Commutative Algebra and Algebraic Geometry
Organizer: David Eisenbud
Date: Tuesday, November 13
3:45: Luke Oeding (Berkeley): Eigenvectors of tensors and Waring decomposition
Waring’s problem for polynomials is to write a given polynomial as a minimal sum of powers of linear forms. The minimal number of summands required in a Waring decomposition (the Waring rank) is related to secant varieties. I will explain recent work of Landsberg and Ottaviani that unified and generalized many constructions for equations of secant varieties via vector bundle techniques. With Ottaviani we have turned this construction into effective algorithms to actually find the Waring decomposition of a polynomial (provided the Waring rank is below a certain bound). Our algorithms generalize Sylvester’s algorithm for binary forms, using an essential new ingredient – eigenvectors of tensors. Of course a naive algorithm always exists, but is rarely effective. I will explain how computations using linear algebra make our algorithms effective. Given time, I will demonstrate our Macaulay2 implementations.
5:00: Grigory Mikhalkin: Tropical curves and their phases: from face to edge models
We look at the constructions of embedded and immersed real algebraic curves in the plane. We survey and compare patchworking and tropical construction. As an example we consider topological classification of rational quintics in RP2. The first half of the talk will contain a short introduction to tropical curves and their phases.