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Organizing Committee
Texas Tech University
Syracuse University
University of California, San Diego
University of Connecticut and Jagiellonian University, Cracow, Poland
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:
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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Projects
Resolutions of length 3: licci conjecture, genericity of Schubert examples
Jerzy Weyman (University of Connecticut and Jagiellonian University, Cracow, Poland)
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
Ela Celikbas (West Virginia University )
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.
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
Oana Veliche (Northeastern University )
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.
related to the material covered in arXiv:1812.11552.
Macaulay 2 software packages
Lars Christensen (Texas Tech University)
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.
several existing pieces of code the group will develop robust and documented code that will be published as M2 packages.
Schubert examples
Jacinta Torres (Institute of Mathematics of the Polish Academy of Sciences), Sara Angela Filippini (Jagiellonian University)
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.
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
Claudiu Raicu (University of Notre Dame)
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.
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.
Lecture Videos
Ela Celikbas
West Virginia University
August 3, 2020 • 12:00 AM
Jacinta Torres
Institute of Mathematics of the Polish Academy of Sciences
August 3, 2020 • 12:00 AM
Adela Vraciu
University of South Carolina
August 3, 2020 • 12:00 AM
Oana Veliche
Northeastern University
August 4, 2020 • 12:00 AM
András Lőrincz
Humboldt-Universität zu Berlin
August 4, 2020 • 12:00 AM
Sara Angela Filippini
Jagiellonian University
August 4, 2020 • 12:00 AM
Kyu-Hwan Lee
University of Connecticut
August 4, 2020 • 12:00 AM
Claudiu Raicu
University of Notre Dame
August 5, 2020 • 12:00 AM
Sema Güntürkün
Amherst College
August 5, 2020 • 12:00 AM
Amy Huang
Texas A&M University
August 5, 2020 • 12:00 AM
Jai Laxmi
University of Connecticut
August 6, 2020 • 12:00 AM
Pedro Marques
University of Évora
August 6, 2020 • 12:00 AM
Weiqiang Wang
University of Virginia
August 6, 2020 • 12:00 AM
Eric Sommers
University of Massachusetts
August 7, 2020 • 12:00 AM
Özgür Esentepe
University of Connecticut
August 7, 2020 • 12:00 AM
Liana Sega
University of Missouri - Kansas City
August 7, 2020 • 12:00 AM