Topology seminars: 2024-25

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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.