Global motivic integration and the cohomology of moduli spaces

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Algebraic geometry, at the crudest level, is the study of the geometry of solutions of polynomial equations in many variables. In order to proceed one must enlarge the collection of solution spaces to a larger collection consisting of spaces that are algebraic gluings of such spaces. We will refer to such spaces as varieties. In the 1950's the Grothendieck school taught us that it is also important to remember the defining equations. The geometry of such solution spaces is usually studied by associating to the space an algebraic invariant. In 1995, inspired by ideas from calculus and number theory, M. Kontsevich proved some conjectures pertaining to some invariants associated to a special class of varieities. The theory of motivic integration was born. The theory has been developed over the last ten years most notably by J. Denef and  F. Loeser. The aim of this research is to further extend this theory. The second part of this proposal aims to study some invariants of the a special variety, the so called moduli space of semistable vector bundles over an algebraic curve.