This workshop will be offered virtually. The in-person meeting has been cancelled due to the COVID-19 outbreak. A schedule for virtual talks will be posted soon. Accepted participants will be notified how to access the virtual presentations.
Organizing Committee
Abstract

The structure of free resolutions plays an important role in analyzing singularities of varieties of low codimension. Codimension 2 Cohen-Macaulay varieties (resp. codimension 3 Gorenstein varieties) come from rank conditions on an n x (n+1) matrix (resp. a skew-symmetric (2n+1) x (2n+1) matrix).

This workshop seeks to push such results to Cohen-Macaulay varieties of codimension 3 and Gorenstein varieties of codimension 4.

This problem turns out to be related to the classification of semi-simple Lie algebras. These new methods allow one to create a ‘map’ of free resolutions of a given format. The calculations that arise are very demanding and require new computational methods involving both commutative algebra and representation theory.

The organizers have shared two sets of notes for attendees to review before the workshop. These are downloadable here:

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Confirmed Speakers & Participants

  • Speaker
  • Poster Presenter
  • Attendee
  • Virtual Attendee

Workshop Schedule

Monday, August 3, 2020
TimeEventLocationMaterials
9:50 - 10:00am EDTWelcome - ICERM Director  
10:00 - 10:40am EDTRees algebras of grade three Gorenstein ideals - Bernd Ulrich, Purdue University 
10:55 - 11:15am EDTThe Family of Perfect Ideals of Codimension 3, of Type 2 with 5 Generator - Ela Celikbas, West Virginia University 
11:25 - 12:05pm EDTIntroduction to Schubert varieties I - Jacinta Torres, Karlsruhe Institute for Technology 
12:10 - 12:30pm EDTAn example of calculating a resolution using Hilbert functions - Adela Vraciu, University of South Carolina 
12:40 - 1:30pm EDTBreak for Lunch/ Free Time  
1:30 - 3:30pm EDTOpen Problem Session  
3:30 - 4:00pm EDTCoffee/ Tea Break  
Tuesday, August 4, 2020
TimeEventLocationMaterials
10:00 - 10:40am EDTLinkage and Tor algebra classes of codepth three perfect ideals - Oana Veliche, Northeastern University 
10:55 - 11:15am EDTMinimal free resolutions of orbit closures of quivers - András Lörincz, Humboldt-Universität zu Berlin 
11:25 - 12:05pm EDTIntroduction to gradings of Lie algebras and Schubert varieties II - Sara Angela Filippinni, Jagiellonian University 
12:10 - 12:30pm EDTSome branching formulas for Kac-Moody Lie algebras - Kyu-Hwan Lee, University of Connecticut 
12:40 - 1:30pm EDTBreak for Lunch/ Free Time  
1:30 - 3:30pm EDTDiscussions in subgroups  
3:30 - 4:00pm EDTCoffee/ Tea Break  
Wednesday, August 5, 2020
TimeEventLocationMaterials
10:00 - 10:40am EDTThe syzygies of some thickenings of determinantal varieties - Claudiu Raicu, University of Notre Dame 
10:55 - 11:15am EDTOn growth of the Hilbert function of a quadratic ideal - Sema Gunturkun, Amherst College 
11:25 - 12:05pm EDTSyzygies of Determinantal Thickenings via gl(mjn) Representations - Amy Huang, Texas A&M University 
12:20 - 1:30pm EDTBreak for Lunch/ Free Time  
1:30 - 3:30pm EDTDiscussions in subgroups  
3:30 - 4:00pm EDTCoffee/ Tea Break  
Thursday, August 6, 2020
TimeEventLocationMaterials
10:00 - 10:40am EDTSpinor structures on free resolutions of codimension four Gorenstein ideals - Jai Laxmi, University of Connecticut 
10:55 - 11:15am EDTOn the structure of short, grade-four, Artinian Goresntein algebras - Pedro Marques, University of Évora 
11:25 - 12:05pm EDTHall algebras and i-quantum groups - Weiqiang Wang, University of Virginia 
12:20 - 1:30pm EDTBreak for Lunch/ Free Time  
1:30 - 3:30pm EDTDiscussions in subgroups  
3:30 - 4:00pm EDTCoffee/ Tea Break  
Friday, August 7, 2020
TimeEventLocationMaterials
10:00 - 10:40am EDTThe geometry of nilpotent orbits via subbundles of the cotangent bundle - Eric Sommers, University of Massachusetts Amherst 
10:55 - 11:15am EDTOn a theorem of Happel's - Özgür Esentepe, University of Connecticut 
11:25 - 12:05pm EDTCellular resolutions of powers of monomial ideals of projective dimension one - Liana Sega, University of Missouri Kansas City 
12:20 - 1:30pm EDTBreak for Lunch/ Free Time  
1:30 - 3:30pm EDTDiscussions in subgroups  
3:30 - 4:00pm EDTCoffee/ Tea Break  

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Lecture Videos

Introduction to Schubert varieties I

Jacinta Torres
Institute of Mathematics of the Polish Academy of Sciences
August 3, 2020

Working Groups

Groups will focus on specific open problems related to each topic. There will be discussion to fill in the background of these problems, and we expect the groups to begin work on small subproblems throughout the week.

Resolutions of length 3: licci conjecture, genericity of Schubert examples (lead: Jerzy Weyman)

A connection between free resolutions of length 3 of so-called Dynkin formats and Schubert varieties is investigated in https://arxiv.org/abs/2005.01253. An open problem is to understand whether or not the examples coming from Schubert varieties are generic examples of such resolutions. A related problem is to understand whether every perfect ideal of codimension 3 with a resolution of Dynkin format is licci.

Resolutions of Gorenstein ideals of codimension 4 (lead: Ela Celikbas)

The group will discuss the existence of spinor coordinates on resolutions of Gorenstein ideals of codimension 4 (see https://arxiv.org/abs/1912.07510) and it implications for classifying such ideals, including the expectation that such classification should be easier for 6,7,8 generators. Also description of spinor coordinates in specific examples will be discussed.

Multiplicative structures on resolutions of perfect ideals of codimension 3 (lead: Oana Veliche)

The group will discuss the classification of multiplicative structures and their feasibility for resolutions of different formats. This is related to the material covered in arXiv:1812.11552.

Macaulay 2 software packages (lead: Lars Christensen)

The group will compute the multiplication tables of free resolutions of length 3 and 4 and related higher structure maps. Taking off from several existing pieces of code the group will develop robust and documented code that will be published as M2 packages.

Schubert examples (leads: Jacinta Torres and Sara Angela Filippini)

Continuing from https://arxiv.org/abs/2005.01253, there are open problems regarding how to extend the Schubert examples in Dynkin cases to non-Dynkin settings. Furthermore, the larger examples in the E_8 case are conjectural and some of the E_7 examples rely on computer calculation and a conceptual understanding is desired.

Equivariant ideals and superalgebras (lead: Claudiu Raicu)

This group will investigate the connection between classical Lie superalgebras and families of equivariant ideals in polynomial rings. The primary example is the gl_n x gl_m equivariant ideals in the ring of polynomials on the space of n x m matrices whose linear strands are representations of gl(m|n), see https://arxiv.org/abs/1411.0151.