Simon Willerton

On the Rozansky-Witten weight systems

Ideas of Rozansky and Witten, as developed by Kapranov, show that a complex symplectic manifold X gives rise to Vassiliev weight systems. In this paper we study these weight systems by using D.X/, the derived category of coherent sheaves on X . The main idea (stated here a little imprecisely) is that D.X/ is the category of modules over the shifted tangent sheaf, which is a Lie algebra object in D.X/; the weight systems then arise from this Lie algebra in a standard way. The other main results are a description of the symmetric algebra, universal enveloping algebra and Duflo isomorphism in this context, and the fact that a slight modification of D.X/ has the structure of a braided ribbon category, which gives another way to look at the associated invariants of links. Our original motivation for this work was to try to gain insight into the Jacobi diagram algebras used in Vassiliev theory by looking at them in a new light, but there are other potential applications, in particular to the rigorous construction of the .1C1C1/‐dimensional Rozansky‐Witten TQFT, and to hyperkahler geometry. 57R56, 57M27; 17B70, 14F05, 53D35, 57R27

The twisted Drinfeld double of a finite group via gerbes and finite groupoids

The twisted Drinfeld double (or quasi-quantum double) of a finite group with a 3‐ cocycle is identified with a certain twisted groupoid algebra. The groupoid is the loop (or inertia) groupoid of the original group and the twisting is shown geometrically to be the loop transgression of the 3‐cocycle. The twisted representation theory of finite groupoids is developed and used to derive properties of the Drinfeld double, such as representations being classified by their characters. This is all motivated by gerbes and 3‐dimensional quantum field theory. In particular the representation category of the twisted Drinfeld double is viewed as the “space of sections” associated to a transgressed gerbe over the loop groupoid. 57R56; 16W30, 18B40

On the asymptotic magnitude of subsets of Euclidean space

Magnitude is a canonical invariant of finite metric spaces which has its origins in category theory; it is analogous to cardinality of finite sets. Here, by approximating certain compact subsets of Euclidean space with finite subsets, the magnitudes of line segments, circles and Cantor sets are defined and calculated. It is observed that asymptotically these satisfy the inclusion-exclusion principle, relating them to intrinsic volumes of polyconvex sets.

On the magnitude of spheres, surfaces and other homogeneous spaces

In this paper we calculate the magnitude of metric spaces using measures rather than finite subsets as had been done previously. An explicit formula for the magnitude of an

Gerbes and homotopy quantum field theories

n

A COMBINATORIAL HALF-INTEGRATION FROM WEIGHT SYSTEM TO VASSILIEV KNOT INVARIANT

-sphere with its intrinsic metric is given. For an arbitrary homogeneous Riemannian manifold the leading terms of the asymptotic expansion of the magnitude are calculated and expressed in terms of the volume and total scalar curvature of the manifold.

Free groups and finite-type invariants of pure braids

This paper is part expository and part presentation of calculational results. The target space of the Kontsevich integral for knots is a space of diagrams; this space has various algebraic structures which are described here. These are utilized with Les theorem on the behaviour of the Kontsevich integral under cabling and with the Melvin-Morton Theorem, to obtain, in the Kontsevich integral for torus knots, both an explicit expression up to degree five and the general coefficients of the wheel diagrams.

Categorifying the magnitude of a graph

For smooth finite dimensional manifolds, we characterise gerbes with connection as functors on a certain surface cobordism category. This allows us to relate gerbes with connection to Turaevs 1+1-dimensional homotopy quantum field theories, and we show that flat gerbes are related to a specific class of rank one homotopy quantum field theories. AMS Classification 55P48; 57R56, 81T70