** 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

**Syllabus**

**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**

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