
Date Time 
Location  Speaker 
Title – click for abstract 

09/08 4:00pm 
BLOC 302 
Georgy Scholten Sorbonne Université 
Global Optimization of Analytic Functions over Compact Domains
In this talk, we introduce a new method for minimizing analytic Morse functions over compact domains through the use of polynomial approximations. This is, in essence, an effective application of the StoneWeierstrass Theorem, as we seek to extend a local method to a global setting, through the construction of polynomial approximants satisfying an arbitrary set precision in Linfty norm. The critical points of the polynomial approximant are computed exactly, using methods from computer algebra. Our Main Theorem states probabilistic conditions for capturing all local minima of the objective function $f$ over the compact domain. We present a probabilistic method, iterative on the degree, to construct the lowest degree possible leastsquares polynomial approximants of f which attains a desired precision over the domain. We then compute the critical points of the approximant and initialize local minimization methods on the objective function f at these points, in order to recover the totality of the local minima of f over the domain. 

09/15 4:00pm 
BLOC 302 
ChongKyu Han Seoul National University 
First integrals, stability of dynamics, and affine control with prescribed invariant subsets
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 quasilinear systems of first order PDEs. In determined cases this method has applications to determining the stability and affine control of the population dynamics of VolterraKolmogorov type. 

09/18 3:00pm 
BLOC 302 
Stephen Karp Notre Dame 
Positivity in real Schubert calculus
Cchubert calculus involves studying intersection problems among linear subspaces of C^n. A classical example of a Schubert problem is to find all 2dimensional subspaces of C^4 which intersect 4 given 2dimensional subspaces nontrivially (it turns out there are 2 of them). In the 1990s, B. and M. Shapiro conjectured that a certain family of Schubert problems has the remarkable property that all of its complex solutions are real. This conjecture inspired a lot of work in the area, including its proof by MukhinTarasovVarchenko in the 2000s, based on a correspondence between solutions of such a Schubert problem and eigenspaces of a family of commuting operators. I will present a strengthening of this correspondence which explicitly solves the Schubert problem inside the group algebra of the symmetric group. This implies a positive version of the ShapiroShapiro conjecture, which thereby resolves some conjectures of Sottile, Eremenko, MukhinTarasov, and myself. This is joint work with Kevin Purbhoo. 

09/22 4:00pm 
BLOC 302 
Timothy Duff U. Washington Seattle 
Geometry of two, three, or four cameras
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 wellstudied 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. LoucksTavitas, 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 nonperspective 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) 

09/29 4:00pm 
BLOC 302 
Klemen Sivic University of Ljubljana 
Applications of Borel fixed point theorem to linear algebra
The Borel fixed point theorem says that an action of a solvable algebraic group on a projective variety has a fixed point. In the talk we will show how this theorem can help us to determine the upper bound on dimension of certain matrix spaces. In particular, we will consider spaces of matrices with bounded number of eigenvalues and spaces of matrices with commutators of bounded rank. 

10/06 4:00pm 
BLOC 302 
Stephen McKean Harvard 
Real bitangents to plane quartics
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 nonnegative 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. 

10/20 4:00pm 
BLOC 302 
Gal Yehuda Yale 
Slicing and Covering the Hypercube
How many hyperplanes are needed in order to slice all the edges of the Boolean hypercube? How many hyperplanes are required in order to cover all the vertices of the cube, such that: (1) all variables are included in the cover, and (2) no element is redundant? We will discuss these problems, motivation, and connections to convex geometry. The talk is based on joint work with Amir Yehudayoff. 

11/06 4:00pm 
BLOC 302 
Thomas Brazelton Harvard University 
TBA 

11/10 4:00pm 
BLOC 302 
Julia Lindberg The University of Texas 
TBA 

11/17 4:00pm 
BLOC 302 
Keller VandeBogert Notre Dame 
TBA
TBA 

12/04 3:00pm 
BLOC 302 
Jake Levinson Université de Montréal 
TBA 