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: A survey on mixed multiplicities

The notion of multiplicity in algebra traces back to the work of Samuel in 1951 in connections with intersection theory of algebraic varieties. Given a multigraded stan- dard graded algebra, one can define a finite set of num- bers called mixed multiplicities. These numbers agree with the multidegrees of multi-projective varieties in the case of algebraically closed fields. If one considers this construction for certain multigraded algebras, one ob- tains related notions of multiplicity such as mixed mul- tiplicities of ideals and multiplicity sequences. Mixed multiplicities can also be seen in other fields of mathe- matics and are related to Schubert polynomials, mixed volumes, and projective degrees of rational maps, and maximum likelihood degree. In this series of lectures, we will discuss the history, definition, and properties of mixed multiplicities.

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.