Seminar

Yairon Cid-Ruiz: "Numerical criteria for integral dependence and their behavior in families"

A classical theorem of Rees tells us that, in an equidimensional and universally catenary Noetherian local ring, two zero-dimensional ideals I ⊂ J have the same integral closure if and only if they have the same Hilbert-Samuel multiplicity. This seminal result sparked much interest and has become an important research topic in commutative algebra, singularity theory, and algebraic geometry. For instance, an important consequence is Teissier’s principle of specialization of integral dependence. We will present new criteria for the integrality and birationality of an extension of graded algebras in terms of the general notion of polar multiplicities due to Kleiman and Thorup. We will discuss the behavior in families of ideals of certain invariants studied by Gaffney and Gassler: the polar multiplicities and Segre numbers of ideals. I will report on ongoing joint work with Claudia Polini and Bernd Ulrich.

Benjamin Briggs: "A-infinity tricks in local algebra"

This talk will be mostly expository: the plan is to explain what an A-infinity algebra is and how you might use them to prove things about commutative rings. I'll focus on how they can be used the construct Iyengar and Burke's bar resolution that achieves Serre's bound on the Betti numbers of a module, what they have to do with the Eisenbud-Shamash resolution as well, and, if there's time, I'll talk about some new analogues of these resolutions, also using A-infinity tricks. This last part is joint work with James Cameron, Janina Letz, and Josh Pollitz on something we call Koszul homomorphisms.