Luca Sodomaco

My mathematical research is in Applied Nonlinear Algebra. I am particularly interested in Metric Algebraic Geometry, Algebraic Optimization, and Tensors. 

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

The core of my doctoral thesis addresses the problem of studying the distance function from a variety of rank-one partially symmetric tensors, namely, a Segre-Veronese variety. One motivation for this problem comes from the need, e.g., in certain constrained optimization problems, to approximate a given tensor by its closest rank-one tensor with respect to a given distance function. This is the so-called best rank-one approximation problem for real tensors. In this context, the singular vector tuples and the singular values of a tensor play important roles, generalizing the notions of eigenvectors and eigenvalues of a matrix.

During my postdoctoral experiences, I studied new interactions among Algebraic Geometry, Distance Optimization, and Game Theory. In my research, I like using software for symbolic computation, such as Macaulay2, the Julia package HomotopyContinuation.jl, and Sage.