Luca Sodomaco

My mathematical research is at the intersection between Real Algebraic Geometry, Optimization Theory and Multilinear Algebra. 

A relevant part of my doctoral research has been dedicated to the study of metric invariants of real algebraic varieties, with a particular interest in varieties in tensor spaces. A strong relationship between classical Algebraic Geometry and Multilinear Algebra is established by the notion of tensor rank. From a more geometric point of view, the problem of computing the rank of a tensor translates to a membership problem to a certain secant variety of a Segre product of projective spaces.

The core of my doctoral thesis deals with the problem of studying the distance function from a variety of rank-one partially symmetric tensors, namely a Segre-Veronese variety. One of the motivations of this problem comes from the need, e.g. in some constrained optimization problems, to approximate a given tensor to its closest rank-one tensor, with respect to a certain distance function. This is the so-called best rank-one approximation problem for real tensors. In this context, an important role is played by the singular vector tuples and the singular values of a tensor, which generalize the notions of eigenvector and eigenvalue of a matrix.

After joining Aalto University, I started studying new interactions between the Algebraic Geometry of Tensors and Algebraic Statistics. In my research, I like using software for symbolic computation such as Macaulay2 and Sage.