Codimension one foliations in homogeneous spaces
In this talk I will talk about the (moduli) space of foliations of small degree on certain homogeneous varieties. The (historical) example that will lead the talk is the case of the projective space: every codimension one minimal degree foliation on the projective space is obtained as the fibers of a linear projection from P^n to P^1. This implies that the moduli space of such foliations is a Grassmannian. By using equivariant techniques, we will see that an analogous result is true for a certain class of homogeneous varieties called cominuscule Grassmannians; among these one can find for instance ordinary Grassmannians of lines and other more exotic (exceptional) Grassmannians. This is a joint work with Daniele Faenzi and Alan Muniz.
Euclidean Distance degree of Segre-Veronese varieties for general inner products
The algebraic complexity of the rank-1 approximation problem of a tensor is measured by an invariant called the Euclidean Distance (ED) degree. I will discuss a conjecture asserting that, seen as a function of an inner product on the space of tensors, the ED degree achieves its minimal value at Frobenius inner product.
The Q-algebraicity problem in real algebraic geometry
In 2020, Parusinski and Rond proved that every algebraic set V \subset R^n is homeomorphic to a \bar{Q}^r-algebraic set V' \subset R^n, where \bar{Q}^r denotes the field of real algebraic numbers. The aim of this talk is to introduce a new approach to real algebraic geometry with equations over Q in order to provide some classes of algebraic sets that positively answer the following open problem:
Q-Algebraicity problem: (Parusinski, 2021) Is every algebraic set V \subset R^n homeomorphic to some Q-algebraic set V' \subset R^m, with m \geq n?
