The Junior Algebra Seminar is a seminar at 果冻传媒 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 , 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!
Term 1
(Birmingham) - Homotopy Type Theory
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-L枚f 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 鈭� 1) = (a 鈭� 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.
(Cambridge) - Characteristic subgroup growth
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.
(Oxford) - Derived deformation theory of schemes and infinity categories
I will give an introduction to derived deformation theory 鈥� i.e. the use of homotopy-theoretic ideas to study how objects deform 鈥� 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鈥檒l 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鈥檒l discuss the additional structures that appear in positive characteristic, such as restricted Lie algebras, the p-centre, and reduced enveloping algebras.
