Hochschild cohomology, non-commutative deformations and mirror symmetry (HHNCDMIR)

Our research programme addresses several interesting current issues in non-commutative algebraic geometry, and important links with symplectic geometry and algebraic topology. Non-commutative algebraic geometry is concerned with the study of algebraic objects in geometric ways.

One of the basic philosophies is that, in analogy with (derived) categories of (quasi-)coherent sheaves over schemes and (derived) module categories, non-commutative spaces can be represented by suitable abelian or triangulated categories. This point of view has proven extremely useful in non-commutative algebra, algebraic geometry and more recently in string theory thanks to the Homological Mirror Symmetry conjecture.

One of our main aims is to set up a deformation framework for non-commutative spaces represented by "enhanced" triangulated categories, encompassing both the noncommutative schemes represented by derived abelian categories and the derived-affine spaces, represented by dg algebras. This framework should clarify and resolve some of the important problems known to exist in the deformation theory of derived-affine spaces. It should moreover be applicable to Fukaya-type categories, and yield a new way of proving and interpreting instances of "deformed mirror symmetry".

This theory will be developed in interaction with concrete applications of the abelian deformation theory developed in our earlier work, and with the development of new decomposition and comparison techniques for Hochschild cohomology.

By understanding the links between the different theories and fields of application, we aim to achieve an interdisciplinary understanding of non-commutative spaces using abelian and triangulated structures.