Geometry of Real Tensors and Phylogenetics

This thesis is devoted to the study of special algebraic varieties arising from the theory of tensors and phylogenetics. The main motivations for analyzing these objects come from both pure and applied mathematics. In the context of tensors, the Waring problem is a classical question going back to the work of Hilbert, Scorza, Sylvester, and many other geometers and algebraists of the nineteenth century. We contribute to the solution of the real and complex Waring problem for some classes of homogeneous polynomials. More specifically, we classify the Waring ranks of real and complex reducible cubics, extending a result by Segre. Furthermore, we give upper bounds for the real Waring rank of any monomial; as a by-product we characterize monomials whose least exponent is one as the only ones whose real and complex Waring ranks coincide. Finally, for plane curves of low degree we introduce the space of real sums of powers, parametrizing real Waring decompositions and we analyze the real rank boundary of such curves. We also study an intriguing invariant of abelian groups from algebraic geometry applied to computational phylogenetics. This invariant constitutes an upper bound for the degree of equations of toric varieties; the latter describe the group-based models on n taxa. We show that this invariant is finite for any abelian group, thus proving that such an upper bound on degree exists. We achieve this result by the means of the combinatorial structure of special matrices, corresponding to binomials in the ideals of these toric varieties. This solves a conjecture by Sturmfels and Sullivant.