Georgia Institute of Technology College of Sciences

This is joint work with Aaron Landesman. There are a number of difficult open questions around representations of free and surface groups, which it turns out are accessible to methods from Hodge theory and arithmetic geometry. For example, I'll discuss applications of these methods to the following concrete theorem about surface groups, whose proof relies on non-abelian Hodge theory and the Langlands program:

Theorem. Let $\rho: \pi_1(\Sigma_{g,n})\to GL_r(\mathbb{C})$ be a representation of the fundamental group of a compact orientable surface of genus $g$ with $n$ punctures, with $r<\sqrt{g+1}$. If the conjugacy class of $\rho$ has finite orbit under the mapping class group of $\Sigma_{g,n}$, then $\rho$ has finite image.

This answers a question of Peter Whang. I'll also discuss closely related applications to the Putman-Wieland conjecture on homological representations of mapping class groups. 

For polynomials in 1 variable, Matt Baker and Oliver Lorschied were able to connect results about roots of polynomials over valued
fields (Newton polygons) and over real fields (Descartes's rule) by looking at factorization of polynomials over the tropical and signed
hyperfields respectively. In this talk, I will describe some ongoing work with Andreas Gross about extending these ideas to two or more
variables. Our main tool is the use of resultants to transform questions about 0-dimensional systems of equations to factoring a single
homogeneous polynomial.

Projective space, rational maps, and other notions from algebraic geometry appear naturally in the study of image formation and various camera models in computer vision. Considerable attention has been paid to multiview ideals, which collect all polynomial constraints on images that must be satisfied by a given camera arrangement. We extend past work on multiview ideals to settings where the camera arrangement is unknown. We characterize various "image formation ideals", which are interesting objects in their own right. Some nice previous results about multiview ideals also fall out from our framework. We give a new proof of a result by Aholt, Sturmfels, and Thomas that the multiview ideal has a universal Groebner basis consisting of k-focals (also known as k-linearities in the vision literature) for k in {2,3,4}. (Preliminary report based on ongoing joint projects with Sameer Agarwal, Max Lieblich, Jessie Loucks Tavitas, and Rekha Thomas.)

Given an affine space of matrices L and a matrix Θ ∈ L, consider the problem of computing the closest rank deficient matrix to Θ on L with respect to the Frobenius norm. This is a nonconvex problem with several applications in control theory, computer algebra, and computer vision. We introduce a novel semidefinite programming (SDP) relaxation, and prove that it always gives the global minimizer of the nonconvex problem in the low noise regime, i.e., when Θ is close to be rank deficient. Our SDP is the first convex relaxation for this problem with provable guarantees. We evaluate the performance of our SDP relaxation in examples from system identification, approximate GCD, triangulation, and camera resectioning. Our relaxation reliably obtains the global minimizer under non-adversarial noise, and its noise tolerance is significantly better than state of the art methods.