Math 591: Real Algebraic Geometry and Convex Optimization (Spring 2019)

Tuesday and Thursday 3:00 - 4:15pm in 1218 SAS Hall

Instructor: Cynthia Vinzant (3260 SAS, email)
Offce Hours: Tuesday, Wednesday 11am-noon or by appointment

Description: Real algebraic geometry is the study of solutions to polynomial equalities and inequalities over the real numbers. It provides powerful tools to understand the geometry behind many problems in optimization and develop new methods for solving them. This class will introduce basic notions and techniques in real algebraic geometry, convexity, and conic optimization. Further topics include semidefinite programming, sums of squares, moment problems, hyperbolic and stable polynomials, and several applications in polynomial and combinatorial optimization.

Final Project
Students will write a final paper (due Tuesday, May 7) on a topic related to real algebraic geometry and convex optimization and give a presentation in class at the end of the semester. (More details.)


Homework 1 (.pdf .tex) due Thursday, January 24
Homework 2 (.pdf .tex) due Thursday, February 7
Homework 3 (.pdf .tex) due Friday, February 22
Homework 4 (.pdf .tex) due Thursday, March 7

Oral Presentations

Thurs. April 11
Jane Coons on Geometry of maximum likelihood estimation in Gaussian graphical models, after Uhler
Cash Bortner, Owen Coss, Ben Hollering on Gaussian Graphical Models: An Algebraic and Geometric Perspective, after Uhler

Tues. April 16
Christian Smith on Cuts, matrix completions and graph rigidity, after Laurent
Wesley Nelson on Nonnegative Polynomials and Circuit Polynomials, after Wang

Thurs. April 18
Saran Ahluwalia, Geneva Collins, Jonathan Dunay on Typical and Generic Ranks in Matrix Completion, after Bernstein, Blekherman, and Sinn
Ella Pavlechko on A Riemannian approach to convex optimization, after Mishra, Meyer, and Sepulchre

Tues. April 23
Eric Geiger on Symmetry groups, semidefinite programs, and sums of squares, after Gatermann and Parrilo
Jordan Almeter on Symmetric Sums Of Squares Over k-Subset Hypercubes after Raymond, Sanderson, Singh and Thomas

Thurs. April 25
Georgy Scholten on The Chow of Reciprocal Linear Space, after Kummer and Vinzant
Katherine Harris on The Euclidean Distance Degree of an Algebraic Variety, after Draisma, Horobet, Ottaviani, Sturmfels, and Thomas