Math 582G: Convex Algebraic Geometry (Winter 2022)
Convex algebraic geometry involves the study of convex objects defined by real polynomial inequalities using the interplay of their convex and algebraic structure, with a special emphasis on those appearing in convex optimization. This class will introduce basic notions and techniques in real algebraic geometry, convexity, and conic optimization. Topics include semidefinite programming, sums of squares, moment problems, hyperbolic and stable polynomials, and applications in combinatorics and optimization.

Class: MWF 3:00-3:50pm, PDL C-401 (first week on Zoom)
Instructor: Cynthia Vinzant (PDL C-526, email)
Office Hours: Monday 4-5pm, Thursday 2-3pm in PDL C-526 or at the class Zoom link
Class Syllabus

Zoom link for online and hybrid class

Assignments (to be submitted on Canvas)
Homework 1 (.tex, .pdf) due Friday, Jan. 21
Homework 2 (.tex, .pdf) due Friday, Feb. 4
Homework 3 (.tex, .pdf) due Friday, Feb. 18
Homework 4 (.tex, .pdf) due Wednesday, March 9

Barvinok, A Course on Convexity
Blekherman, Parrilo, Thomas, Semidefinite Optimization and Convex Algebraic Geometry

Schedule (Tentative)
Jan. 3Introduction pdf, video Blekherman, Parrilo, Thomas, Ch. 1
Jan. 5 Real Algebraic Geometry Basics pdf, video Cox, Little, O'Shea, Ch. 1, Ch. 4.1
Jan. 7 Convex Geometry Basics pdf, video Barvinok Ch. I, II
Jan. 10 Convex Duality pdf Barvinok Ch. III, IV
Jan. 12 The PSD cone pdf, video Barvinok II.12
Jan. 14 Convex Optimization pdf, video Barvinok IV.5-7
Jan. 17MLK Jr. Day - no class
Jan. 19 Sums of squares and SDPs pdf Blekherman, Parrilo, Thomas, Ch. 3
Jan. 21 The dual perspective -- moments pdf, video Blekherman, Parrilo, Thomas, Ch. 3.5
Jan. 24Applications to combinatorial optimization (MAXCUT) pdf, video Blekherman, Parrilo, Thomas Ch. 7
Jan. 26 Applications to combinatorial optimization II (Stable sets and theta bodies) pdf, video Blekherman, Parrilo, Thomas Ch. 7
Jan. 28 Convex hulls and computations -- online only! pdf, video SOS Computations
Examples: m2, Mathematica
Jan. 31Sums of squares modulo an ideal pdf, video Blekherman, Sinn, Smith, Velasco
Feb. 2 Spectrahedra and PSD matrix completion pdf, video Barvinok II.13
Feb. 4 PSD matrix completion and applications pdf, video Barvinok II.13 , 14, 15
Feb. 7Convex duality and the numerical range of a matrix pdf Barvinok II.14, Henrion
Feb. 9Algebraic Boundaries pdf, video Blekherman, Parrilo, Thomas Ch. 5, Henrion
Feb. 11Hyperbolic polynomials I pdf, video Renegar
Feb. 14Hyperbolic polynomials II pdf, video Renegar
Bauschke, Güler, Lewis, Sendov
Feb. 16Hyperbolic programming pdf, video Güler, Nesterov and Nemirovskii Renegar I, II
Feb. 18 Determinantal Representations pdf, video Vinnikov
Feb. 21Presidents' Day - no class
Feb. 23 Stable polynomials I pdf, video Wagner, Pemantle
Feb. 25 Stable polynomials II pdf, video Wagner, Leake
Feb. 28Stable polynomials and negative dependence pdf, video Wagner, Borcea, Brändén, Liggett
March 2Log-concave polynomials I pdf, video Brändén , ALOV
March 4Log-concave polynomials II pdf, video ALOV, Brändén, Huh
March 7 Connections to matroids pdf, video Gurvits, ALOV, Huh, Schröter, Wang
March 9Applications in sampling algorithms pdf, video Kaufman, Oppenheim, ALOV
March 11Open Problems pdf, video Gurvits, Cordero-Erausquin, Klartag, Merigot, Santambrogio, Alimohammadi, Anari, Shiragur, Vuong