Hilbert functions of Artinian algebras

My research plan is centered in commutative algebra and algebraic geometry.

The overall purpose is the description of the algebraic and geometric consequences forced on algebras by conditions on the Hilbert function and vice-versa.

The long-term goal is to determine the Lefschetz properties of Artinian Gorenstein algebras such that their Hilbert functions are the so-called SI-sequences and make contributions to the solution of Stanley’s unimodality conjecture.

A series of concrete problems have been developed for this proposal, which, solving them enrich the knowledge about the Hilbert functions, and in particular they provide partial answers to the proposed problems.

During the first year, I will study problems that are more feasible with my current knowledge, simultaneously, I will be benefited from the geometric and computational knowledge of my host advisor. The host advisor will be a key person for the implementation of the more geometric aspects of the project.

The goals of this project will be achieved by: investigating the Jordan types that are finer invariant than the Lefschetz properties, using the method of higher Hessians for Artinian Gorenstein algebras, applying the new method of rank matrices which have been introduced in my Ph.D. thesis. This method has been already used in my Ph.D. thesis as well as the ongoing joint project.

Improving these methods and introducing new techniques to study Hilbert functions of Artinian Algebras is another goal of this project.