Junior Algebra Seminar (Algebraic Topology and Group Theory)
The Junior Algebra Seminar is a new seminar at Warwick for young researchers in algebra, with a particular focus on algebraic topology, group theory, and associated areas.
In Term 2 the seminar will be held on Mondays 2pm-3pm in B3.02, Zeeman Building, except in weeks 2 and 3.
In Term 3, the seminar will be held on Thursdays 3pm-4pm in MS.05, Zeeman Building, except in week 2.
Roughly, odd weeks will focus on group theory and related areas, whilst even weeks will focus on algebraic topology and related areas.
The organisers of the seminar are Nathan Lockwood / Mak Pehar and Dan Roebuck (algebraic topology), and Michael Cavaliere (group theory). Please get in touch with the relevant organisers if you would like to give a talk or suggest a speaker!
This seminar replaces the previous AGATA seminar, which ran in term 3 of the 2022/23 academic year. That seminar focused on algebraic geometry alongside algebraic topology and general algebra; the junior seminar at Warwick focusing on algebraic geometry now is JAWS.
Initially discovered by Gaschutz in the 1950s, before later being extended by Dalla Volta and Lucchini in the 1990s, the theory of crowns in finite groups has a long and rich history, with numerous applications. Central to this theory is the observation that establishing generation results for a particular class of groups, known as crown-based powers, is often sufficient to derive corresponding results for all finite groups. In this talk, we will explore how this framework can be applied to a range of generation problems, highlighting in particular how the structure of a group’s chief factors determines its generation behaviour.
Yorick Fuhrmann (Warwick): Lifting heavy-weight structures on stable infinity-categories
Introduced by Bondarko, weight structures play an important role in the toolbox when studying stable infinity-categories. In this talk I will give an introduction to the subject and then explain how they contribute to the study of a linear version of equivariant spectra, namely modules over the Eilenberg-MacLane spectrum of the constant Mackey functor. Ultimately, we will consider the Picard group of this category.
Josh Bridges (Birmingham) - Small essential 2-subgroups in fusion systems
A (saturated) fusion system on a p-group P contains data about conjugacy within P, the typical case being the system induced by a group on its Sylow p-subgroup. Fusion systems are completely determined by looking at their essential subgroups, which must admit an automorphism of order coprime to p. For p=2, we describe two new methods that address the question: given an essential subgroup E<P of a fusion system on P, what can we say about P? In particular, one method gives us sufficient conditions to deduce that E is normal in P, while the other explores cases where we have strong control over the normaliser tower of E in P.
Jean Paul Schemeil (Nottingham) - Voevodsky motives and their tensor-triangular geometry
The category of Voevodsky motives provides an algebraic-geometric analogue of the derived category of abelian groups in topology; however, the richness of the algebraic and geometric data it encodes makes it a substantially more complicated object to study. A promising framework for organising this complexity is through Balmer’s tensor-triangular (tt) geometry: the goal of this program is to obtain a classification of motives up to the tensor-triangulated structure of their ambient category via the computation of its Balmer spectrum. In this talk, I aim to first provide a general overview of the subject: we will review the bounds obtained by Vishik through the detection of isotropic points, as well as the lower bounds established by Balmer–Gallauer through the computation of the spectrum of Artin(-Tate) motives. I will conclude by focusing on the tt-geometry of motives of real quadrics, discussing how the computation of this spectrum fits within these established bounds.
Let G be a transitive permutation group on a set Ω with point stabiliser H. Recall that a base for G is a subset of Ω whose pointwise stabiliser in G is trivial. The minimal size of a base is called the base size of G, denoted b(G), and this classical invariant has been widely studied for many years, finding many applications.
Observe that b(G) coincides with the smallest positive integer k such that G has a regular orbit on the Cartesian product (G/H)^k. As a natural generalisation, we can consider the minimal number r such that G has a regular orbit on (G/H_1) x … x (G/H_r) for any core-free subgroups H_1,..., H_r of G. We call this invariant the regularity number of G.
In this talk, we will explain how a combination of probabilistic, combinatorial and computational methods can be used to study the regularity number of finite almost simple and simple algebraic groups, as well as several other related problems. In particular, we will report on a generalisation of Cameron's base size conjecture and some recent progress on similar problems for simple algebraic groups.
Jake Saunders (Sheffield) - Higher algebra in characteristic 2
Abstract: In higher algebra, commutativity is a structure rather than a property, with different strengths of commutativity parametrised by En operads. Homotopy-commutative ring spectra of characteristic 2 are characterised by their homology, making calculations in this context a bit more convenient. However, such a ring spectrum may admit multiple inequivalent En refinements. In this talk I will introduce the relevant theory and discuss some examples of this phenomenon, as well as calculations which can be used to construct algebraic models of geometrically-constructed ring spectra.
Lenny Greenfield (Leeds) - Orbits on Graph Colourings
We consider the set of all possible colourings of a finite graph using a finite number of colours. We define an action of the automorphism group of the graph on the set of all colourings where two colourings are equivalent if there exists an automorphism that moves the colours of vertices in one colouring to the colours of vertices in the other colouring. In this talk we will discuss methods for calculating the number of distinct colourings when the automorphism group action on the graph is transitive.
A twofold symmetric monoidal category is a category with two symmetric monoidal products, with the same unit, such that one product laxly distributes over the other. To make this definition work in the setting of infinity-categories, one may hope for a simple combinatorial category that controls the theory, similar to how finite sets control the theory of symmetric monoidal infinity-categories. Recently, Barwick has identified this category with a certain subcategory of the category of finite graphs. In my talk, I will give an overview of these ideas and describe some related constructions I'm thinking about.
Term 3
Jimmy Bryden (Manchester) - Commuting Involution Graphs of L_2(q) and Other Simple Groups
In the study of finite simple groups, involutions (elements of order 2) play an important role. Two often cited examples of this are the Feit-Thompson and Brauer-Fowler theorems which tell us that every non-abelian finite simple group contains an involution and that the involution centralisers control lots of the group structure. Another example, more in line with this talk, is the construction of some of the sporadic simple groups via the automorphism groups of commuting involution graphs. If G is a finite group and X is a conjugacy class of involutions of G, the graph with vertex set X in which two distinct vertices are joined if and only if they commute in G is known as the commuting involution graph of G on X. Much is known about the connectivity properties of commuting involution graphs for simple groups. However, outside of the few examples alluded to above, almost nothing is known about the automorphism groups of such graphs. In this talk I hope to communicate some interesting recent developments in this area, particularly relating to the simple group L_2(q).
Eunike Setiawan (Imperial College London) - Title TBC
Malcolm Chen (Manchester) - Covering numbers of groups, rings, and modules
A cover of an algebraic structure A is a collection of proper substructures whose set-theoretic union equals A, and the covering number of A is the cardinality of minimal such cover. While this notion is motivated by group theory and has been studied extensively for groups, many problems remain open. In contrast, recent work of Swartz and Werner [J. Algebra 639, 249–280 (2024)] gives a complete classification of rings admitting finite covers by subrings, as well as their covering numbers.
We will begin the talk with a brief survey of known results on covering numbers of groups and rings. The main focus of the talk will then be on my joint work with Swartz and Werner [Comm. Algebra, in press], in which we obtain a complete classification of rings covered by ideals, as well as modules covered by submodules. We will also highlight the key similarities and differences between these results on groups, rings, and modules. The talk will be entirely self-contained, with no technical background assumed beyond a first course in algebra.
