## Columbia Mathematics Department Colloquium

Picard-Lefschetz theory

and hidden group actions

and hidden group actions

by

# Paul Seidel

# (MIT)

Abstract:

Classical Picard-Lefschetz theory (from about 1920) provides a way of computing the

homology of affine algebraic varieties by dimensional induction. It

determines not just the homology groups themselves, but also the intersection pairing.

Recent advances in symplectic topology indicate that a certain class of affine varieties admits hidden symmetries (not realized by geometric group actions), which

allows one to define q-deformed intersection pairings. We adapt the framework of

Picard-Lefschetz theory to construct and compute such pairings.

homology of affine algebraic varieties by dimensional induction. It

determines not just the homology groups themselves, but also the intersection pairing.

Recent advances in symplectic topology indicate that a certain class of affine varieties admits hidden symmetries (not realized by geometric group actions), which

allows one to define q-deformed intersection pairings. We adapt the framework of

Picard-Lefschetz theory to construct and compute such pairings.