Metric and Probabilistic Aspects of Grassmann and Tensor Geometry

Metric Algebraic Geometry is an emerging field that connects algebraic and differential geometry through the study of metric properties of real algebraic varieties, motivated by applications to the sciences. In this thesis we address three different problems in tensor and Grassmann geometry that showcase the interplay between differential, analytic and algebraic aspects typical of Metric Algebraic Geometry. The first problem deals with estimating the probability that a random real symmetric tensor is close to rank–one tensors. We show how this problem is equivalent to estimating the volume of a neighborhood of the real Veronese variety. We study metric invariants of the Veronese variety and give explicit formulae for its reach and curvature coefficients with respect to the Bombieri–Weyl metric. A description of its second fundamental form is also obtained in terms of matrices from the Gaussian Orthogonal Ensemble. We give a closed formula for the probability of being close to rank–one and show that in the case of rational normal curves it has an exponential decay with respect to the order of the tensor. In the second part we study typical ranks of real m×n×ℓ tensors. We introduce a new geometric framework through which we express rank probabilities for Gaussian tensors in terms of the probabilities of having enough points in the intersection between a random linear subspace and the Segre variety of rank–one matrices. We show that in the range (m−1)(n−1)+1 ≤ ℓ ≤ mn typical ranks are contained in {ℓ, ℓ+1} and ℓ is always typical. Based on asymptotic results on the average number of real points in a random slice of the Segre variety with a complementary dimension subspace, we give some heuristics on how the rank probabilities behave. For 3 × 3 × 5 tensors, for which typical ranks are known to be 5 and 6, we relate the rank probabilities to the probability that a random determinantal cubic surface contains only real lines. As a by–product, we obtain bounds on the expected number of real lines on such a cubic. The last problem constitutes the main project carried out during my PhD studies. Motivated by the notion of Euclidean Distance Degree, which is a measure of the complexity of the nearest point problem to an algebraic variety in a Euclidean space, we introduce the concept of Grassmann Distance Complexity (GDC) of a subanalytic set in the Grassmannian. This measures the complexity of solving the nearest point problem when the subset is subanalytic and the distance used is the Riemannian one. As this distance is not smooth nor semialgebraic, we employ tools from subgradient theory for locally Lipschitz functions and o–minimial geometry. We prove key properties of the GDC, such as its finiteness, and compute explicit bounds for real algebraic hypersurfaces. In the last part we explicitly solve the nearest point problem for a class of simple Schubert varieties through a non–linear version of Eckart–Young theorem. The research we present in this thesis can be found in the following publications and preprints: • A. Cazzaniga, A. Lerario, A. Rosana, What Is the Probability That a Random Symmetric Tensor Is Close to Rank–One?, SIAM Journal on Applied Algebra and Geometry, 8(2):227-258, 2024. • P. Breiding, S. Eggleston, A. Rosana, Typical Ranks of Random Order–Three Tensors, International Mathematics Research Notices, Volume 2025, Issue 4, 2025. • A. Lerario, A. Rosana, The Grassmann distance complexity, arxiv:2411.16589