Title: Castelnuovo–Mumford Regularity: Im- portance, Bounds, and Open Problems

Castelnuovo–Mumford regularity is a central invari- ant in commutative algebra and algebraic geometry, measuring the complexity of a graded module through the shifts in its minimal free resolution. Its importance is underscored by two major recent developments: the resolution of Stillman’s Conjecture by Ananyan and Hochster, and the disproof of the Eisenbud–Goto Conjecture by McCullough and Peeva. Both results, rather than settling questions about regularity itself, reveal how fundamental this invariant is to the field. They leave open what may be the central remaining problems: does a double-exponential bound on the regularity of homogeneous ideals exist in terms of the degrees of their generators alone (independent of the number of variables)? And does an analogous bound hold for prime ideals in terms of their multiplicity? These lectures will develop the classical theory of Castelnuovo–Mumford regularity, trace its role in these two stories, and focus on the open problems and directions that remain.

Title: Polynomial interpolation and the ideal contain- ment problem

We will discuss polynomial interpolation in several vari- ables; particularly, resulting in Chudnovsky’s and De- mailly’s conjectures. We will introduce an algebraic ap- proach to these conjectures through the ideal contain- ment problem. We will discuss the proofs of Chud- novsky’s conjecture for a general set of points and De- mailly’s conjecture for a general set of sufficiently many points.

Title: Modules of finite length and finite projective di- mension over local rings

In these talks, I will discuss various special properties (known and conjectural) of modules of finite length and finite projective dimension. The focus will be on conjec- tured lower bounds on the length, and also the Loewy length, of such modules. Part of the lectures will be based on recent work of Nawaj K.C. and Pollitz, and also Ma, Walker and me.

Title:Differential operators in commutative algebra

We will discuss various questions and results regarding rings of differential operators, particularly in the context of commutative algebra. There will be a special focus on differential operators on hypersurfaces, and on classical invariant rings such as determinantal rings and related families.

This course is devoted to some of the rich interactions between commutative algebra and combinatorics. We will study two families of ideals: edge ideals and homogeneous simplicial toric ideals. The first ones are monomial ideals while the second are binomial. 

Our main goal is to show how combinatorics can be used to compute algebraic invariants such as graded Betti numbers, Castelnuovo–Mumford regularity, projective dimension, and Hilbert function. Those invariants are related to the combinatorics of some underlying graphs in the setting of edge ideals, while in the toric setting they are closely related to sumsets and additive combinatorics.

The course will also illustrate how the interaction between commutative algebra and combinatorics is genuinely bidirectional: combinatorial structures often lead to new algebraic results, while algebraic methods can be used to solve combinatorial problems. Most of the material presented is based on recent joint work with Ignacio García-Marco, Mario González-Sánchez, and Sara Asensio.