# Computational Algebraic Geometry and Applications to Computer Vision

Many models in science and engineering are described by polynomials. Computational algebraic geometry gives tools to analyze and exploit algebraic structure. In this talk, we offer a user-friendly introduction to some of these notions, including dimension (formalizing degrees of freedom), degree (formalizing the number of solutions to a polynomial system) and 0-1 laws in algebraic geometry (solution sets to polynomial systems exhibit similar behavior for all but a measure 0 subset of problem instances). We will also mention algorithms, based on Gröbner bases (symbolic techniques) and homotopy continuation (numerical techniques).

Applied examples are drawn from the structure-from-motion problem in computer vision, where the task of building a 3D model from multiple 2D images leads to nontrivial polynomial systems.

References include:

J. Kileel, Minimal Problems for the Calibrated Trifocal Variety, SIAM Journal on Applied Algebra and Geometry 1 (2017) 575-598.

J. Kileel, Z. Kukelova, T. Pajdla and B. Sturmfels, Distortion Varieties, Foundations of Computational Mathematics, first online.

*Joe Kileel is a postdoctoral research associate at Princeton’s Program in Applied and Computational Mathematics, and a member of the Simons Collaboration on Algorithms and Geometry. He received his PhD in May 2017 from UC Berkeley, advised by Bernd Sturmfels, where his thesis was awarded the Bernard Friedman Memorial Prize for best in applied mathematics. Joe's current publication list is available at http://web.math.princeton.edu/~jkileel/. *