Algebra, geometry and applications

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I am interested in the interplay between geometry and algebra and their applications to representation theory, number theory and mathematical physics. In my work I use the language of algebraic geometry and the methods of homological algebra. Modern algebraic geometry is a branch of mathematics which vastly generalizes our geometric intuition to encompass seemingly purely algebraic problems. On the other hand homological algebra can be viewed as a tool designed to extract, in an algebraic form, essential information from geometric and also algebraic mathematical objects in the guise of invariants. Taken together they provide very powerful techniques for dealing with a wide variety of problems. Namely they allow the different branches of mathematics: geometry and algebra, to interact and complement each other. Thus a question in one area may be solved by transforming it into a question in another area where a previously intractable problem may become very accessible to the methods already available.