Testing for isomorphism between algebraic structures is a fundamental problem in computational algebra, and the most challenging cases occur between nilpotent structures. A common strategy is to identify characteristic substructures that are invariant under automorphisms, since doing so can lead to dramatic reductions in search spaces. However, even with a ready supply of these substructures, a number of perplexing computational issues stand in the way of an effective, systematic approach to isomorphism testing.

In this talk I will present recent results that formulate the “characteristic” property— which seems inherently to be a local property defined in terms of automorphisms of the parent structure — as a global property of the ambient category. Moreover, the categorical objects associated with characteristic substructures can be described constructively in a way that can, in principle, help to resolve some of the aforementioned computational issues in isomorphism testing. 

This is a preliminary report on ongoing joint work with Heiko Dietrich, Josh Maglione, Eamonn O’Brien, and James Wilson.

Let $W$ be a finite simply laced Weyl group. In some cases, the maximal subsets of mutually orthogonal roots of $W$ can themselves be regarded as a discrete structure that spans an irreducible $W$-module. This structure itself behaves very much like a root system, and satisfies an interesting set of three-term linear relations, generalizing the famous Ptolemy relations. I will explain how all this relates to noncrossing partitions, the Fano plane, and Steiner quadruple systems.

I will show, in a concrete example, how computer calculations can show that a certain group generated by given matrices over a ring $R$ has infinite index in $GL(n,R)$. 

I will present some new conjectures about asymptotic weight multiplicities involving finite-dimensional irreducible modules for finite-dimensional simple Lie algebras (over C). These conjectures arise while analyzing coloured Jones polynomials of torus knots.

Let G be an algebraic group over an algebraically closed field K, and let V be a finite-dimensional G-module. In Weyl’s terminology, a first fundamental theorem of invariant theory (FFT) for the pair (G,V) is a set of generators for the ring of invariants K[U]^G, where U is a direct sum of copies of V and its dual G-module V^*. A second fundamental theorem of invariant theory (SFT) for (G,V) is a generating set for the ideal of relations among these generators. In this talk, I will discuss the arc space analogues of Weyl’s FFT and SFT for the standard representations of the classical groups GL_n, SL_n, and Sp_{2n}. These results have several applications to vertex algebras which I will also discuss. This is a joint work with Bailin Song.

Partial groups are group-like structures where instead of a binary product, one has total product defined only on a subset of words of the underlying set. They were introduced by Chermak ultimately for the purpose of studying the $p$-local structure of a finite group, which can itself be distilled into a very special type of partial group called a locality.  Gonzalez showed that a partial group can be viewed as a particular type of simplicial set. We explain how to view the category of partial groups as a reflective (full) subcategory of presheaves on the category of nonempty finite sets and all functions. This leads to a concrete way for computing colimits of partial groups.

Modular forms played a fundamental role in the birth (McKay 1978) and proof (Borcherds 1992) of the Monstrous Moonshine conjecture. Building on this bridge between number theory and infinite dimensional Lie algebras, Zhu established in 1996 the modular invariance for the graded traces of any vertex operator algebra satisfying certain finiteness and semisimplicity conditions. In the non-semisimple but finite setting, Miyamoto more recently showed that in addition to the graded traces, one needs to incorporate other functions, called graded pseudo-traces, to obtain an SL(2, \mathbb{Z}) invariant space of (pseudo-)characters. In this talk I will present our recent results regarding the existence of graded pseudo-traces in the non-finite cases of the Heisenberg and Virasoro vertex algebras. This talk is based on joint work with Barron, Batistelli and Yamskulna.

A generalization of Alexander quandles, principal quandles are built from a group G together with an automorphism of G. The case in which G is abelian was characterized by Holmes, and this result was subsequently generalized to principal quandles over certain nonabelian groups by Higashitani and Kurihara. In this talk, I will discuss results of Petr Vojtechovsky and myself detailing the extension of this characterization to principal quandles over any group.

A refined version of the McKay Conjecture, as proposed by Navarro, suggests that the bijection between the sets of irreducible characters with p'-degree of G and the normalizer of a Sylow p-subgroup remains unchanged under specific Galois automorphisms. 

Similar to the McKay Conjecture, the proof has been reduced to inductive conditions by Navarro, Späth, and Vallejo.I will introduce these inductive conditions and present some progress on their verification for finite groups of type A.

The famous Moufang Theorem states that if three elements associate in a Moufang loop, they generate an associative subloop, that is, a group. This is an instance of a general phenomenon that we call propagating equations. I will give a brief introduction to propagating equations. As an application, we construct varieties of loops in which the analog of the Moufang Theorem also holds, solving an open problem. This is joint work with Ales Drapal.