Richard Rimanyi 

Title: Some new directions in enumerative algebraic geometry

Abstract:
An effective tool in algebraic geometry is associating characteristic classes to subvarieties 
of a smooth ambient space. A classical example is associating the so-called cohomological 
Schubert classes to the Schubert varieties of a Grassmannian. In the talk I will argue for 
generalizations of this example in many directions. The cohomology theory will be generalized 
to K theory, and further to elliptic cohomology. The usual "fundamental class" characteristic 
class will be $\hbar$-deformed. The usual setting of homogeneous spaces (e.g. Grassmannians, 
flag manifolds) will be generalized to Nakajima quiver varieties, and further to Cherkis bow 
varieties. As a result, relations will be established to representation theory (quantum integrable 
systems) and superstring theory (3d mirror symmetry).