The Briançon-Skoda Theorem states that \(\overline{J^{n + \lambda-1}} \subseteq J^{\lambda}\) for all integers \(\lambda\geq 1\) and \(J=(f_1,\ldots, f_n)\subseteq \mathbb{C}[x_1,\ldots, x_m],\) thus giving a relationship between the usual powers of an ideal and the integral closure of its powers. Unfortunately, this result does not hold for arbitrary rings. However, there have been numerous generalizations up to a closure operation on \(J^{\lambda}\). For example, Hochster and Huneke proved generalizations using tight closure and +-closure.

The general motto of noncommutative algebraic geometry is that any category sufficiently close to the derived category of a variety should be regarded as a noncommutative variety. Using this principle, pioneering work of Artin-Tate-Van den Bergh-Zhang extends important aspects of projective geometry to the noncommutative setting. I’ll talk about extensions of this theory to noncommutative spaces associated to dg-algebras with a focus on how it feeds back into the commutative setting.

Let \(A = \{a_0=0<a_1<\dots<a_{n-1}=d\}\) be a finite set of relatively prime integers. For all \(s\in \mathbb{N}\), the \(s\)-fold sumset of \(A\) is the set \(sA\) of integers obtained by collecting all sums of \(s\) elements in \(A\). On the other hand, given a field \(k\), one can associate with \(A\) the projective monomial curve \(\mathcal{C}_A\) parametrized by (A\), i.e., the Zariski closure of \( \{(v^d:u^{a_1}v^{d-a_1}:\cdots:u^{a_{n-2}}v^{d-a_{n-2}}:u^d) \mid (u:v) \in \mathbb{P}_k^1\} \subset \mathbb{P}_k^{\, n-1} \, .\)

Given a sequence of relatively prime integers \(a_0 = 0 < a_1 < \dots < a_n = d\) and a field \(k\), consider the projective monomial curve \(\mathcal{C}\subset\mathbb{P}_k^{\,n}\) of degree \(d\) parametrically defined by \(x_i = u^{a_i}v^{d-a_i}\) for all \(i \in \{0,\ldots,n\}\) and its  coordinate ring \(k[\mathcal{C}]\). The curve \(\mathcal{C}_1 \subset \mathbb A_k^n\) with parametric equations \(x_i = t^{a_i}\) for \(i \in \{1,\ldots,n\}\) is an affine chart of \(\mathcal{C}\) and we denote its coordinate ring by \(k[\mathcal{C}_1]\).

Given a valuation over a singularity, it is a fundamental problem whether it has finitely generated associated graded rings. This problem has deep connection with the theory of K-stability and moduli, where the finite generation of certain minimizing valuations were shown. For klt singularities, we propose the study of Kollár valuations, which are valuations with finitely generated associated graded rings that induces klt degenerations. We show that the locus of Kollár valuations is path connected. We discuss some open questions and examples.

Let \(R\) be a Noetherian local ring  of dimension \(d\).   We define an intersection product on schemes \(Y\) which are birational and projective over \(Spec(R)\)  which allows us to interpret multiplicity in \(R\) as an intersection product, generalizing a theorem of Ramanujam and others. We use this to give new geometric formulations and proofs of some classical theorems about multiplicity. In particular, we give a new geometric proof of a celebrated theorem of Rees about degree functions.

We discuss work on the epsilon multiplicity which shows that the epsilon multiplicity exists and is a limit of Amao multiplicities in the most general possible case. These are results from joint work with Cutkosky and more recent work from last month.