My research is in the areas of algebra, algebraic geometry and category theory. These areas are the source of powerful conceptual tools for understanding the concept of space in a wide sense, from very classical to very abstract.
For example, the classical subject of hypergeometric functions was developed (in my joint works with I.M. Gelfand and A.V. Zelevinsky) to include period integrals of algebraic hypersurfaces in toric varieties. This led us to the discovery of secondary polytopes, which are combinatorial geometric objects governing both hypergeometric functions and discriminant polynomials in singularity theory. These concepts are now widely used in Mirror Symmetry.
Other directions of algebraic geometry which I am interested in (and have worked) include non- commutative geometry (study of the neighborhood of the commutative regime), derived and infinite- dimensional geometry (algebro-geometric study of spaces of formal loops and paths).
Category theory provides a unified background for all these areas. In addition, various flavors of category theory (triangulated categories, higher categories, theory of operads) lead to contexts where algebraic expressions themselves become objects with non- trivial geometric structure, not really expressible on a (1-dimensional) line. This additional interface of algebra and geometry seems necessary in order to approach truly higher-dimensional problems.
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