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]

Papers now in review

The working group participants have produced sixteen research articles. Fourteen of them have an ArXiv page and are listed below. Thanks to everybody for participating, and thanks to the authors for submitting their articles.

All submissions are now in peer-review. Successful submissions will be published in a special issue of the journal Le Matematiche on Linear Spaces of Symmetric Matrices. The editors of the special issue are Orlando Marigliano, Mateusz Michałek, Kristian Ranestad and Tim Seynnaeve. [om, mm, kr, ts]


  1. January 7, 17h: LSSM Retrospective. Online session.
  2. October 22, 16h: LSSM seminar. Online session.
  3. October 8, 14:30h: LSSM seminar. Online session.
  4. July 27, 18h: Claudia Fevola, Yelena Mandelshtam and Bernd Sturmfels. Two-dimensional LSSM. Online session.
  5. July 17, 14h: Mateusz Michałek. Enumerative geometry of LSSM. Seminar talk. [link]
  6. July 6, 16h: Tim Seynnaeve. LSSM and combinatorial algebraic geometry. Online session. [slides, talk]
  7. June 29, 16h: Orlando Marigliano. LSSM and statistics. Online session.
  8. 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]
  7. Abeer Al Ahmadieh, Mario Kummer and Miruna-Stefana Sorea. A generalization of the space of complete quadrics. [link]
  8. Taylor Brysiewicz, Khazhgali Kozhasov and Mario Kummer. Nodes on quintic spectrahedra. [link]
  9. Carlos Améndola, Lukas Gustafsson, Kathlén Kohn, Orlando Marigliano and Anna Seigal. The Maximum Likelihood Degree of Linear Spaces of Symmetric Matrices. [link]
  10. Kathlen Kohn, Rosa Winter and Yuhan Jiang. Linear Spaces of Symmetric Matrices with Non-Maximal Maximum Likelihood Degree. [link]
  11. Serkan Hoşten, Isabelle Shankar and Angélica Torres. The degree of the central curve in semidefinite, linear, and quadratic programming. [link]
  12. Isobel Davies and Orlando Marigliano. Coloured Graphical Models and their Symmetries. [link]
  13. Christopher Eur, Tara Fife, José Alejandro Samper and Tim Seynnaeve. Reciprocal maximum likelihood degrees of diagonal linear concentration models. [link]
  14. Carlos Améndola and Piotr Zwiernik. Likelihood Geometry of Correlation Models. [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.