Regularity and algebraic invariants of projective forms

Abstract

Identifying and exploiting intrinsic properties is a paramount and fascinating goal in mathematics, which often leads to understanding the fundamental nature of the objects under consideration. Not only does this underlie deep theoretical breakthroughs, but it also fosters advances in computational routines applied to real-world problems. In this talk, I will provide a gentle overview of recent results I have obtained, together with several collaborators, on structural invariants of projective forms, and their computational facets.
The first part of the presentation will be devoted to the study of minimal additive decomposition of forms, which can be phrased as decomposing symmetric tensors into simple components. This challenging problem has rich algebraic foundations and has attracted considerable interest from different communities due to its multifaceted interpretation in terms of osculating varieties, Artinian Gorenstein algebras and extension of linear operators, which I will briefly connect.
In the second part, I will present arithmetic results on smooth plane projective cubics, namely elliptic curves, which are employed in several classical cryptographic schemes. On the one hand, I will report on structural results and computational challenges arising from such objects, highlighting their concrete relevance for modern protocols. On the other hand, I will discuss novel regularities exhibited by these curves, pointing at potential future developments.

Lo speaker si presenta al Dipartimento in qualità di candidato al bando “Rita Levi Montalcini”.