3:45PM: Projective duality and matrix Schubert varieties
Alen KnutsonIf X is the cone over a projective variety PX in PV, then its conormal variety in V x V^* projects to the cone in V^* over the "projective dual" of PX. If G acts on V with finitely many orbits, this gives a very tricky bijection between G-orbits on V and on V^*. I'll explain this bijection in the case of B_+ x B_- orbits acting on V = matrix space, whose orbit closures are "matrix Schubert varieties". Generalizing the definition slightly, I'll recall the pipe dream degeneration of matrix Schubert varieties, and state some conjectures about the corresponding degenerations of their conormal varieties. In brief, the nil Hecke algebra behind ordinary pipe dreams is replaced by the Temperley-Lieb algebra for a new kind of pipe dream.