LSSM logo under a Gaussian curve

Linear Spaces of Symmetric Matrices

A collaboration project at MPI Leipzig and worldwide


The aim of this project is to advance the understanding of linear spaces of symmetric matrices. Looking at these seemingly simple objects from many different perspectives gives rise to many interesting mathematical questions. We collected them in a list aspirationally named "3264 Questions about Symmetric Matrices." The more than 40 participants in our working group aim to address these questions in a series of papers.

Here are some of the lenses that we use when studying LSSM: combinatorics, intersection theory, algebraic geometry, semidefinite programming, algebraic statistics, matroid theory, nonarchimedian analysis, optimization, likelihood geometry.

The project is coordinated by Orlando Marigliano, Mateusz Michałek, Kristian Ranestad, Tim Seynnaeve, and Bernd Sturmfels. [om, mm, kr, ts, bs]

Call for papers

We are happy to announce a call for papers for a special issue of the research journal Le Matematiche on Linear Spaces of Symmetric Matrices. To contribute a paper, please send an email with a title and list of authors by September 15, 2020 to one of the editors of the special issue: Orlando Marigliano, Mateusz Michałek, Kristian Ranestad and Tim Seynnaeve. [om, mm, kr, ts]

The final article is to be submitted by November 30, 2020. It should be prepared using the style file below and should not exceed 15 pages in length. All submitted articles will undergo a rigorous peer-review process. Each person may be a coauthor of at most two contributions. We recommend two or three authors per submission.


  1. October 22, 16h: LSSM seminar. Online session.
  2. October 8, 14:30h: LSSM seminar. Online session.
  3. July 27, 18h: Claudia Fevola, Yelena Mandelshtam and Bernd Sturmfels. Two-dimensional LSSM. Online session.
  4. July 17, 14h: Mateusz Michałek. Enumerative geometry of LSSM. Seminar talk. [link]
  5. July 6, 16h: Tim Seynnaeve. LSSM and combinatorial algebraic geometry. Online session. [slides, talk]
  6. June 29, 16h: Orlando Marigliano. LSSM and statistics. Online session.
  7. June 22, 16h: Bernd Sturmfels. Semidefinite optimization. Online session.


  1. Bernd Sturmfels. 3264 Questions about Symmetric Matrices. [pdf]
  2. LaTeX style file for the papers. [cls]
  3. Mateusz Michałek, Leonid Monin and Jarosław Wiśniewski. Maximum likelihood degree, complete quadrics and C*-action. [link]
  4. Special Issue on Twenty-Seven Questions about the Cubic Surface. Le Matematiche. [link]
  5. Laurent Manivel, Mateusz Michalek, Leonid Monin, Tim Seynnaeve and Martin Vodiča. Complete quadrics: Schubert calculus for Gaussian models and semindefinite programming. [link, code]

LSSM Articles

  1. Claudia Fevola, Yelena Mandelshtam and Bernd Sturmfels. Pencils of quadrics: old and new. Sample for usage of the style file and suggested notation. [link]
  2. Tobias Boege, Jane Ivy Coons, Christopher Eur, Aida Maraj and Frank Röttger. Reciprocal Maximum Likelihood Degrees of Brownian Motion Tree Models. [link]
  3. Arthur Bik, Henrik Eisenmann and Bernd Sturmfels. Jordan Algebras of Symmetric Matrices. [link]
  4. Taylor Brysiewicz, Claudia Fevola and Bernd Sturmfels. Tangent Quadrics in Real 3-Space. [link]
  5. Yairon Cid-Ruiz. Equations and multidegrees for inverse symmetric matrix pairs. [link]
  6. Stefan Dye, Kathlén Kohn, Felix Rydell and Rainer Sinn. Maximum Likelihood Estimation for Nets of Conics. [link]


MPI Leipzig, Le Matematiche.

There are 104 quadratic surfaces that contain 3 given points, are tangent to 3 given lines, and are tangent to 3 given planes.