Courses
Emil Horobeț Low-rank approximation of tensors, an algebraic optimization approach (3 slots)Cezar Joița Introduction to algorithmic desingularization; applications (2 slots)Khazhgali Kozhasov Real solutions in Euclidean Distance minimization problems (3-4 slots)Laurențiu Maxim Singularities and optimization (3 slots) Dirk Siersma and Mihai Tibăr Topological tools for studying the Euclidean Distance problems (4 slots) Abstracts
Emil Horobeț Low-rank approximation of tensorsAbstract In the first part of this series of presentations we will learn about the algebraic geometry of the variety of rank-one regular tensors (the Segre variety), rank-one symmetric tensors (the Veronese variety), the k-th secant variety of the Segre and Veronese varieties (these are the tensors of (border) rank less than or equal to k) and we well learn about the Grassmanian variety in the Plücker embedding. We will understand the parametric construction of these varieties and study their dimensions and degrees if these are known, finishing with describing the tangent space structure at regular points to these varieties. In the second part of this series, we will set up the problem of finding the closest point to these varieties. This problem corresponds to the problem of the best low-rank approximation, used in data analysis, machine learning, signal and image processing, etc. We will study the singularities of the distance function from a generic point, restricted to these varieties and we will develop formulas for the number of critical points (isolated singularities). Finally, we will try to understand the discriminant locus of this problem, the so-called hyperdeterminant.
Cezar Joița Introduction to algorithmic desingularization; applicationsAbstract Presentation of the algorithmic desingularization of plane curves, with applications to classical and more recent problems.Khazhgali Kozhasov Real solutions in Euclidean Distance minimization problems. Abstract I will survey works, and state some open problems related to understanding the optimality of the ED degree bounds on the number of (real) critical points of the distance function.Laurențiu Maxim Singularities and optimization Abstract I will explain how various facets of singularity theory can be used to understand the algebraic optimization degrees for linear optimization and the nearest point problem.Dirk Siersma and Mihai Tibăr Topological tools for studying the Euclidean Distance problemsAbstract I'll describe how algebraic and differential topological tools apply to enumerative algebraic geometry problems, including some of the Euclidean Distance problems. The discriminant of the Euclidean Distance will be largely detailed in case of plane curves.