Wednesday, February 19 at 3:30pm to 4:30pm
Kaprielian Hall (KAP), 414
3620 South Vermont Avenue, Los Angeles, CA 90089
Allen Knutson, Cornell University
Abstract: How many points in the plane lie on two generic lines? One. How many lines in 3-space intersect four generic lines? Two. Hilbert's 15th problem was to put calculations of such numbers (via principles of Schubert) onto a rigorous footing, and motivated the definition of cohomology ring (here, of the "Grassmannian" of lines in 3-space).
I'll talk about many generalizations of this original problem -- going beyond cohomology to equivariant cohomology, quantum cohomology, K-theory, or any combination; also going
beyond Grassmannians to other moduli spaces -- all of which have answers that are easy to compute, but only a few of which have manifestly positive formulae (some very recent). Parts of this talk are joint work with Terry Tao, Paul Zinn-Justin, and Iva Halacheva.