This thesis is divided into two parts, each part exploring a different topic within
the general area of nonlinear algebra. In the first part, we study several applications of tensors. First, we study tensor networks, and more specifically: uniform
matrix product states. We use methods from nonlinear algebra and algebraic geometry to answer questions about topology, defining equations, and identifiability
of uniform matrix product states. By an interplay of theorems from algebra, geometry, and quantum physics we answer several questions and conjectures posed
by Critch, Morton and Hackbusch. In addition, we prove a tensor version of the
so-called quantum Wielandt inequality, solving an open problem regarding the
higher-dimensional version of matrix product states.
Second, we present new contributions to the study of fast matrix multiplication. Motivated by the symmetric version of matrix multiplication we study the
plethysm S^k(sl_n) of the adjoint representation sl_n of the Lie group SL_n . Moreover, we discuss two algebraic approaches for constructing new tensors which
could potentially be used to prove new upper bounds on the complexity of matrix
multiplication. One approach is based on the highest weight vectors of the aforementioned plethysm. The other approach uses smoothable finite-dimensional
algebras.
Finally, we study the Hessian discriminant of a cubic surface, a recently introduced invariant defined in terms of the Waring rank. We express the Hessian
discriminant in terms of fundamental invariants. This answers Question 15 of the
27 questions on the cubic surface posed by Bernd Sturmfels.
In the second part of this thesis, we apply algebro-geometric methods to
study matroids and flag matroids. We review a geometric interpretation of the
Tutte polynomial in terms of the equivariant K-theory of the Grassmannian. By
generalizing Grassmannians to partial flag varieties, we obtain a new invariant of
flag matroids: the flag-geometric Tutte polynomial. We study this invariant in
detail, and prove several interesting combinatorial properties.
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa:de:qucosa:73286 |
| Date | 08 January 2021 |
| Creators | Seynnaeve, Tim |
| Contributors | Universität Leipzig, Max-Planck-Institut für Mathematik in den Naturwissenschaften |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | info:eu-repo/semantics/acceptedVersion, doc-type:doctoralThesis, info:eu-repo/semantics/doctoralThesis, doc-type:Text |
| Rights | info:eu-repo/semantics/openAccess |
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