connected to the Chern-Simons class. Also, Verlinde derived a formula for Riemann-Roch numbers of a line bundle over a moduli space of bundles over a Riemann surface, using ideas from conformal field theory. All these facts point to the need for new types of geometry that include generalizations of the concept of a vector bundle or a principal bundle. For this purpose, Brylinski has developed a theory of the differential geometry of groupoids and gerbes, which he is applying to the study of gauge groups, moduli spaces of vector bundles, Quillen line bundles, and the space of knots. Although the basic concepts he employs are of an abstract nature, they can often lead to concrete formulas. For instance, he has obtained a concrete description of Quillen metrics on some geometric determinant line bundles connected with line bundles over Riemann surfaces. This research is expected to result in a better understanding of the geometric underpinnings of mathematical physics. ***
