EWM-EMS Summer School: Identifiability in tensor decompositions: geometric criteria, statistical models, and computations

Tensor decompositions lie at the intersection of algebraic geometry, representation theory, and computation. Rank-one tensors form Segre (or Segre–Veronese) varieties, and decompositions correspond to expressing a point as a sum of points on these varieties, i.e. to the geometry of secant varieties (dimension/defectivity, singularities, degenerations), involving rank/border rank, tangential constructions, flattenings/catalecticants, and limits.

They are also central in quantum information: multipartite states are tensors, local basis changes act by product groups, and entanglement types correspond to orbits and orbit closures; rank/border rank measure entanglement complexity, while degenerations encode asymptotic/approximate notions. In quantum state tomography, identifiability asks whether measurement data determine the underlying state uniquely.

The school focuses on identifiability: when decompositions are unique (generic vs special points and exceptional loci), how uniqueness can be certified (tangent-space/Terracini criteria, Kruskal-type conditions, flattenings/catalecticants, apolarity, equations of secant varieties), and how geometric insight and computational/numerical certification inform each other. It bridges (i) geometry of decompositions and 0-dimensional schemes/cactus phenomena and (ii) identifiability in algebraic models (algebraic statistics and quantum state models) with practical computational testing and certification.

The school will feature lectures by Kaie Kubjas (Aalto University, Finland) and Pierpaola Santarsiero (Università Politecnica delle Marche, Italy).