3:45: Geometric Littlewood-Richardson rules (equivariant cohomology, K-theory, and flag varieties): degenerations and matroid shiftingRavi VakilLittlewood-Richardson numbers have many interpretations, including as structure coefficients of the cohomology rings of Grassmannians. The geometric Littlewood-Richardson rule is a direct geometric interpretation of these coefficients by degenerations. Allen Knutson and I extend this rule by reinterpretating the degenerations simply in terms of linear equations in Plucker coordinates. Then degenerations correspond to shifting of matroids (a notion from extremal combinatorics), and this turns out to be a fruitful way to package the subtle geometry of the degenerations. This allows us to give an equivariant geometric interpretation of the puzzles of Knutson and Tao, to prove a conjectural rule in K-theory, to conjecture a rule in equivariant K-theory, and hopefully to tackle the general case of the flag variety (which might have interesting interpretations in terms of Coskun's Mondrian tableaux). I will work through an example illustrating the key idea of shifting in action. This is work in progress. |
PDF notes (of non-slide-based portion of talk), courtesy of Bjorn Poonen
5:00: Castelnuovo-Mumford regularity and linkageBernd UlrichWe give bounds on the Castelnuovo-Mumford regularity of varieties in projective space in terms of the degrees of their defining equations. This generalizes work by Bertram, Ein and Lazarsfeld, who had treated the case of smooth varieties in characteristic zero. Instead, we allow mild singularities and have no assumption on the characteristic. Our proofs use linkage, Kodaira Vanishing, and a result by Karen Smith in positive characteristic.This is joint work with Marc Chardin. |
PDF notes, courtesy of Bjorn Poonen