Hochschild theory in algebraic geometry

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There are two themes to this project. The first is the study ofHochschild structures that arise naturally in algebraic geometry.Hochschild homology and cohomology are essential invariants in bothalgebra and geometry, and the PI proposes to study them from analgebraic-geometric point of view. The second is the study ofexplicit example of spaces which have the extra structure of {\emgenus one fibration}. These spaces are ubiquitous in classical andmodern algebraic geometry, and the PI expects that by using thetechniques of derived categories new insights can be gained into theirgeometry.There are three projects contained in the present proposal.The first project is concerned with finding explicit formulas for themultiplication in the Hochschild cohomology ring of orbifolds, withapplications to the Ruan conjecture. The second aims at understandingthe 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 withmulti-sections of small degree. It has applications to the study ofstable singularities in string theory, as well as to the understandingof the Torelli problem for Calabi-Yau threefolds.Algebraic geometry, which is the geometric study of solutions ofpolynomial equations, has seen in the last few years majordevelopments, especially in terms of its applications in other fieldsof science. Of particular importance are applications to moderntheoretical physics, in particular in string theory. The presentproject will increase our general understanding of the geometry ofsome of the spaces that are important in algebraic geometry andphysics.