Commutative Algebra and Algebraic Geometry
Tuesdays, 3:45-6pm in Evans 939
Organizer: David Eisenbud
Date: October 23
3:45 Sam Payne: Tropicalization of the moduli space of curves
Tropical geometry allows a systematic study of algebraic curves over valued fields in terms of the marked dual graphs of special fibers of models of the curve over the valuation ring. In the past several years, a number of researchers, including Caporaso, Gathmann, Kozlov, Mikhalkin, and their collaborators, have introduced and studied moduli spaces for these marked graphs, which are often called tropical curves, and estabilshed various analogies to moduli spaces of curves. I will present work that explains and extends these analogies, canonically and functorially, by applying a new generalized tropicalization map to the Deligne-Mumford compactification of the moduli space of stable curves. Berkovich spaces appear in the construction of this new tropicalization map in a natural and elementary way, but no tropical or nonarchimedean analytic background is assumed. This is joint work with D. Abramovich and L. Caporaso.}
5:00 Bernd Sturmfels: A Hilbert Scheme in Computer Vision
Multiview geometry is the study of two-dimensional images of three-dimensional scenes, a foundational subject in computer vision. We determine a universal Groebner basis for the multiview ideal of n generic cameras. As the cameras move, the multiview varieties vary in a family of dimension 11n-15. This family is the distinguished component of a multigraded Hilbert scheme with a unique Borel-fixed point. We present a combinatorial study of ideals lying on that Hilbert scheme. This is joint work with Chris Aholt and Rekha Thomas.