Math 591: Final Project

Important dates
Proposed topic and group: due by Friday, March 22, 5pm (via email)
Oral presentations: last 2-3 weeks of class (April 11, 16, 18, 23, 25)
Final papers due: Tuesday, May 7, 5pm

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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

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Potential Projects
The final project should be an in depth exploration of some topic related to the course material. This could be an exposition of a research paper or book chapter. Some suggested topics/papers are below, but you are welcome to propose your own!

Groups
You may collaborate for your final projects in groups of sizes one, two, or three, with the expectation that group members contribute equally. Both the oral presentations and final papers may be done as a group.

Oral presentations
You will be expected to give a 30 minute presentation in class on your project, with an additional 5 minutes for questions. This should provide an introduction to the topic that is accessible to the other students in the class and showcases some theorem, example, or computation (or all of the above).

Final Papers
The final paper should single spaced, produced in LaTeX, and at least two pages long per group member. It can be longer! This should accessible to another student in the class, meaning that it should include any definitions of any terms that have not been used in class, clearly explain any main results, and include examples with pictures (if at all possible). The final paper should include some new content, meaning an example, computation, or theorem(!) that is not in the original source.

Some suggested topics
Approximating real-rooted and stable polynomials, with combinatorial applications, Barvinok (2018)
Hyperbolic Polynomials and Convex Analysis, Bauschke, Guler, Lewis, Sendov (2001)
Sums of squares on the hypercube, Blekherman, Gouveia, Pfeiffer (2014)
Sums of squares and varieties of minimal degree, Blekherman, Smith, Velasco (2013)
Negative dependence and the geometry of polynomials, Borcea, Brändén, Liggett (2007)
Obstructions to determinantal representability, Brändén (2010)
A Positivstellensatz for Sums of Nonnegative Circuit Polynomials, Dressler, Iliman, de Wolff (2016)
On spaces of matrices containing a nonzero matrix of bounded rank, Falikman, Friedland, Loewy (2002)
Symmetry groups, semidefinite programs, and sums of squares, Gatermann, Parrilo (2004)
Semidefinite geometry of the numerical range, Henrion (2010)
Moment Matrices, Border Bases and Real Radical Computation, Lasserre Laurent, Mourrain, Rostalski (2011)
Cuts, matrix completions and graph rigidity, Laurent (1997)
Hyperbolic Polynomials Approach to Van der Waerden/Schrijver-Valiant like Conjectures : Sharper Bounds, Simpler Proofs and Algorithmic Applications, Gurvits (2005)
The Solution of The Kadison-Singer Problem, Marcus, Srivastava (2017)
Spectrahedral Lifts of Convex Sets, Thomas (2018)
Gaussian graphical models: an algebraic and geometric perspective, Uhler (2017)