Abstract: We study the cohomology of subsheaves of the tangent bundle over the projective space, providing a classification of locally free distributions of low degree. We also describe the corresponding singular loci.

Abstract: A Waring decomposition of a polynomial is a minimal decomposition as sum of powers of linear forms. We will overview this topic, starting from Alexander-Hirschowitz Theorem, stressing on its uniqueness. When uniqueness holds, it gives a convenient canonical form of the polynomial.

Abstract: I will report on recent results on monads on an ACM variety. More precisely, results on their existence, simplicity, stability and on the algebraic structure of the set of pairs of morphisms which define these monads. This is joint work with Pedro Macias Marques and Simone Marchesi.

Abstract; The rank of a homogeneous polynomial F of degree d is the minimal number of summands when it is written as a sum of powers of linear forms. In terms of apolarity the rank is the minimal length of a smooth finite apolar subscheme, i.e. a subscheme whose homogeneous ideal is contained in the annihilator of the form in the ring of differential operators. We define the cactus rank of F as the minimal degree of any scheme apolar to F (not necessarily smooth). The cactus variety of degree d forms is the closure of the family of degree d forms with cactus rank r. Bernardi and Ranestad proved that the cactus rank of a general cubic form F is at most 2n+2 and conjectured that this upper bound is attained for n>=8. In a joint work with these authors and Jelisiejew, we bound the dimension of the cactus variety of cubic forms, thus giving an approach to computing the cactus rank of a general cubic.