DC 4 – Cohomology of tensor varieties

Project Title: Cohomology of tensor varieties
Advisors: M. Michałek (UKON), G. Ottaviani (UniFI) Mentor: M. Schweighofer (UKON)
Objectives: Tensors provide unified, powerful tools across various areas of mathematics. They represent e.g. probability distributions in statistics, quantum states in physics, multilinear algorithmic problems in computer science and linear systems in algebraic geometry. As such diverse tools getting basic information about tensors (like rank, border rank, eigenvalues) is on the one hand very important, on the other hand highly nontrivial. In the project new invariants of tensors, called characteristic numbers, are introduced. These are based on associating to tensor a cohomology class in the space of complete collineations, by first looking at a contraction map. The approach generalizes e.g. maximum likelihood degree of linear concentration models, chromatic polynomial of graphs, characteristic numbers of systems of quadrics, algebraic degree of semidefinite programming and Euler characteristic of determinantal hypersurfaces. In particular, characteristic numbers compute the number of critical points of the likelihood function for a class of multivariate Gaussian distributions. The precise objectives are: to find new methods to efficiently compute those new invariants (symbolically and numerically), to find and prove theorems relating the new invariants to those already known and to apply them in practice to obtain new information e.g. about tensor rank.
Expected Results: New algorithms to compute characteristic numbers of tensors. Relations of characteristic numbers to tensor border rank, proving new bounds on tensor border rank. Computation of characteristic numbers for tensors of central importance.
Planned secondment(s): with Giorgio Ottaviani at UniFI to work on the representation theoretic and algebro-geometric methods for characteristic numbers (M21-33).
Joint degree: University of Konstanz, University of Florence

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