The traveling salesman problem is one of the best-known examples of an algorithmically hard problem, but what does that mean formally? It turns out that a solution to this problem would immediately give a solution to any other NP problem, and in this sense we say it is "NP-complete." In this talk, we will give a more formal definition of what it means for a problem to be NP-complete, develop the machinery needed to prove NP-completeness, and then use this machinery to prove that the traveling salesman problem (and a few others) is indeed NP-complete.

The Springer resolution of the nilpotent cone of a semisimple Lie algebra has important

applications in representation theory, and in particular was used by Springer to give a geometric construction of the irreducible representations of Weyl groups.  This talk concerns a generalization of the Springer resolution constructed with the use of toric varieties.  We will discuss how this is connected in type A with Lusztig's generalized Springer correspondence, as well as an analogue of an affine paving of the fibers.  Part of this talk is joint work with Martha Precup and Amber Russell.