Notes
Recording
Abstract: The goal of my talks is to introduce the basic theory of local Euler obstructions, and explain their relationship to D-modules, Kazhdan-Lusztig theory, and intersection theory. I will then illustrate the theory by computing the local Euler obstructions in two cases of interest: (1) for the determinantal varieties of general, symmetric and skew-symmetric matrices, which is based on joint work with András Lőrincz; (2) for the Schubert stratification of the Lagrangian Grassmannian, which is based on joint work with Paul LeVan.
Notes
Recording
Abstract: The Bernstein-Sato polynomial (or b-function) of a hypersurface in a smooth variety is an important invariant of that hypersurface which contains information about its singularities. Kashiwara showed how to understand the roots of the b-function using numerical data coming from a resolution of singularities of the hypersurface, and Lichtin refined this argument to give even more constraints on the shape of the roots. The plan of the two talks is to describe Kashiwara's proof for existence of b-functions, and explain a generalization of Lichtin's proof on the shape of the roots to more general b-functions, which is joint work with Mircea Mustaţă.
In the first talk, I will give an overview of the necessary concepts from the theory of D-modules and introduce the D-module N_f and explain how it is useful in the study of b-functions. In the second talk, I will explain Kashiwara's proof of existence of the Bernstein-Sato polynomial, define more general b-functions, and sketch the proof giving the shape of their roots in terms of numerical data from a resolution of singularities. Everything will be done over the complex numbers and on a smooth ambient variety.
Notes (parts 1 and 2)
This is a continuation from the talk on 09/04.
Notes
Abstract: These two talks will be an introduction to homological properties of rings of differential operators: by Auslander-Buchsbaum-Serre a commutative ring is regular if and only if it has finite global dimension. Now, for a noncommutative ring (such as a ring of differential operators), how can we determine if this ring is "regular"?
While the standard definition of regularity for commutative rings does not carry over to noncommutative ones, the definition of global dimension does, and so may be used as a measure of regularity for these rings.
I will give an overview on homological dimensions of rings of differential operators and motivate some related endomorphism rings of finite global dimension, so-called noncommutative resolutions of singularities.
Finally, I will discuss the hands-on example of toric rings, where we can describe noncommutative resolutions coming from the Frobenius homomorphism and (in positive characteristic) show that the rings of differential operators on these toric rings have finite global dimension (this is joint work with G. Muller and K.E. Smith).
Notes
Abstract: In this talk, I will give some history behind the classic Lipman-Zariski Conjecture and the generalized Lipman-Zariski questions of Graf. Then I’ll give some results on the torsion and cotorsion of exterior powers of the module of Kaehler differentials over complete intersection rings, and say how these are used to prove a generalized Lipman-Zariski result under certain conditions. This is joint work with Sophia Vassiliadou.
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Abstract: I will discuss some classic constructions and facts in homological algebra as background for the second talk. I’ll also explain how these come into play for the module of Kaehler differentials and its exterior powers, and, if time, for their torsion and cotorsion.
Slides
Abstract: In this second talk we explore some consequences of the results presented in the previous talk. We first characterize the k-torsion freeness of the module of high order differentials of a hypersurface in terms of the singular locus of the corresponding ring. This is a joint work with Hernán de Alba. Secondly, we present some recent results on Nash blowups in positive characteristic that were initially motivated by the study of the module of differentials of high order. This is a joint work with Luis Núñez Betancourt.
Slides
Abstract: In this first talk we discuss some basic aspects of the module of differentials of high order. We start by giving an explicit presentation of this module in the case of a finitely generated algebra via a higher order Jacobian matrix. Then we apply this presentation to study some algebraic aspects of this module. This is a joint work with Paul Barajas.
This is a continuation of the talk from 07/24.
Abstract: Consider a smooth complex variety on which a linear algebraic group acts with finitely many orbits. In this situation, the theory of equivariant D-modules provides a framework and techniques to study local cohomology with support in subvarieties preserved by the group action, as well as related invariants including Lyubeznik numbers and Hodge ideals. In these lectures, we will introduce this theory via the example of generic determinantal varieties. As an application, we will discuss recent joint work with Claudiu Raicu on Hodge ideals for the generic determinant and mixed Hodge structure on local cohomology with support in determinantal varieties.
Slides
This is a continuation of the talk from 07/10
Slides
Abstract: The main objective will be to describe primary ideals with the use of differential operators.
These descriptions involve the study of several objects of different nature, so to say; the list includes: differential operators, differential equations with constant coefficients, Macaulay’s inverse systems, symbolic powers, Hilbert schemes, and the join construction.
As an interesting consequence, we will introduce a new notion of differential powers which coincides with symbolic powers in many interesting non-smooth settings, and so it could serve as a generalization of the Zariski-Nagata Theorem. I will report on some joint work with Roser Homs and Bernd Sturmfels.
Notes (parts 1 and 2)
This is a continuation of the talk from 06/26.
Notes (parts 1 and 2)
Abstract: A-hypergeometric systems are examples of holonomic D-modules that have a deep connection to the theory of toric varieties. I will give an introduction to this theory and report on recent progress (joint with Jens Forsgård) towards computing monodromy of A-hypergeometric systems.
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This is a continuation of the talk from 06/12
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Abstract: Bernstein-Sato polynomials are a central object in the theory of D-modules with many applications in Analysis, Representation Theory, Singularities or Commutative Algebra. They have been extensively studied over the last decades but they are not completely understood and only a few explicit examples can be found in the literature. The aim of these talks is to give a gentle introduction to the theory of Bernstein-Sato polynomials. We will discuss its existence in regular rings as well as some recent development in the singular case. We will also present some basic properties of roots of the Bernstein-Sato polynomial and we will relate these roots to other known invariants of singularities. In order to illustrate the theory we will consider the case of plane curves that has received a lot of attention recently.
Notes
This was a continuation of the talk from 05/29.
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Abstract: Differential operators on singular rings have found many applications, but are generally quite hard to describe explicitly. One question you can ask is "how transitive" is the action of the differential operators on the ring, and specifically, is there a nontrivial submodule? If not, the ring is called D-simple. In many cases, D-simplicity corresponds to "more mild" singularities: for example, smooth rings are D-simple, as are their invariant subrings under finite group actions. Moreover, in positive characteristic D-simplicity is related to the well-studied notion of (strong) F-regularity. This has led to the expectation that in characteristic 0 D-simplicity should correspond to a class of singularities called klt singularities (the characteristic 0 analogue of strong F-regularity). In this talk, we'll show that this is unfortunately not the case: klt singularities may fail to be D-simple, thus illustrating the different behavior of differential operators in characteristic zero and characteristic p. In the first of the two talks, we'll define the basic notion of D-simplicity, give some of the known examples and consequences, and state our counterexample and some of the implications. In the second talk, we'll discuss the proof that this is actually a counterexample, which involves a nice connection between the differential operators on a graded ring R (with isolated singularities) and the global geometry of Proj R.