Topology seminars: 2024-25

A family of diagram algebras called the Temperley-Lieb algebras has been an important topic of study for several decades in areas such as knot theory, statistical mechanics and representation theory. In recent years, Boyd and Hepworth pioneered the study of such families of algebras using homological algebra, one important result being that the Temperley-Lieb algebras satisfy homological stability. In this talk, I'll give an overview of the study of the (co)homology of these algebras, and I will discuss joint work in progress with Guy Boyde and Daniel Graves on the (co)homology of a related family called the blob algebras.

This is joint work with Henry Rice and Pierre-Louis Guillot.

Much of chromatic homotopy theory is related to the Brown-Peterson spectrum $BP$ with $\pi_*(BP)=\mathbb{Z}_{(p)}[v_1,v_2,v_3,\dotsc]$ (where everything implicitly depends on a chosen prime number $p$, and $|v_k|=2(p^k-1)$). 

There are also truncated versions called $BP\langle n\rangle$ with  $\pi_*(BP\langle n\rangle)=\mathbb{Z}_{(p)}[v_1,\dotsc,v_n]$.  The chromatic redshift conjecture of Ausoni and Rognes (various formulations of which have now been proved) implies that the algebraic $K$-theory spectrum $K(BP\langle n\rangle)$ should share various properties with $BP\rangle n+1\rangle$.  Most proofs about this kind of thing proceed by using the topological cyclic homology functor $TC(R)$ as an approximation to $K(R)$.  There are also other spectra called $THH(R)$, $TC^-(R)$, $TP(R)$ and $TR(R)$ which fit into the same circle of ideas.  I will discuss various calculations for the case $R=BP\langle n\rangle$.  There is a complicated story about what has been proved (sometimes up to a specified approximation) or conjectured in the literature.

The study of cohomology theories leads to the study of spectra, their representing objects.  These are complicated objects with complicated structures.  However, they become much simpler, and in fact algebraic, when we work rationally.  We will use rational algebraic models to look at the equivariant cohomology theories representing real and complex vector bundles.

In non-equivariant topology, the ordinary homology of a point is described by the dimension axiom and is quite simple - namely, it is concentrated in degree zero. The situation in G-equivariant topology is different. This is because Bredon homology - the equivariant counterpart of the ordinary homology - is naturally graded over RO(G), the ring of G-representations. Whereas the equivariant dimension axiom describes the part of the Bredon homology of a point graded over trivial representations, it does not put any requirements on the rest of the grading - in which the homology may be quite complicated.

The RO(G)-graded Bredon homology theories are represented by G-Eilenberg-MacLane spectra, and thus the Bredon homology of a point is the same as coefficients of these spectra. During the talk, I will present the method of computing the RO(C2)-graded coefficients of C2-Eilenberg-MacLane spectra based on the Tate square. As demonstrated by Greenlees, the Tate square gives an algorithmic approach to computing the coefficients of equivariant spectra. In the talk, we will discuss how to use this method to obtain the RO(C2)-graded coefficients of a C2-Eilenberg-MacLane spectrum as a RO(C2)-graded abelian group. We will also present the multiplicative structure of the C2-Eilenberg-MacLane spectrum associated to the Burnside Mackey functor. Time permitting, we will further discuss how to extend this knowledge towards groups of prime power order.

The Madsen-Weiss theorem may be viewed as a calculation of the homology of the compactly-supported mapping class group of the infinite-genus surface L sometimes called the "Loch Ness monster surface". In contrast, the homology of the full (not necessarily compactly-supported) mapping class group Mod(L) of L is much less well-understood. I will talk about joint work with Xiaolei Wu in which we prove that the homology of Mod(L) is uncountably generated in every positive degree, but that the dual Miller-Morita-Mumford classes vanish on Mod(L). I will also discuss the analogous questions for other infinite-type surfaces, including a complete calculation of the homology of Mod(S) when S is the plane minus a Cantor set.

A transfer system is a graph on a lattice satisfying certain restriction and composition properties. They were first studied on the lattice of subgroups of a finite group in order to examine equivariant homotopy commutativity, which then unlocked a wealth of links to combinatorial methods.

On a finite total order [n], transfer systems can be used to classify different homotopy theories on [n].  The talk will involve plenty of examples and not assume any background knowledge. 

The 1-point compactification of the space of unordered configurations of n points in a compact manifold M is well-known to be homotopy invariant in M. In fact the collection of all these spaces have further structure: they form a commutative monoid object, by superposition of configurations. I will show how this commutative monoid object has a simple (derived) presentation, and explain various consequences for the homology of configuration spaces which can be derived from this.

The concept of 'brave new algebras' tells us that E_infty-rings should play the role of commutative rings in higher algebra. With this in mind, we should be able to extend concepts from commutative algebra to higher categories; this is not only a fun procedure but a fruitful one, such as the role of spectral algebraic geometry and chromatic homotopy theory in studying the homotopy groups of spheres. This talk will act as an introduction to one such categorification - that of Galois theory - and discuss it's use in Chromatic homotopy theory in studying the Morava stabiliser group. No knowledge of spectra or chromatic homotopy theory is required, but some familiarity with higher categories will help. 

In this talk, we will give an overview of Lagrangian Floer theory. We will cover the definition, mention some applications, and discuss its relation to low-dimensional topology and algebraic topology. I will then explain some of my recent work on this. No prior knowledge of symplectic manifolds is assumed.

The past decade has seen a proliferation of homology theories for graphs. In particular, large literatures have grown up around magnitude homology (due to Hepworth and Willerton) and path homology (Grigor’yan, Lin, Muranov and Yau). Though their origins are quite separate, Asao proved in 2022 that in fact these homology theories are intimately related. To every directed graph one can associate a certain spectral sequence - the magnitude-path spectral sequence, or MPSS - whose page E^1 is exactly magnitude homology, while path homology lies along a single axis of page E^2. In this talk, based on joint work with Richard Hepworth, I will describe the construction of the sequence and argue that each one of its pages deserves to be regarded as a homology theory for directed graphs, satisfying a Künneth theorem and an excision theorem, and with a homotopy-invariance property that grows stronger as we turn the pages of the sequence. The ‘nested’ family of homotopy theories associated to the pages is not yet well understood. But I will describe a new cofibration category structure on the category of directed graphs, associated to page E^2 of the MPSS, and speculate on possible relations to the ‘nested’ homotopy theories of filtered chain complexes described in recent papers by Whitehouse and collaborators.

Typically operads are defined to have actions of the symmetric groups on their sets of n-ary operations, which are then equivariant with respect to the operadic composition. We can also define operads without such actions or with actions governed by other families of groups, such as braid groups. In this talk I will introduce action operads before describing how their structure can be used to govern the actions of other operads. I.e., starting with a suitable family of groups which constitute an action operad, we can retrieve familiar flavours of operad, but also see these in a general setting with even more examples.

I will try to describe some simple results about the basic algebra of such operads, as well as relate this back to the familiar setting of symmetric operads. Time permitting, I will also describe how Cat-enriched versions of these operads with general groups of equivariance give rise to various flavours of monoidal category, as well as where these ideas have been utilised by others working in more topological settings.

Joint work with Nick Gurski.

The notion of a group action compatible with an order-reversing involution can be neatly captured by taking a group with a homomorphism to the cyclic group of order two (as seen in James Cranch's seminar last week). This has a preposterous number of different names in the literature (the humorous caricature of "group with oriented parity sign" is due to Sarah Whitehouse). In this talk I will report on joint work-in-progress with Sarah on how such a structure can be built into Hochschild homology. I will start with background on Hochschild homology and functor homology before describing some new results and conjectures.

I'll report on joint work in progress with Dan Graves (University of Leeds), which extends old work of Teimuraz Pirashvili.

PROPs are very general objects which encode many types of algebraic structure, including algebras, coalgebras, bialgebras, Hopf algebras, .... Operads are simpler objects which encode algebras (but definitely not bialgebras or Hopf algebras) nicely. I'll explain how one can go about assembling some interesting PROPs from operadic ingredients.

I'll try to give this talk at a level appropriate for all (including, in particular, the new PhD students).

When considering (co)limits of categories, one might ask for (co)cones to only commute up to natural isomorphism, or for universal properties to only hold up to equivalences of categories. In a general bicategory K such universal properties are modelled by the notion of a bicategorical (co)limit, where equations/relations are only ever imposed on data in the highest available dimension. However, these notions can also be modelled up to equivalence via ordinary (co)limits enriched over V= Cat, provided that one restricts their attention to weights that are well behaved with respect to the canonical monoidal model structure on V.

In this talk I will explain how the above story adapts to the setting involving (co)limits *of* two-dimensional categories (or more generally, *in* three-dimensional categories). This involves homotopically well-behaved (co)limits enriched over the base V now given by Lack's monoidal model structure on the category of 2-categories and 2-functors. Running examples for motivation include Kleisli and Eilenberg-Moore constructions for pseudomonads, including for those on monoidal bicategories, as well as strictification constructions on bicategories and pseudo-double categories. This talk is based on my recent preprint.