I work in geometric representation theory, and love all things vaguely geometric. I spend most of my time thinking about:
- perverse sheaves and the decomposition theorem,
- modular representation theory, particularly from a
geometric perspective, torsion in cohomology,
- weights (e.g. Frobenius action and Weil conjectures or Hodge
theory) and especially their interactions with perverse sheaves,
- Soergel bimodules, their Hodge theory and relations to
- combinatorial models for perverse sheaves (e.g. Soergel bimodules,
moment graphs, "intersection cohomology" of polytopes etc.),
- algebra that looks geometric
(e.g. the coinvariant algebra of H3),
- categorification, diagrammatic algebra, "generators and relations",
- higher categories (usually stopping at 2 or 3!), especially monoidal categories and their module categories,
- Kazhdan-Lusztig theory and its modular versions
(p-canonical basis etc.)
- braid group actions and link homology,
- microlocal approaches to perverse sheaves, characteristic cycles.
- derived equivalence (in representation theory, mirror symmetry, ...),
- Deligne-Lusztig theory and Broué's conjecture,
- Langlands philosophy, particularly geometric Langlands and mod p versions,
- real reductive Lie groups and their representation theory,
- character sheaves, characters of finite groups of Lie
- topology of algebraic varieties, singularity theory,
- algebraic geometry, especially interactions with complex
geometry and Hodge theory, period domains etc.,
- knot theory, low dimensional topology.
- The decomposition theorem and the
topology of algebraic maps
Notes from lectures given by Luca Migliorini in Freiburg in February, 2010.
- Mirror symmetry, Langlands duality and
the Hitchin system
Notes from lectures given by Tamas Hausel in Oxford, Hilary Term, 2010. (Notes by Greg Berczi, Michael Groechenig and myself.)
- Six lectures on Deligne-Lusztig theory
Notes from lectures given by Raphael Rouquier in Oxford, Hilary Term, 2010. (Notes from Lecture 5 by David Craven.)
- Character sheaves, tensor categories and
non-abelian Fourier transform
Notes from lectures given by Victor Ostrik in Luminy, September 2010. 2010.
- Koszul duality and applications in representation theory
Notes from lectures given by Wolfgang Soergel in Luminy, September 2010.
Here are some passages I enjoy.