Joint UOttawa/Carleton
Algebra Seminar
Fall 2025
November 28 (Carleton University, HP4351 at 13:00)
Jesse Huang (University of Waterloo)
Title: Long Live the King’s Conjecture!
Abstract: King’s conjecture proposed that every smooth projective toric variety admits a full strong exceptional collection of line bundles – equivalently, a tilting bundle composed of line bundles. Although the original conjecture is known to fail in general, recent advances on resolution of diagonal of toric varieties inspired by homological mirror symmetry suggest a new perspective: it can be proved that a birational reformulation of the King’s conjecture is indeed true!
In this talk, we will discuss the new “birational King”, and its new implications on the coherent-constructible correspondence and homological properties of Bondal-Thomson monads. This talk is based on joint works with Favero, and Ballard-Berkesch-Brown-Cranton Heller-Erman-Favero-Ganatra-Hanlon.
November 21 (UOttawa)
Rui Xiong (University of Ottawa)
Title: An ADE classification of Hodge-Tate hyperplanes in Grassmannians
Abstract: In this talk, I will present a new ADE classification of Hodge–Tate hyperplanes in Grassmannians and discuss its potential connections to representation theory. This is joint work with Sergey Galkin, Naichung Conan Leung, and Changzheng Li.
Special time: November 18 (Carleton, 1:00 – 2:30, HP 4351)
Jason Bell (University of Waterloo)
Title: Enveloping algebras of derivations of associative algebras
Abstract: Let k be a field of characteristic zero and let L be a Lie algebra over k. It is a long-standing conjecture that the enveloping algebra of L is noetherian if and only if L is finite-dimensional over k. For many years, the Lie algebra deemed most likely to give a counterexample to this conjecture was the Lie algebra of derivations of the ring of Laurent polynomials over k in one variable. Sue Sierra and Chelsea Walton gave a spectacular proof that the enveloping algebra of this Lie algebra is non-noetherian and hence not a counterexample. It is natural to consider whether one can extend the Sierra-Walton result to cover Lie algebras of derivations of other associative k-algebras. We show that this is indeed the case for all finitely generated associative k-algebras: if A is a finitely generated k-algebra and L is the Lie algebra of k-linear derivations of A then the enveloping algebra is noetherian if and only if L is finite-dimensional over k. This is joint work with Lucas Buzaglo.
October 24 (UOttawa)
Hadi Salmasian (University of Ottawa)
Title: Counting F_q-points, chromatic polynomials, and tableaux
Abstract: Let X be the intersection of a nilpotent GL_n(F_q)-conjugacy class in gl(n,F_q) with an ad-nilpotent ideal of the Borel subalgebra. We give an explicit formula for the number of points of X, which is polynomial in q with positive coefficients. Along the way, we explain the connection between this formula and the (petting) zoo of symmetric functions. This talk will be accessible to advanced undergraduate students. Based on ongoing project with M. Bardestani, K. Karai, and S. Ram.
September 19 (UOttawa)
Cameron Ruether (IMAR, Romania)
Title: The Canonical Quadratic Pair on Clifford Algebras
Abstract: Quadratic pairs on central simple algebras over a field were introduced in The Book of Involutions in order to manage the problems which arise with orthogonal involutions in characteristic 2. They are used throughout the book to give a characteristic agnostic treatment of semisimple groups of type D. However, a notable exception occurs in the final chapters on triality where the base field is assumed to be of characteristic different from 2. The authors make this assumption because, as they write, “we did not succeed in giving a rational definition of the quadratic pair on [the Clifford algebra].” This problem was solved by Dolphin and Quéguiner-Mathieu, who defined the canonicalquadratic pair while working over fields of characteristic 2. The notion of quadratic pairs has been extended by Calmés and Fasel beyond working over fields to the setting over schemes. We show that there exists a canonical quadratic pair on Clifford algebras in this setting as well, which extends the definition of Dolphin and Quéguiner-Mathieu. However, due to some quirks of working over schemes, the approach of Dolphin and Quéguiner-Mathieu requires some modification. We will review the basic definitions and properties of quadratic pairs, over fields and over schemes, and then describe the modified construction of the canonical pairon Clifford algebras. We will outline the key properties which earn it the name “canonical” and which allow for the theory of triality to proceed over schemes.
September 12 (UOttawa)
Susanne Pumpluen (University of Nottingham, UK)
Title: Nonassociative cyclic algebras
Abstract: Nonassociative algebras are not very well known, but have recently been used repeatedly in several surprising applications, ranging from space-time block coding to Learning with Errors (LWE). The LWE problem is the foundation of modern lattice-based cryptography.
Mendelsohn and Cong (2005) just developed a non-associative version of LWE based on nonassociative cyclic algebras and their lattices that is provably secure, and seems to have greater freedom in choices of parameters.
The structure of nonassociative algebras and of their automorphisms is very similar to the structure of their well known associative cousins.
In this talk, I will introduce the structure of these algebras, give ideas how to work with their lattices, and time-permitting also present their automorphisms. (Part of this work is joint with M Nevins).
August 7 – Special summer session (UOttawa)
15:30-16:20 Cameron Ruther (IMAR, Romania)
Deformations of Azumaya Algebras with Quadratic Pair
The deformation theory of Azumaya algebras, or more generally of torsors for an algebraic group G, is well understood from the works of Grothendieck, Illusie, and others. Azumaya algebras with quadratic pair are the characteristic independent analogue of Azumaya algebras with orthogonal involution and thus they correspond to PGO-torsors. The general formalism says that for a given Azumaya algebras with quadratic pair over a base scheme, and a given infinitesimal thickening of the base scheme, a deformation of the algebra with quadratic pair exists if and only if an obstruction class in the second cohomology of the Lie algbera is zero. However, there is of course a forgetful map where one forgets the quadratic pair and simply asks about the deformation theory of the underlying Azumaya algebra. We ask: does there exist an Azumaya algebra with quadratic pair whose deformation obstruction is non-zero but such that the underlying Azumaya algebra does have deformations? We show that this cannot happen if 2 is invertible over the base scheme, but, more interestingly, we outline the construction of an example where this phenomenon does occur over an Igusa surface, which is a scheme in characteristic 2.
16:30-17:20 Susanne Pumpluen (University of Nottingham, UK)
How to classify skew polycyclic codes using algebra isomorphisms that preserve Hamming distance.
(Part two of this talk is about joint work with M Nevins.)
Cyclic and constacyclic codes, and the more general polycyclic codes, have been used to derive quantum codes and are promising candidates for code-based postquantum cryptography. Skew constacyclic codes, and more generally skew polycyclic codes (also called skew quasi-cyclic codes (SQPCs) in Bag, Panario (2025)) are built from skew polynomials and also candidates for quantum codes.
We show how to employ algebra isomorphisms to define different equivalence relations between skew polycyclic codes to avoid duplicating existing codes. Codes in the same equivalence class will have the same performance parameters (length, dimension, and minimum distance). This approach was first exploited by Chen, Fan, Lin, Liu (2012) for constacyclic codes over finite fields, and recently generalized by Ou-azzou, Najmeddine, and Nuh Aydin (2025) to skew constacylic codes and by Ou-azzou, Horlemann, Aydin to skew polycyclic codes over finite fields. We will choose more general notions of equivalence and isometry. For skew constacyclic codes, we prove with combinatorial methods that the notions of isometry and equivalence defined by Ou-azzou et all coincide when the associated algebras are not associative.