Topology seminars: 2025-26

(Joint work with Barrero, Barthel, Pol and Williamson)

Let $G$ be a finite group, and let $\text{Orb}_G$ be the category of $G$-sets on which $G$ acts transitively, with the morphisms being equivariant bijections. Then the homotopy category of rational $G$-spectra is equivalent to the category of functors from $\text{Orb}_G$ to rational vector spaces, and one can understand everything about this quite explicitly.  

For many purposes it is useful to work equivariantly with respect to a family of finite (or sometimes profinite) groups rather than just a single group.  In this talk I will fix a prime $p$ and consider the family $\mathcal{A}$ of finite abelian $p$-groups (but in fact we can prove many things for larger or smaller families as well).  We make $\mathcal{A}$ into a category using the surjective homomorphisms as morphisms.  In this context, the homotopy category is equivalent to the derived category of the abelian category of contravariant functors from $\mathcal{A}$ to rational vector spaces. Our main result is to determine the lattice of thick ideals in the subcategory of compact objects in the homotopy category, and thus to determine the Balmer spectrum.

In noncommutative geometry, one typically equips an associative algebra A with further structure such that it can then be regarded as encoding a “noncommutative space”. A general axiomatic approach is via the notion of a differential calculus (aka exterior algebra), introduced by Woronowicz. Unfortunately, such an approach is, in general, insufficient to yield existence or uniqueness of a compatible notion of form, and such procedures are typically not functorial. In this talk, we aim to address this problem treating the geometry of a category E as a relative notion: it will emerge when E is equipped with an isofibration E-->Mon(V) into the category of monoids in a monoidal additive category V. 

We start by considering the notion of first order differential calculi in the setting of monoids internal to a monoidal additive category V and show that standard results extend to this setting. Then, we establish sufficient conditions on a faithful isofibration E-->Mon(V) so that E admits a canonical functor into the category of first order differential calculi in V. 

This talk is based on joint work with Gabriele Lobbia and Keegan Flood, which can be found at arXiv:2512.20742.

The Fukaya category of a symplectic manifold M is a Z-linear category that encodes certain curves between Lagrangian submanifolds of M.

I will explain how, in the presence of an anti-symplectomorphism of M, one can talk about symmetric forms on these Lagrangian submanifolds. This notion is encoded by a so-called Poincaré structure, which is defined in analogy to the visible Poincaré structure on the category of compact modules over the group-ring of a group G.

We will then relate the real Hochschild homology of this Fukaya category to the symplectic cohomology of M, in a way which is analogous (and in fact extends) the relation between the Hochschild homology of the chains on the based loop space with the homology of the free loop space.

This is all joint work with Cheuk Yu Mak.

In this talk, I will discuss finite bijective set-theoretic solutions of the Pentagon Equation and present their complete classification. The main result reveals an unexpected connection between these solutions and matched pairs of groups. After reviewing the necessary definitions, I will turn to the basic building blocks of the theory, the irretractable solutions, and describe their relationship with matched pairs of groups. I will then explain how all bijective solutions arise as lifts of irretractable ones, thereby obtaining the full classification. If times permits, I will also briefly comment on related connections with Hopf algebras.

In this talk, I will present the joint work with Guoqi Yan on the structure of RO(G)-graded homotopy groups of Eilenberg-MacLane spectra over cyclic 2-groups. This structure is already well-known for G=C_2 and was computed by Zeng and Georgakopoulos for G=C_4. I will show how to induce the knowledge of homotopy for lower-order groups to higher order groups. I will also discuss some relations between elements in the coefficients of a Green functor.

The moduli space M𝑔 plays an important role in several areas of maths: as the classifying space for smooth surface bundles of genus 𝑔 in algebraic topology, as the moduli space of curves of genus 𝑔 in algebraic geometry, and as a 𝐾 (𝜋, 1) for the mapping class group of the surface of genus 𝑔 in geometric group theory. From each of these perspectives the (rational) cohomology of M𝑔 is an interesting invariant – for example it gives characteristic classes for surface bundles.

I will present a new approach to studying these cohomology groups, by constructing a space |M| that combines all the genera and considering a “genus filtration” on it. From this we obtain a spectral sequence whose 𝐸1-page consists of the cohomology of all the M𝑔 and that converges to a known term. This perspective on the cohomology of the M imposes some strong constraints, and we can for example read off that 𝐻18 (M6; Q) = 0.

A multicomplex is a variant of a bicomplex and these structures arise naturally in many geometric, topological and algebraic contexts; for example, from filtered simplicial sets. I will explain some recent joint work with Joana Cirici and Muriel Livernet which explores homotopy theories related to the two spectral sequences of a truncated multicomplex. There are potential applications to the study of homotopy types of almost and generalized complex manifolds.

Recently (together with Andrew Fisher and James Cranch) I have thinking about cohomology of diagram algebras. These arise in all sorts of areas of maths but at present I don't understand any of this as much as I would like. In this talk, I'll focus on one particular algebra: the Temperley-Lieb algebra. Time and understanding permitting, I hope to say a little about these algebras in the context of algebra, representation theory, combinatorics, topology, homological algebra and (at least very vaguely) statistical mechanics. Some of what I hope to say can be found in the following recently-revised arXiv submission: https://arxiv.org/pdf/2307.11929

Large-scale geometry (or coarse geometry) is the study of spaces, not through local features, but through global or asymptotic ones, for example through the growth function or an asymptotic notion of dimension. These ideas have proven powerful in understanding metric spaces and beyond, but most notably in geometric group theory, where results like the Švarc–Milnor lemma allow us to study the structure of finitely generated groups via the large-scale geometry of their Cayley graphs.

In this talk, I will discuss two coarse-geometric tools that describe how spaces behave “at infinity,” and explore how they relate. The first is the classical notion of ends, introduced by Freudenthal, which records the number of distinct ways an observer may head off to infinity in a given space. The second is the more recent concept of coarse path components, arising from the coarse homotopy theory developed by Mitchener, Norouzizadeh, and Schick. I will outline both notions, and present examples illustrating when they coincide and when they differ.