The Junior Algebra Seminar is a seminar at Warwick for young researchers in algebra, with a particular focus on algebraic topology, group theory, and associated areas.
In Term 1, the seminar will be held on Mondays 2pm-3pm in B1.01, Zeeman Building.
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 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!
Proofs and constructions in (higher) category theory can be unintuitive when written ordinarily; so we reach for alternative systems of notation, such as commutative diagrams or string diagrams. (Homotopy) type theory provides another such system, in which we imagine consuming given resources to produce others. In this talk, we will motivate homotopy type theory by considering each part of the slogan "homotopy type theory is Martin-Löf dependent type theory with intensional identity types, univalence, and higher inductive types".
Jessica Anzanello (Milano-Bicocca) - Fixed-point-free elements in two-orbit permutation groups
A classical theorem of Jordan states that every non-trivial finite transitive permutation group contains a derangement, that is, an element with no fixed points. A deep result of Fein, Kantor, and Schacher adds that one can even find a derangement of prime-power order. In this talk, I will discuss a natural extension of this result to permutation groups G with two orbits on n > 2 points. In this setting, we show that either G contains a derangement, or it contains an element of prime-power order with exactly one fixed point. As a corollary, if the two orbits have size a and b, with a + b > 2 and (a, b − 1) = (a − 1, b) = 1, then G has a derangement. This confirms and generalizes a conjecture of Ellis and Harper. This is joint work with Sean Eberhard.
Given a finitely generated residually finite group $\Gamma$, its subgroup growth type is the rate of growth of the function $s_n(\Gamma)$, counting the number of finite index subgroups in $\Gamma$ of index at most $n.$ Various properties of the group $\Gamma$ can be read off of its subgroup growth type.
Let $s_n^{\mathrm{ch}}(\Gamma)$ denote the number of finite index characteristic subgroups of $\Gamma$, of index at most $n.$ Answering a question of Igor Rivin, we compute the characteristic subgroup growth type of free groups, and more generally of large groups.
This talk is based on joint work with Alex Lubotzky.
Samuel Moore (Oxford) - Derived deformation theory of schemes and infinity categories
I will give an introduction to derived deformation theory – i.e. the use of homotopy-theoretic ideas to study how objects deform – following work of Brantner, Mathew, Lurie and Pridham. I will focus on the examples of deforming derived schemes and linear infinity categories, and explain my results relating these two situations in arbitrary characteristic.
Lewis Groves (Bath) - Lie Algebras in Positive Characteristic
The representation theory of Lie algebras over fields of positive characteristic behaves quite differently to the characteristic zero case. For example, in positive characteristic, the dimension of all simple modules is finite and bounded. In this talk, we’ll begin by briefly recalling the classification of finite simple representations of sl_2, and then explore how this changes when we move to the positive characteristic setting. Along the way, we’ll discuss the additional structures that appear in positive characteristic, such as restricted Lie algebras, the p-centre, and reduced enveloping algebras.
Dan Roebuck (Warwick) - Completing categories of metric spaces
Metric spaces admit lots of different notions of homomorphism. Well-known examples include continuous, uniformly continuous, Lipschitz, bounded, and distance non-increasing maps. However, the categories of metric spaces with these maps tend to be quite badly behaved: important functors out of them often fail to be continuous. For some notions of homomorphism, there are classical extensions of the naive categories that are better behaved. For example, the category of topological spaces extends the category of metric spaces and continuous maps. In this talk I will present a general procedure for 'correcting' the naive categories by formally adding certain limits and provide an overview of the resulting theory.
Jasmine Hewitt (Manchester) - Fusion systems: from finite groups to exotica
Given a finite group $G$ and a Sylow $p$-subgroup $S$, the fusion system of $G$ on $S$ *essentially* tells us which elements and subgroups of $S$ are conjugate in $G$. This idea of “fusion” can be traced back to results of Alperin, Burnside and Frobenius.
Nowadays, fusion systems are studied in their own right. Abstractly, a fusion system on a finite $p$-group $S$ is a category whose objects are the subgroups of $S$ and whose morphisms are injective group homomorphisms subject to some axioms. Every finite group gives rise to a (saturated) fusion system, but the converse fails: we call them “exotic” fusion systems.
This talk will be introductory and should be accessible to those who know some group theory and representation theory (towards the end). I will conclude the talk by discussing some conjectures surrounding (exotic) fusion systems and some recently “proved” results like Alperin’s Weight Conjecture.
