There are two themes to this project. The first is the study of Hochschild structures that arise naturally in algebraic geometry. Hochschild homology and cohomology are essential invariants in both algebra and geometry, and the PI proposes to study them from an algebraic-geometric point of view. The second is the study of explicit example of spaces which have the extra structure of {em genus one fibration}. These spaces are ubiquitous in classical and modern algebraic geometry, and the PI expects that by using the techniques of derived categories new insights can be gained into their geometry.There are three projects contained in the present proposal. The first project is concerned with finding explicit formulas for the multiplication in the Hochschild cohomology ring of orbifolds, with applications to the Ruan conjecture. The second aims at understanding the abstract Mukai pairing on the Hochschild homology of smooth varieties, and its relationship to the Poincar'e pairing on differential forms. This would provide a new perspective on the Riemann-Roch theorem. The third project is a study of genus one fibrations with multi-sections of small degree. It has applications to the study of stable singularities in string theory, as well as to the understanding of the Torelli problem for Calabi-Yau threefolds.
Algebraic geometry, which is the geometric study of solutions of polynomial equations, has seen in the last few years major developments, especially in terms of its applications in other fields of science. Of particular importance are applications to modern theoretical physics, in particular in string theory. The present project will increase our general understanding of the geometry of some of the spaces that are important in algebraic geometry and physics.
