VIRTUAL ONLY: Dmodules, Group Actions, and Frobenius: Computing on Singularities
Institute for Computational and Experimental Research in Mathematics (ICERM)
August 9, 2021  August 13, 2021
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Monday, August 9, 2021

9:45  10:00 am EDTWelcomeVirtual
 Brendan Hassett, ICERM/Brown University

10:00  10:40 am EDTOn the Hodge filtration on local cohomologyVirtual
 Speaker
 Mircea Mustaţă, University of Michigan
 Session Chair
 Uli Walther, Purdue University
Abstract
The local cohomology sheaf of a smooth complex variety along a closed subvariety comes endowed with a Hodge filtration, via Saito's theory of mixed Hodge modules. I will discuss some properties of this filtration, based on joint work with Mihnea Popa.

10:50  11:00 am EDTBreakCoffee Break  Virtual

11:00  11:40 am EDTExtremal Singularities in Positive CharacteristicVirtual
 Speaker
 Karen Smith, University of Michigan
 Session Chair
 Uli Walther, Purdue University
Abstract
What is the most singular possible singularity? What can we say about its geometric and algebraic properties? This seemingly naive question has a sensible answer in characteristic p. The ""Fpure threshold,"" which is an analog of the log canonical threshold, can be used to ""measure"" how bad a singularity is. The Fpure threshold is a numerical invariant of a point on (say) a hypersurfacea positive rational number that is 1 at any smooth point (or more generally, any Fpure point) but less than one in general, with ""more singular"" points having smaller Fpure thresholds. We explain a recently proved lower bound on the Fpure threshold in terms of the multiplicity of the singularity. We also show that there is a nice class of hypersurfaceswhich we call ""Extremal hypersurfaces""for which this bound is achieved. These have very nice (extreme!) geometric properties. For example, the affine cone over a non Frobenius split cubic surface of characteristic two is one example of an ""extremal singularity"". Geometrically, these are the only cubic surfaces with the property that *every* triple of coplanar lines on the surface meets in a single point (rather than a ""triangle"" as expected)a very extreme property indeed.

11:50 am  1:00 pm EDTLunch/Free TimeVirtual

1:00  1:40 pm EDTPure subrings of polynomial ringsVirtual
 Speaker
 Anurag Singh, University of Utah
 Session Chair
 Uli Walther, Purdue University
Abstract
Let G be a linearly reductive group over a field K, with a linear action on a polynomial ring over K. Then the invariant ring is a pure subring of the polynomial ring; many key properties of classical invariant rings including finite generation and the CohenMacaulay property, as in the HochsterRoberts theorem, follow from purity. Now let A denote either a field, or the ring of integers, or a ring of padic integers. When is a given finitely generated Aalgebra a pure subring of a polynomial ring over A? We will discuss how this can be addressed via Dmodules, Group Actions, and Frobenius! The recent Computing on Singularities is joint work with Jack Jeffries.

1:50  2:00 pm EDTBreakCoffee Break  Virtual

2:00  2:40 pm EDTPrimary decomposition with differential operatorsVirtual
 Yairon CidRuiz, Ghent University
Abstract
We introduce differential primary decompositions for ideals in a commutative ring. Ideal membership is characterized by differential conditions. The minimal number of conditions needed is the arithmetic multiplicity. Minimal differential primary decompositions are unique up to change of bases. Our results generalize the construction of Noetherian operators for primary ideals in the analytic theory of Ehrenpreis Palamodov, and they offer a concise method for representing affine schemes. The case of modules is also addressed. This is joint work with Bernd Sturmfels.

3:00  5:00 pm EDTGathertown Afternoon CoffeeCoffee Break  Virtual
Tuesday, August 10, 2021

10:00  10:40 am EDTNonabelian transformationsVirtual
 Speaker
 Botong Wang, University of Wisconsin
 Session Chair
 Claudiu Raicu, University of Notre Dame
Abstract
As the constructible counterpart of the FourierMukai transformation, the nonabelian Mellin transformation of a constructible complex can be considered as taking the hypercohomology of the complex twisted by all possible local systems simultaneously. We will explain a texactness result about the nonabelian Mellin transformation, generalizing a theorem of GabberLoeser on affine torus. We will also discuss some local vanishing properties of the Sabbah's specialization functor, which is a key step in the proof of the texactness result.

10:50  11:00 am EDTBreakCoffee Break  Virtual

11:00  11:40 am EDTBernstein's inequality and holonomicity for certain singular ringsVirtual
 Speaker
 Jack Jeffries, University of NebraskaLincoln
 Session Chair
 Claudiu Raicu, University of Notre Dame
Abstract
We prove the Bernstein inequality and develop the theory of holonomic Dmodules for rings of invariants of finite groups in characteristic zero, and for strongly Fregular finitely generated graded algebras with FFRT in prime characteristic. In each of these cases, the ring itself, its localizations, and its local cohomology modules are holonomic. We also show that holonomic Dmodules, in this context, have finite length. This is based on joint work with Josep Àlvarez Montaner, Daniel J. Hernández, Luis NúñezBetancourt, Pedro Teixeira, and Emily E. Witt.

11:50 am  1:00 pm EDTLunch/Free TimeVirtual

1:00  1:40 pm EDTTBAVirtual
 Speaker
 Mireille Boutin, Purdue University
 Session Chair
 Claudiu Raicu, University of Notre Dame

1:50  2:00 pm EDTBreakCoffee Break  Virtual

2:00  3:00 pm EDTProblem SessionVirtual

3:00  5:00 pm EDTGathertown Afternoon CoffeeCoffee Break  Virtual
Wednesday, August 11, 2021

10:00  10:40 am EDTBernsteinSato ideals and RiemannHilbert correspondence for Alexander complexesVirtual
 Speaker
 Lei Wu, KU Leuven
 Session Chair
 Linquan Ma, Purdue University
Abstract
Using Vfiltrations, Kashiwara and Malgrange constructed RiemannHilbert correspondence for nearby and vanishing cycles along a single holomorphic functions. Sabbah then constructed multi Vfiltrations along a finite set of holomorphic functions and thus obtained the multivariate BernsteinSato polynomials. However, Sabbah's method also indicates that the method of Kashiwara and Malgrange can not be generalized to the multivariate case in general. In this talk, first I will explain the construction of BernsteinSato ideals and Alexander complexes by using Mellin transformations. Then, I will focus on the construction of RiemannHilbert correspondence for Alexander complexes (the multivariate generalization of nearby cycles) by using BernsteinSato ideals and relative holonomic Dmodules.

10:50  11:00 am EDTBreakCoffee Break  Virtual

11:00  11:40 am EDTCharacteristicfree definition of holonomicityVirtual
 Speaker
 Gennady Lyubeznik, University of Minnesota,
 Session Chair
 Linquan Ma, Purdue University
Abstract
Most of the theory of Dmodules has been developed only in characteristic zero. This includes holonomic modules. Some candidates for holonomic modules in characteristic p>0 have been proposed using definitions specific to characteristic p>0. The first characteristicfree definition of holonomicity was given in 2010 by the speaker, but only for modules over polynomial rings. In the talk I am going to describe an extension of this definition to arbitrary nonsingular varieties. This is joint work with Wenliang Zhang.

11:50 am  1:00 pm EDTLunch/Free TimeVirtual

1:00  1:40 pm EDTStudying singularities using closure operationsVirtual
 Speaker
 Rebecca R.G., George Mason University
 Session Chair
 Linquan Ma, Purdue University
Abstract
A number of the innovations used in studying singularities in commutative algebra have come from the study of tight closure and its test ideal in rings of equal characteristic. In replicating these results in rings of mixed characteristic, it has been useful to find closure operations that share key properties with tight closure. By studying the shared structure of common closure operations in commutative algebra, we show that many tight closure properties, in particular the structure of the test ideal, hold for a much larger set of closure operations, including (big CohenMacaulay) module closures and mixed characteristic closures. In this talk, I will describe the structures that these closure operations have in common and share some of the results on test ideals that have come out of this theory. Parts of this research are joint with subsets of Neil Epstein, Janet Vassilev, Felipe Pérez, and Zhan Jiang.

1:50  2:00 pm EDTBreakCoffee Break  Virtual

2:00  3:00 pm EDTLightning TalksVirtual

3:00  5:00 pm EDTGathertown Afternoon CoffeeCoffee Break  Virtual
Thursday, August 12, 2021

10:00  10:40 am EDTComputing with equivariant DmodulesVirtual
 Speaker
 András Lőrincz, HumboldtUniversität zu Berlin
 Session Chair
 Claudia Miller, Syracuse University
Abstract
In this talk, I will discuss some results and tools concerning equivariant Dmodules, with a focus on representations of reductive groups having finitely many orbits. In particular, I provide explicit descriptions of: categories of equivariant Dmodules as quivers, Dmodule structures of local cohomology modules supported in orbit closures, Lyubeznik numbers, BernsteinSato polynomials of holonomic functions, character formulas.

10:50  11:00 am EDTBreakCoffee Break  Virtual

11:00  11:40 am EDTSymbolic powers in mixed characteristicVirtual
 Speaker
 Eloísa Grifo, University of Nebraska  Lincoln
 Session Chair
 Claudia Miller, Syracuse University
Abstract
In a polynomial ring over a perfect field, the symbolic powers of a radical ideal consist of the polynomials that vanish to order n on the corresponding variety, and can be described via differential operators. If we replace the field with a DVR, we need both differential operators and Joyal and Buium's notion of a pderivation to give an analogous result. As an application, we will discuss an explicit Chevalley lemma for the symbolic powers of prime ideals in direct summands of polynomial rings. This is joint work with Alessandro De Stefani and Jack Jeffries.

11:50 am  1:00 pm EDTLunch/Free TimeVirtual

1:00  1:40 pm EDTAsymptotic vanishing of local cohomology modulesVirtual
 Speaker
 Wenliang Zhang, University of Illinois at Chicago
 Session Chair
 Claudia Miller, Syracuse University
Abstract
In this talk, I will survey some recent results on asymptotic vanishing of cohomology of lci varieties and explain an approach to extending these results to graded rings over a field. If time permits, I will explain an application of our approach to the study of rings of prime characteristic.

2:00  3:00 pm EDTTBD Virtual

3:00  5:00 pm EDTGathertown Afternoon CoffeeCoffee Break  Virtual
Friday, August 13, 2021

10:00  10:40 am EDTMultiplicity sequencesVirtual
 Speaker
 Claudia Polini, University of Notre Dame
 Session Chair
 Christine Berkesch, University of Minnesota
Abstract
I will report on joint work with Trung, Ulrich, and Validashti

10:50  11:00 am EDTBreakCoffee Break  Virtual

11:00  11:40 am EDT(Irregular) Hodge theory of GKZ systemsVirtual
 Speaker
 Thomas Reichelt, Universität Heidelberg
 Session Chair
 Christine Berkesch, University of Minnesota
Abstract
GKZ hypergeometric systems were introduced by Gelfand, Kapranov and Zelevinsky as a generalization of Gauss hypergeometric differential equation. It can be shown that for certain parameters the GKZsystems carry the structure of an irregular mixed Hodge module, a category recently defined by Claude Sabbah. We will discuss the Hodge and weight filtration of these Dmodules.

11:50 am  1:00 pm EDTLunch/Free TimeVirtual

1:00  1:40 pm EDTTBAVirtual
 Speaker
 Emily Witt, University of Kansas
 Session Chair
 Christine Berkesch, University of Minnesota

1:50  2:00 pm EDTBreakCoffee Break  Virtual

2:00  3:00 pm EDTProblem SessionVirtual

3:00  5:00 pm EDTGathertown Afternoon CoffeeCoffee Break  Virtual
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