For any finite graph I define certain algebras, including deformed Hecke type ones. All these algebras contain commutative subalgebras generated by either additive or multiplicative, or elliptic Dunkl elements. I'm planing to explain why these commutative subalgebras can be identified with classical and (small) quantum cohomology (and that of K-theory) of certain graph varieties, including partial flag and Hessenberg varieties.

This talk will take place at RIMS 006.