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Kickoff for Linear Spaces of Symmetric Matrices Study Group

Published Jul 31, 2020

Scientists at MiS and other institutes around the globe have just established a collaborative project that aspires to advance the understanding of linear spaces of symmetric matrices (LSSM). These objects appear simple at first, but a closer investigation from many contrasting perspectives prompts a variety of intriguing mathematical questions.

The more than 40 participants of the study group compiled them in a list aspirationally named “3264 Questions about Symmetric Matrices” and they plan to engage these problems in a series of papers. The project is coordinated by Orlando Marigliano, Mateusz Michałek, Kristian Ranestad, Tim Seynnaeve, and Bernd Sturmfels.

Schubert triangle with corresponding quadratic surfaces on spheres
The triangle shows numbers found by Hermann Schubert in his 1879 book on Enumerative Geometry. There are 104 quadratic surfaces, like the blue sphere, that contain three given points, which are tangent to three given lines and to three given planes.⁠ ⁠ The red number 128 occurs for two points, five lines and two planes. The first row 1,3,9,17,21, .... arises in statistics, namely in maximum likelihood estimation for Gaussian random variables.

Some of the aspects under which LSSM will be scrutinized include

  • combinatorics
  • intersection theory
  • algebraic geometry
  • semidefinite programming
  • algebraic statistics
  • matroid theory
  • non-archimedian analysis
  • optimization
  • likelihood geometry