I work mostly in geometric representation theory and in noncommutative geometry.
Geometric representation theory tries to apply the methods of algebraic geometry for studying representations of various algebras important from the representation theoretic perspective. Typical examples include:
- Classification of irreducible representations of Hecke algebras (Deligne-Langlands-Lusztig conjecture) in terms of K-theory and perverse sheaves;
- Applications of D-modules and perverse sheaves to representations of complex or real reductive groups and to semisimple Lie algebras (Kazhdan-Lusztig conjecture);
- The study of integrable representations of quantum groups using the geometry of quiver varieties (Nakajima);
- Geometric Langlands program.
To get more details I suggest to look at the Intro in our book: Chriss-Ginzburg, Representation Theory and Complex Geometry (Birkhauser Boston, 1997), or at my survey article Geometric Methods in Representation Theory of Hecke Algebras and Quantum Groups.
During the last 5-10 years, I've also got interested in what may be called noncommutative geometry. Some of the inspiration comes from the theory of quivers (I teach a course on quivers quite frequently). Another source of inspiration comes from Mirror symmetry (e.g., Calaby-Yau categories) and, more generally, from the mathematics appearing in string theory. To get a rough idea of what I mean, you may want to look at the following papers:
- V. Ginzburg, Lectures on Noncommutative Geometry
- V. Ginzburg, Non-commutative Symplectic Geometry, Quiver varieties, and Operads, Math. Res. Lett. 8 (2001), no. 3, 377-400
- P. Etingof, V. Ginzburg, Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism, Invent. Math. 147 (2002), no. 2, 243-348
I have 7 graduate students at the moment; all of them choose their own favorite topic for research, not necessarily directly related to what I'm doing myself. However, I do have joint projects with some of my students.