Daniel Bath: D-modules and (Twisted) Logarithmic Comparison Theorems: Beyond Free Divisors

A divisor is said to satisfy the Logarithmic Comparison Theorem (LCT) when the natural inclusion of the logarithmic subcomplex into the meromorphic (or rational) de Rham complex is a quasi-isomorphism. In other words, thanks to Grothendieck and Deligne, the logarithmic subcomplex computes the cohomology of the complement with constant coefficients. When the divisor is free (i.e. the syzygies of the Jacobian ideal are free) there is a D-module theoretic formulation of the LCT due to Calderón Moreno and Narváez Macarro. And when the divisor is free and satisfies certain homogeneity properties, the LCT holds by Castro Jiménez, Mond, and Narváez Macarro. But outside the free case this D-module formulation breaks down.

We will discuss a D-module theoretic construction valid for any divisor that implies the LCT and, in classical cases, equates to the LCT. We will do the same for twisted Logarithmic Comparison Theorems, where one aims to compute the cohomology of the complement with coefficients in an arbitrary rank one local system. This will let us give a new proof of that the LCT holds for hyperplane arrangements (a recently resolved conjecture of Terao) and give a complete characterization of when the LCT holds for isolated singularities, the latter completing a story started by Holland and Mond. Further applications, especially to Bernstein-Sato polynomials, will be discussed. Much of the new material is based on joint work by the speaker and Morihiko Saito. 

Eamon Quinlan-Gallego: Test ideals and Bernstein-Sato polynomials in positive characteristic

Given a complex hypersurface V = {f = 0}, its Bernstein-Sato polynomial is a classical invariant that detects subtle properties of V, such as the jumping numbers for multiplier ideals. The goal of this course is to explain a characteristic-p analogue of this story. 

We will begin by explaining the classical picture. After that, we will go through the basics of differential operators in positive characteristic, and use them to construct the so-called test ideals of Hara and Yoshida, which are known to provide characteristic-p analogues for multiplier ideals. We will then construct the Bernstein-Sato polynomial in positive characteristic, and explain how its connection to test ideals is much stronger than in characteristic zero. If time permits we will talk about extensions to arbitrary subvarieties and (Z/p^n)-coefficients. We will mention some open problems.

Ilya Smirnov: Hilbert-Kunz multiplicity as a measure of singularity

The driving force of algebra in characteristic p > 0 is the Frobenius endomorphism. The use of Frobenius in the study of singularities has its origin largely in the work of Kunz, who characterized in 1969 regular rings by the flatness of Frobenius. This opened a way and many new results followed over the years creating, for example, an entire field of F-invariants, i.e., the numerical invariants quantifying properties of Frobenius. 

I will present the theorem of Kunz and will describe how it can be used to build singularity measures. As the main example, I will concentrate on the Hilbert-Kunz multiplicity and describe some of its useful properties, failures, and pressing questions.

Joel Castillo Rey: Hilbert-Kunz theory in characteristic 2

The Watanabe-Yoshida conjecture has been a major open problem in Hilbert-Kunz theory for the past two decades. Roughly speaking, this conjecture claims that the Hilbert-Kunz multiplicity over singularities attains its infimum at the A1 singularities, for each given characteristic and dimension.

Not long after it was stated, Enescu and Shimomoto proved it for odd characteristic complete intersection rings. I present the remaining characteristic 2 case of this theorem. This will gives us an idea why this case is treated differently.

We will notice, however, that we do not need to explicitly compute the invariant to prove this. Surprisingly, an explicit formula for the HK multiplicity of A1 singularities as a function of characteristic and dimension remains unknown. To attack this problem, I will explain the powerful Han-Monsky algorithm and show its complete success in characteristic 2.

Finally, I will introduce the graded Han-Monsky algorithm, an extension of these techniques to the graded realm that will allow us to compute an interesting invariant recently introduced by Mukhophadhyay, the Frobenius-Poincaré function.

Kriti Goel: Hilbert-Kunz multiplicity of powers of an ideal

P. Monsky proved the existence of Hilbert-Kunz multiplicity in 1983. Since then, it has been extensively studied, partly because of its connections with the theory of tight closure and its unpredictable behaviour. Unlike the Hilbert-Samuel multiplicity, the Hilbert-Kunz multiplicity need not be an integer. In this talk, we consider Hilbert-Kunz multiplicity of powers of an ideal, in an attempt to write it as a function of the power of the ideal. This involves a surprising connection with the Hilbert-Samuel coefficients of Frobenius powers of an ideal.

Roger Gómez López: Limit distribution of Hodge spectral exponents of plane curve singularities

K. Saito formulated the question whether a continuous distribution is the limit of the distribution of the Hodge spectral exponents of a hypersurface as this hypersurface moves in a sense that has to be made precise. He proved it for irreducible plane curves with a very specific limit formulation. We focus on the case of irreducible plane curve singularities, in which M. Saito provided an explicit formula for the Hodge spectral exponents. We explore different formulations of achieving the limit of the distribution of these invariants. We also obtain some results on a related question by K. Saito, on whether the continuous distribution is in a certain sense a bound for the distribution of Hodge spectral exponents.

Álvaro González Hernández: Local normal forms of non-commutative potentials

In 1975, Artin described local normal forms for all possible (Du Val) singularities that arise as rational double points on surfaces defined over a field of positive characteristic. In a similar fashion, by studying isomorphism classes of non-commutative Jacobi algebras, it is possible to describe local normal forms corresponding to the contraction algebras of 3-fold flops and divisor-to-curve contractions over the complex numbers. These are incredibly useful as they allow us to understand compound Du Val singularities on 3-folds. I would like to present an introduction to this theory and mention the possible generalisations in the positive characteristic setting.

Nirmal Kotal: F-rationality of blow-up algebras and test ideals

In this talk, we see some sufficient conditions for Cohen-Macaulay normal Rees algebras to be F-rational. Let $(R,\mathfrak{m})$ be a Gorenstein normal local domain of dimension greater than 1 and of prime characteristic. Let $I$ be a $\mathfrak{m}$-primary ideal. We give conditions on the test ideals $\tau (I^n), n\geq 1$ which would imply that the normalization of the Blow-up algebra $R[It]$ is F-rational. 

Christian Muñoz Cabello: Deformations of corank 1 frontals

A smooth mapping $f \colon X \subseteq \mathbb{C}^n \to \mathbb{C}^{n+1}$ is frontal if the image space $f(X)$ admits a smoothly-varying tangent hypersurface. Such a property is preserved via diffeomorphisms in the source and target, meaning that the Thom-Mather theory of smooth deformations can be adapted to work within the framework of frontal map germs. In this joint work with J. J. Nuño Ballesteros and R. Oset Sinha, we define a notion of frontal deformation, stability and codimension, along with tools to compute stable and versal frontal deformations in corank 1. We also introduce the notion of frontal Milnor number, an analogue to D. Mond's image Milnor number, and give a frontal version of the Mond conjecture.