Alessandro Gimigliano

The subject of my research are in the fields of Algebraic Geometry  and Commutative Algebra .
1) Secant varieties of certain projective varieties.
The study of the varieties of  secant spaces to projective varieties is a classic subject of study in Algebraic Geometry which has found a renewed interest in contmporary researchers. 
The main problem under study is how to determine which varieties possess secant varieties whose dimension id different from the "expected one", and the other problem is to give equations for secant varieties.
2) Tensor decomposition.
The problem of  determining how to write a tensor as sum of decomposable ones is related to the generalization of the concept of "rank" of a matrix to tensors.
(tensor rank). This problem is related to 1) because the varieties which parameterize generic tensors, or symmetric ones, or skew symmetric are classically studied varieties (Segre varieties, Veronese varieties and Grassmmannians, respectively). Their secant varieties give the stratification of those tensors with respect to tensor rank.
3) 0-dimensional projective schemes.
The ideals of  0-dimensional projective schemes, an in particular of "fat points" (infinitesimal neighborhoods of points) have been under study by many years in contemporary Algebraic Geometry; their properties are related to several problems in Mathematics (secant varieties, polynomial interpolation). In particular we are interested to determine the dimension of the space of forms of a given degree containing the scheme and the minimal  resolution of their ideal.