Spectrahedra and Hyperbolic Polynomials

• Applicant: Professor Dr. Mario Kummer
• Subject Area: Mathematics
• Term: from 2019 to 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 421473641

• Final Report Year: 2023

Final Report Abstract

Semidefinite programming is a branch of convex optimization that has aroused great interest both theoretically and practically. Typical applications include polynomial optimization or combinatorial optimization, such as the Max-Cut Problem. With the help of interior-point method one can solve a semidefinite program for fixed precision in a time that is polynomial in the program description size. A question of fundamental interest is that of characterizing the sets which are the feasible sets of semidefinite programming, namely the so-called spectrahedra. The content of the generalized Lax conjecture is such a presumed characterization. The results achieved in this project include several positive partial results towards this conjecture as well as the development of some techniques that might lead to a counter example.

Publications

• Spectral linear matrix inequalities. Advances in Mathematics, 384, 107749.
  Kummer, Mario
  
• Hyperbolic secant varieties of M-curves. Journal für die reine und angewandte Mathematik (Crelles Journal), 2022(787), 125-162.
  Kummer, Mario & Sinn, Rainer
  
• Linear Principal Minor Polynomials: Hyperbolic Determinantal Inequalities and Spectral Containment. International Mathematics Research Notices, 2023(24), 21346-21380.
  Blekherman, Grigoriy; Kummer, Mario; Sanyal, Raman; Shu, Kevin & Sun, Shengding
  
• Matroids on Eight Elements with the Half-Plane Property and Related Concepts. SIAM Journal on Discrete Mathematics, 37(3), 2208-2227.
  Kummer, Mario & Sert, Büşra
  
• Positive Ulrich sheaves. Canadian Journal of Mathematics, 76(3), 881-914.
  Hanselka, Christoph & Kummer, Mario
  
• Spectrahedral shadows and completely positive maps on real closed fields. Journal of the European Mathematical Society (2024, 7, 10).
  Bodirsky, Manuel; Kummer, Mario & Thom, Andreas