Starting with the classical theory of first integrals I will talk about the notion of weak first integral and how to find implicit solutions of quasi-linear systems of first order PDEs. In determined cases this method has applications to determining the stability and affine control of the population dynamics of Volterra-Kolmogorov type.

I will introduce a line of work that aims to characterize the set of all valid algebraic constraints that relate any number of perspective cameras, 3D points, and their 2D projections. More formally, this framework involves the study of certain multigraded vanishing ideals. This leads to several new results, as well as new proofs of old results about the well-studied multiview ideal. For example, a "folklore theorem" from geometric computer vision states: "all algebraic constraints on the 2D projections of 3D points can be obtained from those involving 2, 3, or 4 cameras." (joint w/ S. Agarwal, E. Connelly, J. Loucks-Tavitas, R. Thomas) I will also discuss a complementary line of work focused on practical estimation methods. Incremental 3D reconstruction systems usually focus on estimating the relative orientation of two cameras. This in turn requires solving systems of algebraic equations with (very) special structure. I will describe recent progress extending the domain of such solvers to problems involving three or four cameras, including non-perspective cameras with lens distortion. The key players are numerical homotopy continuation methods, and the Galois/monodromy groups that capture their inherent complexity. (joint w/ P. Hruby, K. Kohn, V. Korotynskiy, V. Larsson, A. Leykin, L. Oeding, T. Pajdla, M. Pollefeys, M. Regan)

Using tools from the enriched enumerative geometry program, Larson and Vogt gave a signed count for the number of real bitangents to a plane quartic. This signed count depends on the choice of an auxiliary “orienting” line. Larson and Vogt proved that the signed count is always non-negative and conjectured an upper bound of 8. Mario Kummer and I proved this conjecture using some real algebraic geometry and a few basic facts about quadrilaterals. In this talk, I will explain the background leading up to Larson and Vogt’s conjecture, as well as my joint work with Kummer.

TBA