# TTT90: Sheffield

The University of Sheffield.

Supported by the London Mathematical Society.

## Date

16 December 2013

## Speakers

### James Walton (University of Leicester)

Topological invariants of tiling spaces

Topological invariants of tiling spaces

Given some aperiodic tiling (of Euclidean space, say), a fruitful approach to understanding its properties is to associate to it a moduli space of 'locally isomorphic' tilings, and to then study the topology of this 'tiling space'. A common topological invariant to consider in this context is the Čech cohomology.

I will describe how, using a Poincaré Duality like result, one may describe these groups in a very geometric way using cellular chains (although non-compactly supported ones) of the Euclidean space which are 'pattern equivariant' (PE) with respect to the tiling. I will show how, with this perspective, one may give a simple method to compute these groups for hierarchical tilings.

I will also discuss the rotationally invariant PE complexes, which seem to capture extra information about the rotationally invariant tilings in the tiling space to the Čech cohomology groups. These groups can be incorporated into a spectral sequence converging to the cohomology of the tiling space of rigid motions of a tiling.

### Tom Bridgeland (University of Sheffield)

Symmetry groups of derived categories of coherent sheaves

Symmetry groups of derived categories of coherent sheaves

Given a smooth projective variety, one can try to compute the group of auto-equivalences of its derived category of coherent sheaves. I will give a gentle introduction to what is known about this tricky problem, and try to explain how ideas from mirror symmetry can help in guessing the answer.

### Andy Tonks (London Metropolitan University)

Incidence algebras and Möbius inversion for decomposition spaces

Incidence algebras and Möbius inversion for decomposition spaces

Joint work with Imma Gálvez and Joachim Kock.

The classical theory of incidence (co)algebras and Möbius inversion for (locally finite) posets may be generalised in two directions. Firstly posets may be replaced by categories, as in Leroux's theory, and secondly the numerical data involved can be seen as arising from the (homotopy) cardinality of basic combinatorial and algebraic objects, as suggested by the work of Lawvere and Menni.

In this talk we introduce further generalisations, to what we call decomposition spaces, fundamental examples of which are provided by weak category objects in (infinity-)groupoids. In particular we obtain the Connes-Kreimer Hopf algebra from a decomposition space of combinatorial forests, and derived Hall algebras from the Waldhausen S-dot construction of a stable infinity-category.