Luca Sodomaco

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

More specifically, I study algebraically the problem of minimizing or maximizing a rational objective function over a set defined by polynomial equalities and inequalities. Prominent examples include minimizing the distance between a data point and an algebraic variety; maximizing the log-likelihood function for discrete and Gaussian algebraic exponential families based on a data sample; and, in Game Theory, simultaneously maximizing expected payoffs in a finite-player game, leading to Nash equilibria. In these scenarios, determinantal varieties and low-rank tensor varieties are central. The key invariants related to these problems are the (Euclidean) Distance degree, the Maximum Likelihood degree, and the number of (totally mixed) Nash equilibria.

Furthermore, all previous optimization problems depend on parameters, corresponding to "data". For example, this could be a data tensor for which one aims to compute its best low-rank approximation. Or a data sample arising from a statistical experiment, for which one wants to learn the best estimator. Finally, the payoff tensors of the given finite-player game. The choice of parameters influences the geometry of the solutions associated with these polynomial systems.

A sufficiently "generic" or "random" selection of parameters often yields solution sets with expected properties, such as the expected dimension or cardinality, or their reducedness.

In contrast, nongeneric parameters give rise to algebraic subsets known as data loci. Each data locus is formed by imposing additional conditions on the solution sets. I like studying several types of data loci in algebraic optimization. In distance optimization, possible data loci are evolutes, offset hypersurfaces, medial axes, Voronoi cells, conditional distance loci, Kalman varieties of matrices and tensors, and higher-order distance loci. In Game Theory, interesting data loci are Nash discriminants and Nash loci.

In my research, I like using software for symbolic computation, such as Macaulay2, the Julia package HomotopyContinuation.jl, and Sage.