# TTT 117: online hosted by Sheffield

## Date

Wednesday 14th December 2022, starting at 14:00 GMT

## Speakers/programme

14:00

Jordan Williamson (Prague)

Definability and duality in triangulated categories

Abstract: Classifying objects in triangulated categories up to isomorphism is generally far too hard, so we often seek to classify objects up to some operations, saying that two objects are equivalent if they can built from each other using these operations. Operations one might take here are coproducts and cones, which gives rise to the idea of thick and localizing subcategories. Instead, one can take operations from purity, which has a long history in algebra, model theory, and representation theory, leading to the notion of definable subcategories. I will explain how one can generalise and axiomatise various aspects of this algebraic theory so that it applies in triangulated categories, and then provide some applications to homotopy theory and representation theory. This is joint work with Isaac Bird.

15:00

Severin Bunk (Oxford)

Functorial field theories from differential cocycles

Abstract: In this talk I will demonstrate how differential cocycles give rise to (bordism-type) functorial field theories (FFTs) on manifolds. I will present some background on smooth FFTs (geometric refinement of TQFTs) and differential cohomology. In particular, I will use higher gerbes with connection as a geometric model for differential cocycles and explain the idea of how these induce smooth FFTs. In the second part, I will focus on the two-dimensional case. Here I will present a concrete, geometric construction of two-dimensional smooth FFTs on background manifolds, starting from gerbes with connection.

16:30

Luciana Basualdo Bonatto (MPI, Bonn)

Decoupling Moduli of Configuration Spaces on Surfaces

Abstract: In this talk, we will discuss the Moduli Spaces of Configurations of Points on Manifolds, focusing specifically on the case of oriented surfaces. These moduli spaces have deep connections to many areas of current research. For instance, they give different perspectives on the study of braid groups on manifolds and on moduli of punctured manifolds. These moduli spaces are also convenient constructions to study generalised configuration spaces, for instance ones that include labels, and even partially summable labels, where particles can collide if their labels are composable. This type of configuration spaces have appeared since the 70’s in McDuff’s configurations of positive and negative particles, and more recently generalised to constructions such as Factorization Homology.

These moduli of generalised configuration spaces on manifolds have two natural maps: one to the moduli space of the underlying manifold (which forgets the configuration) and one to the space of labelled configurations in the infinite euclidean space (which forgets the underlying manifold instead). One would not expect these forgetful maps to retain much information about the original moduli of configurations, but we will show that this is indeed the case for surfaces: in a range which increases with the genus, the product of these maps gives a homology isomorphism. We interpret this as a splitting result where the configuration gets decoupled from the underlying manifold.

As consequences, we show how this leads to a homological stability result in the context of Factorization Homology and also to explicit ways of computing the stable (co)homology.

## Further information

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