Seminar

Intersection theory grew out of very basic questions like: "given four generic lines in \mathbb{P}^3 - how many lines intersect all four of them?" Questions like that are one of the simplest examples of an application of Schubert Calculus, a very important tool in combinatorial representation theory. It is used to describe cohomology rings, but also to construct categories which describe representation theoretic problems geometrically. It serves as the basis of more sophisticated methods of enumerative geometry, like Gromov-Witten theory and Quantum Cohomology. An amazing fact is that the same combinatorics also occurs when decomposing tensor products of representations of a semisimple complex Lie algebra.

This talk will describe some of these basic ideas and use them to explain a connection between quantum fusion products and quantum cohomology, relating Verlinde algebras and quantum cohomology rings. All this is related to questions arising in commutative and noncommutative algebraic geometry, integrable systems, representation theory, combinatorics ...