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

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

Schedule (Tentative)

Lecture	Topic	Notes	Resources
Jan. 3	Introduction	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. 17	MLK 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. 24	Applications 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. 31	Sums 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. 7	Convex duality and the numerical range of a matrix	pdf	Barvinok II.14, Henrion
Feb. 9	Algebraic Boundaries	pdf, video	Blekherman, Parrilo, Thomas Ch. 5, Henrion
Feb. 11	Hyperbolic polynomials I	pdf, video	Renegar
Feb. 14	Hyperbolic polynomials II	pdf, video	Renegar Bauschke, Güler, Lewis, Sendov
Feb. 16	Hyperbolic programming	pdf, video	Güler, Nesterov and Nemirovskii Renegar I, II
Feb. 18	Determinantal Representations	pdf, video	Vinnikov
Feb. 21	Presidents' Day - no class
Feb. 23	Stable polynomials I	pdf, video	Wagner, Pemantle
Feb. 25	Stable polynomials II	pdf, video	Wagner, Leake
Feb. 28	Stable polynomials and negative dependence	pdf, video	Wagner, Borcea, Brändén, Liggett
March 2	Log-concave polynomials I	pdf, video	Brändén , ALOV
March 4	Log-concave polynomials II	pdf, video	ALOV, Brändén, Huh
March 7	Connections to matroids	pdf, video	Gurvits, ALOV, Huh, Schröter, Wang
March 9	Applications in sampling algorithms	pdf, video	Kaufman, Oppenheim, ALOV
March 11	Open Problems	pdf, video	Gurvits, Cordero-Erausquin, Klartag, Merigot, Santambrogio, Alimohammadi, Anari, Shiragur, Vuong