UC Berkeley

Commutative Algebra and Algebraic Geometry Seminar

September 13, 2016

939 Evans Hall

3:45PM: Cameras and their Distortions

Bernd Sturmfels

We discuss two current projects at the interface of computer vision and algebraic geometry. Work with Joe Kileel, Zuzana Kukelov and Tomas Pajdla introduces the distortion varieties of a given projective variety. These are parametrized by duplicating coordinates and multiplying them with monomials. We study their degrees and defining equations. Exact formulas are obtained for the case of one-parameter distortions, the case of most interest for modeling cameras with image distortion. Work with Jean Ponce and Matthew Trager develops multi-view geometry for algebraic cameras that are represented by congruences in the Grassmannian of lines in 3-space.

5:00PM: Affine-compact functors

Joseph Gubeladze

Methods and techniques from algebraic topology and geomerty are often used to solve important open problems in combinatorics. A natural question is whether the category of polytopes and affine maps, the vehicle of geometric and algebraic combinatorics, can be subjected to a categorial/homological analysis. Several well known polytopal constuctions are examined from the functorial point of view, with emphasis on representable functors. The constructions include the hom-polytope, tensor polytope, Minkowski sum, Billera-Sturmfels fiber polytope, cokernel polytopes, and the Mond-Smith-van Straten space of sandwiched simplices.

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