Schedule, Notes/Handouts & Videos

Wiles, in his work on modularity lifting, discovered a numerical criterion for a map R-->T of noetherian complete commutative local rings over a fixed discrete valuation ring O, and of relative dimension zero, to be an isomorphism of complete intersections. The criterion is in terms of the "congruence module" of T attached to an augmentation T-->O and the cotangent module of the composite map R-->O. Diamond generalized this result to a numerical criteria for a module over R to be free, again involving suitable congruence modules and cotangent modules. In my talk, I will present extensions of these results to higher relative dimension. These have applications in number theory, but I will focus mostly on the commutative algebra aspects. This is based on joint work with Khare, Manning, and Urban.

Computing Frobenius closures of ideals is a classically difficult problem. Starting in 2006 with work of Katzman-Sharp the problem is made tractable for Cohen-Macaulay rings.  Since roughly 2019, starting from work of Quy and Maddox, there has been an explosion of activity in computing Frobenius closure of parameter ideals. This has come by studying F-nilpotent singularities. In this talk, we will review the recent aspects of this research, applications of F-nilpotence to computing Frobenius test exponents, and the significant recent advances in F-nilpotent singularities that arose from this.

We study the ring D differential operators of low order for an isolated singularity hypersurface ring of the form R=k[x,y,z]/(f) for a field k of characteristic zero and homogeneous polynomial of of degree at least 3 (that is, the cone of a smooth projective plane curve).  For such hypersurfaces, it was long known that there are no negative degree operators by results of Vigué building on the work of Bernstein, Gel’fand, and Gel’fand, the proofs going via investigations of certain sheaf cohomologies, but the explicit operators were not known as far as we know (and further recent results by Mallory provide similar behavior in 4 variables). We find the differential operators of order up to 3 for such hypersurfaces in 3 variables by completely new, homological methods, as well as minimal resolutions of the R-module that they form. 

This result highlights further the contrasting and yet interactive behaviors in the smooth and non-smooth cases. In the smooth case, such rings of differential operators tend to have many operators of negative degree, yielding that certain important D-modules are simple. In addition, when R is a polynomial ring, for example, many important, but infinitely generated, R-modules such as local cohomology become of finite length as D-modules, which has been an important tool for studying them. 

This is joint work with Rachel Diethorn, Jack Jeffries, Nick Packauskas, Josh Pollitz, Hamid Rahmati, and Sophia Vassiliadou. 

This talk concerns the behavior of the functions reg I^{n-1}/I^n, reg R/I^n, reg I^n, where I is a graded ideal in a standard graded algebra R. If I is generated in a single degree, we can describe these functions in a satisfactory manner. We also study the behavior of the function of the saturation degree of I^n.

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This is a talk about a relative version of Koszul duality, and how it can tell us about the asymptotic homological algebra of (seemingly) non-Koszul local rings. It’s all joint work with James Cameron, Janina Letz, and Josh Pollitz.

A homomorphism f of commutative local rings, say S to R, has a derived fibre F (a differential graded algebra over the residue field k of R) and we say that f is Koszul if F is formal and its homology H(F) (the Tor algebra of R and k over S) is a Koszul algebra in the classical sense. I’ll explain why this is a good definition with interesting consequences, and how it is satisfied by many many examples. I might also talk about how to understand these through A-infinity structures.

The main application is the construction of explicit free resolutions over R in the presence of certain Koszul homomorphisms. This construction simultaneously generalises the resolutions of Priddy over a Koszul algebra, the resolutions of Shamash and Eisenbud over a complete intersection ring, and the bar resolutions of Iyengar and Burke over a Golod ring

What is the most singular possible point on any variety in positive characteristic?  In joint work with coauthors, we gave a precise answer to this question for hypersurfaces using a measure of singularity called the F-pure threshold, and we called these “most singular” hypersurfaces extremal hypersurfaces.  These special hypersurfaces only occur in degrees $d=p^e+1$, where $p$ is the characteristic.  In each degree where they occur, there is one particular extremal hypersurface (up to a change of variables) which stands out—it is the cone over a smooth projective hypersurface.  In this talk, we’ll focus on this special hypersurface in the 4-variable case—that is, we’ll focus on an extremal hypersurface which is the cone over a smooth projective surface, and we’ll discuss some of its surprising geometric properties.  In particular, we highlight a conjecture which further justifies the name “extremal,” which would answer a longstanding classical question about smooth projective surfaces of any degree.

A density function for an algebraic invariant is a measurable function on R which measures the invariant on an R-scale, that is, a density function for a given algebraic invariant, say , is an integrable function f: R −→ R, which gives a new measure μf on R such that the integration R E f on a subset E ⊂ R is ‘the measure of the invariant  on E’.
This function carries a lot more information related to the invariant without seeking extra data (than needed to study the invariant itself). The HK density function, which was introduced by the speaker to study Hilbert-Kunz multiplicity, turned out to be a useful tool in answering some long standing open questions. In this talk we discuss the existence of density function for some well known and some elusive algebraic invariants and its applications. A part of the talk is based on some work with K.I. Watanabe, and some with S.Das and S.Roy.

The major open question in the theory of Gorenstein liaison is whether or not every arithmetically Cohen--Macaulay subscheme of $\mathbb{P}^n$ can be G-linked to a complete intersection.  Migliore and Nagel showed that, if such a scheme is generically Gorenstein (e.g., reduced), then indeed it can be after re-embedding so that it is viewed as a subscheme of $\mathbb{P}^{n+1}$. Motivated by this result, we consider techniques for constructing G-links on a scheme from G-links on a closely-related reduced scheme.

Polarization is a tool for producing a squarefree monomial ideal from an arbitrary monomial ideal.  After reviewing the key aspects of liaison and polarization, we will describe conditions that allow for the lifting of a basic double G-link induced from a vertex decomposition of the Stanley--Reisner complex of the polarization of an ideal $I$ to a basic double G-link of $I$ itself.  We will introduce a notion of polarization of a Gr\"obner basis of an arbitrary homogeneous ideal and give a result for geometric vertex decomposition and elementary G-biliaison that is analogous to our result on vertex decomposition and basic double G-linkage. Time permitting, we will give an application to vertex decompositions of Stanley--Reisner complexes of polarizations of stable Cohen--Macaulay monomial ideals and of artinian monomial ideals, extending a recent result of Fl{\o}ystad and Mafi.

The work described in this talk is joint with Sara Faridi, Jenna Rajchgot and Alexandra Seceleanu. 

In various contexts, one encounters sequences of ideals or modules over polynomial rings with a rich algebraic structure, indicated, for example, by invariance under the action of a symmetric group. It is natural to expect that the properties of related symmetric modules eventually become predictable even when the number of variables increases as one moves along the sequence. We discuss a framework for studying all but finitely many objects in such a sequence simultaneously. It has been used to establish asymptotic results for invariants of the objects such as the Hilbert function and multiplicity as well as graded Betti numbers and Castelnuovo-Mumford regularity. However, the boundaries for these stabilization results are not clear, and there are many open questions.