|Location:||UC Berkeley, 60 Evans Hall|
I will discuss a combinatorial problem which comes from algebraic geometry. The problem, in general, is to show that two theories for "counting" curves in a complex three-dimensional space X (Pandharipande–Thomas theory and reduced Donaldson–Thomas theory) give the same answer. I will prove a combinatorial version of this correspondence in a special case (X is toric Calabi–Yau), where the difficult geometry reduces to a study of the ``topological vertex\'\' (a certain generating function) in these two theories. The combinatorial objects in question are plane partitions, perfect matchings on the honeycomb lattice and related structures.
There will be many pictures. This is a combinatorics talk, so no algebraic geometry will be used once I explain where the problem is coming from.