We establish connections between: the maximum likelihood degree (MLdegree) for linear concentration models, the algebraic degree of semidefinite programming (SDP), and Schubert calculus for complete quadrics. We prove a conjecture by Sturmfels and Uhler on the polynomiality of the ML-degree. We also prove a conjecture by Nie, Ranestad and Sturmfels providing an explicit formula for the degree of SDP. The interactions between the three fields shed new light on the asymptotic behaviour of enumerative invariants for the variety of complete quadrics. We also extend these results to spaces of general matrices and of skew-symmetric matrices.
Matroids are ubiquitous in modern combinatorics. As discovered by Gelfand, Goresky, MacPherson and Serganova there is a beautiful connection between matroid theory and the geometry of Grassmannians: realizable matroids correspond to torus orbits in Grassmannians. Further, as observed by Fink and Speyer general matroids correspond to classes in the K-theory of Grassmannians. This yields in particular a geometric description of the Tutte polynomial. In this review we describe all these constructions in detail, and moreover we generalise some of them to polymatroids. More precisely, we study the class of flag matroids and their relations to flag varieties. In this way, we obtain an analogue of the Tutte polynomial for flag matroids. 4. Flag varieties: geometry 12 4.1. Representations and characters 12 4.2. Grassmannians 13 4.3. Flag varieties 13 5. Representable matroids: combinatorics and geometry 16 6. Introduction to flag matroids 18 6.1. Flag matroids: Definition 18 6.2. Matroid quotients 19 6.3. Representable flag matroids 20 6.4. Flag matroid polytopes 20 6.5. Flag matroids and torus orbits 21
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