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Speaker:
Prof. Dr. Kathlén Kohn (KTH Stockholm)

This event is part of an event series „Konstanz Women in Mathematics“.

We study the maximum likelihood (ML) degree of multivariate Gaussian models that are described by linear conditions on the concentration matrix. We obtain new formulae for the ML degree, one via Schubert calculus, and another using Segre classes from intersection theory. This allows us to characterize the extreme cases on the ML degree spectrum: models with ML degree zero and models with maximal ML degree. It turns out that models with non-maximal ML degree are (up to Zariski closure) exactly those linear spaces for which strong duality in semidefinite programming fails. The subvariety of the Grassmannian formed by these linear spaces is a union of certain coisotropic hypersurfaces of determinantal varieties. We illustrate our results and the underlying geometry in the case of trivariate models: here we give a full, finite list of geometric types of linear subspaces in the space of symmetric 3x3 matrices incl. their ML degrees.

This talk is based on 3 joint works with 1) C. Améndola, L. Gustafsson, O. Marigliano, A. Seigal; 2) Y. Jiang, R. Winter; 3) S. Dye, F. Rydell, R. Sinn.