Organizing Committee
 Christine Berkesch
University of Minnesota  Linquan Ma
Purdue University  Claudia Miller
Syracuse University  Claudiu Raicu
University of Notre Dame  Uli Walther
Purdue University
Abstract
The adoption of Dmodule techniques has transformed the interface between commutative algebra and algebraic geometry over the last two decades. The discovery of interactions and parallels with the Frobenius morphism has been an impetus for many new results, including new invariants attached to singularities but also D and Fmodule based algorithms for computing quantities that used to be unattainable.
Our goal for this workshop is to discuss computational aspects and new challenges in singularity theory, focusing on special varieties that arise from group actions, canonical maps, or universal constructions. By bringing together geometers, algebraists, and invariant theorists, we will address problems from multiple perspectives. These will include comparisons of composition chains for D and Fmodules, the impact of group actions on singularity invariants, and the structure of differential operators on singularities in varying characteristics.
Confirmed Speakers & Participants
Talks will be presented virtually or inperson as indicated in the schedule below.
 Speaker
 Poster Presenter
 Attendee
 Virtual Attendee

Zainab AlMaamari
Sultan Qaboos University

Josep Àlvarez Montaner
Universitat Politècnica de Catalunya

Trevor Arrigoni
University of Kansas

Daniel Bath
KU Leuven

Christine Berkesch
University of Minnesota

Neelima Borade
Princeton University

Mireille Boutin
Purdue University

Anna Brosowsky
University of Michigan

Laura Brustenga i Moncusi
University of Copenhagen

Qianyu Chen
Stony Brook University

Yairon CidRuiz
Ghent University

SUTIRTHA DATTA
Chennai Mathematical Institute

Alessandro De Stefani
Università di Genova

Rachel Diethorn
Yale University

Bradley Dirks
University of Michigan

Shanna Dobson
University of California at Riverside

Florian Enescu
Georgia State University

Eleonore Faber
University of Leeds

Francesca Gandini
Kalamazoo College

Jack Garzella
University of California, San Diego

Eloísa Grifo
University of Nebraska  Lincoln

Justin Hilburn
Perimeter Institute for Theoretical Physics

Mee Seong Im
United States Naval Academy

Jack Jeffries
University of NebraskaLincoln

Minyoung Jeon
The Ohio State University

SeungJo Jung
Jeonbuk National University

Moty Katzman
University of Sheffield

Jennifer Kenkel
University of Michigan

Adam LaClair
Purdue University

Jonghyun Lee
University of Michigan

Xiaobin Li
Southwest Jiaotong University

David Lieberman
University of NebraskaLincoln

András Lőrincz
HumboldtUniversität zu Berlin

Gennady Lyubeznik
University of Minnesota,

Linquan Ma
Purdue University

Saikat Maity
University of Calcutta

Devlin Mallory
University of Michigan

Jason McCullough
Iowa State University

Claudia Miller
Syracuse University

Mircea Mustaţă
University of Michigan

Molena Nguyen
North Carolina State University

Sebastián Olano
University of Michigan

Swaraj Pande
University of Michigan

Vaibhav Pandey
University of Utah

Abraham Pascoe
University of Kansas

Shravan Patankar
University of Illinois at Chicago

Michael Perlman
Queen's University

Sasha Pevzner
University of Minnesota, Twin Cities

McCleary Philbin
University of Minnesota

Claudia Polini
University of Notre Dame

Zhijun (George) Qiao
University of Texas Rio Grande Valley

Ming Hao Quek
Brown University

Eamon QuinlanGallego
University of Michigan

Rebecca R.G.
George Mason University

Claudiu Raicu
University of Notre Dame

Thomas Reichelt
Universität Heidelberg

Sudeshna Roy
Chennai Mathematical Institute

Noussaiba Saadoudi
UMBB & USTHB

Afshan Sadiq
University of Sussex

Mahrud Sayrafi
University of Minnesota

Anurag Singh
University of Utah

JYOTI SINGH
VNIT, Nagpur

Ilya Smirnov
KTH Royal Institute of Technology

Karen Smith
University of Michigan

Daniel Smolkin
University of Michigan

Avi Steiner
University Of Western Ontario

Sheng Tan
Purdue University

Grisha Taroyan
NRU HSE

Will Traves
United States Naval Academy

Bernd Ulrich
Purdue University

Keller VandeBogert
Notre Dame University

Janet Vassilev
University of New Mexico

Sophia Vassiliadou
Georgetown University

Duc Vo
Harvard University

Joe Waldron
Michigan State University

Uli Walther
Purdue University

Yinan Wang
University of Michigan

Botong Wang
University of Wisconsin

Emily Witt
University of Kansas

Lei Wu
KU Leuven

Wenliang Zhang
University of Illinois at Chicago
Workshop Schedule
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 (Virtual)
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 (Virtual)
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 (Virtual)
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
 Speaker
 Yairon CidRuiz, Ghent University
 Session Chair
 Uli Walther, Purdue University (Virtual)
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 Mellin transformationVirtual
 Speaker
 Botong Wang, University of Wisconsin
 Session Chair
 Claudiu Raicu, University of Notre Dame (Virtual)
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 (Virtual)
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 EDTHearing the shape of a room: an unlabeled distance geometry problemVirtual
 Speaker
 Mireille Boutin, Purdue University
 Session Chair
 Claudiu Raicu, University of Notre Dame (Virtual)
Abstract
Suppose that some microphones are placed on a drone inside a room with planar walls/floors/ceilings. A loudspeaker emits a sound impulse and the microphones receive several delayed responses corresponding to the sound bouncing back from each planar surface. These are the firstorder echoes. In this talks, we will discuss the problem of reconstructing the shape of the room from these echoes, an unlabeled distance geometry problem. The time delay for each echo determines the distance from the microphone to a mirror image of the source reflected across a wall. Since we do not know which echo corresponds to which wall, the distances are unlabeled. The problem is to figure out under which circumstances, and how, one can find out the correct distancewall assignments and reconstruct the wall positions. This is joint work with Gregor Kemper.

1:50  2:00 pm EDTBreakCoffee Break  Virtual

2:00  3:00 pm EDTProblem SessionVirtual
 Speakers
 Yairon CidRuiz, Ghent University
 Jack Jeffries, University of NebraskaLincoln
 András Lőrincz, HumboldtUniversität zu Berlin
 Mircea Mustaţă, University of Michigan
 Anurag Singh, University of Utah
 Botong Wang, University of Wisconsin
 Session Chair
 Claudiu Raicu, University of Notre Dame (Virtual)

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 (Virtual)
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 (Virtual)
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 (Virtual)
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
 Speakers
 Daniel Bath, KU Leuven
 Neelima Borade, Princeton University
 Shanna Dobson, University of California at Riverside
 Justin Hilburn, Perimeter Institute for Theoretical Physics
 Mee Seong Im, United States Naval Academy
 Devlin Mallory, University of Michigan
 Session Chair
 Linquan Ma, Purdue University (Virtual)
Abstract
A noncommutative analog of the PeskineSzpiro Acyclicity Lemma.
Daniel Bath
Given a complex of finite Rmodules (R commutative, Noetherian, local) satisfying a sliding projective dimension condition, the PeskineSzpiro Acyclicity Lemma provides a criterion for verifying this complex is exact. We present a noncommutative analog for R an Auslander regular ring. The lemma is especially useful for Dmodules and D[s]modules as the requisite hypotheses are more easily verifiable. Potential use cases will be discussed, time permitting.
Minimal faithful permutation representations of finite groups
Neelima Borade
In this short talk we will introduce the notion of a minimal faithful permutation representation of a finite group and Cayley’s constant p(G) that measures its size. We’ll outline the history of the computation of p(G) for finite groups G and focus primarily on the case when G is a linear group. We'll end with some results for p(G) of a general linear group G based on my undergraduate paper with Dr. TaklooBighash.
Diamonds in Langlands Local Functoriality
Shanna Dobson
We use Scholze's "diamond" construct and Scholze and Fargue's geometrization of the local Langlands correspondence on the FarguesFontaine Curve to investigate Local Functoriality for padic groups and a concomitant diamond universal construction.
Tate's thesis and 3d mirror symmetry
Justin Hilburn
Geometric local Langlands for the multiplicative group G_m, as proved by Beilinson and Drinfeld, is the identification of the monoidal category of Dmodules on the loop group G_m((t)) with with the category of coherent sheaves on the moduli space of G_m local systems on the punctured formal disc. This can be thought of as an infinite dimensional version of the Mellin transform. In joint work with Sam Raskin, I gave a coherent description of the Dmod(G_m((t)))module category of Dmodules on the loop space A*1((t)). This can be thought of as a geometric version of Tate's thesis. If time permits I will explain the statement of this result and how it is related to the physics of 3d mirror symmetry.
Singularities of a modification of the GrothendieckSpringer resolution
Mee Seong Im
I will introduce the extended Borel moment map, which is related to the Springer resolution and the GrothendieckSpringer resolution. The affine quotient of this map has interesting singular locus, which is intimately related to the HilbertChow morphism. I will describe all the important objects mentioned in this abstract, and provide reasons why it is important to study the singular locus of this map.
Differential operators on singular rings
Devlin Mallory
At least since the work of Levasseur and Stafford in the 1980s, the question had been asked of whether one can characterize singularities of rings via certain properties of their rings of differential operators. In particular, one question is whether a ring with mild singularities is a simple module under the action of its ring of differential operators. While an answer in characteristic p had been provided by Karen Smith, no answer had been forthcoming in characteristic 0. We provide a counterexample showing that the expected connection does not exist, through the study of the global geometry of Fano varieties. More specifically, we show that certain del Pezzo surfaces do not have big tangent bundles, and thus their homogeneous coordinate rings are not simple under the action of their rings of differential operators, despite having “mild” singularities. 
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 (Virtual)
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 (Virtual)
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 (Virtual)
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 EDTLightning TalksVirtual
 Speakers
 Swaraj Pande, University of Michigan
 Sudeshna Roy, Chennai Mathematical Institute
 Afshan Sadiq, University of Sussex
 JYOTI SINGH, VNIT, Nagpur
 Session Chair
 Claudia Miller, Syracuse University (Virtual)
Abstract
Multiplicities of jumping numbers
Swaraj Pande
Multiplier ideals are important invariants of singularities of algebraic varieties. Jumping numbers and their multiplicities are numerical invariants arising from multiplier ideals, and are connected to other singularity invariants. In this talk, we'll present some new finiteness results for multiplier ideals of a point scheme. Namely, that the multiplicities of jumping numbers fit into a quasipolynomial. Further, we'll see how the rate of growth of this quasipolynomial is closely related to the valuations that compute jumping numbers.
On derived functors of graded local cohomology modules  II
Sudeshna Roy
This is a joint work with Puthenpurakal and Singh.
On Derived Functors of Graded Local Cohomology Modules
Jyoti Singh
This is the joint work with Prof. Tony J. Puthenpurakal, IIT Bombay.
Primary Decomposition of Binomial Modules
Afshan Sadiq
Let K be a field and R be a polynomial ring in n variables. A binomial module is a submodule of R^m generated by binomials. The aim of this paper is to prove that a binomial module has a primary decomposition into binomial primary modules and the associated primes are binomial ideals. The idea is to generalize the paper of Eisenbud and Strumfels. Eisenbud, D.; Sturmfels, B.: Binomial ideals. Duke Mathematical Journal 84 (No. 1), 145 (1996). 
3:00  5:00 pm EDTGathertown Afternoon CoffeeCoffee Break  Virtual
Friday, August 13, 2021

10:00  10:40 am EDTDifferents of Pfaffians and Determinantal IdealsVirtual
 Speaker
 Claudia Polini, University of Notre Dame
 Session Chair
 Christine Berkesch, University of Minnesota (Virtual)
Abstract
In joint work with Kustin and Ulrich we compute the Kaehler different and the Dedekind different for several classes of ideals. Our techniques include residual intersections and linkage theory. In particular we obtain interesting formulas for determinantal ideals of generic matrices and perfect Gorenstein ideals of height three.

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 (Virtual)
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 EDTThe BernsteinSato polynomial of a simple plane algebroid branchVirtual
 Speaker
 Emily Witt, University of Kansas
 Session Chair
 Christine Berkesch, University of Minnesota (Virtual)
Abstract
We calculate the BernsteinSato polynomial of the irreducible power series in two variables that have the simplest topology, via reduction to characteristic p. To do so, we determine uniform formulas for certain numerical invariants in prime characteristic by constructing, and solving, integer programs.
This is joint work with Daniel Hernández. 
1:50  2:00 pm EDTBreakCoffee Break  Virtual

2:00  3:00 pm EDTProblem SessionVirtual
 Speakers
 Eloísa Grifo, University of Nebraska  Lincoln
 Claudia Polini, University of Notre Dame
 Rebecca R.G., George Mason University
 Thomas Reichelt, Universität Heidelberg
 Emily Witt, University of Kansas
 Lei Wu, KU Leuven
 Wenliang Zhang, University of Illinois at Chicago
 Session Chair
 Christine Berkesch, University of Minnesota (Virtual)

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