Summary of Research Statement: G.L.Jones, EPSRC grant GR/K/75453 Gavin Jones has worked principally on ideas related to the welding problem is as foil-

lows: given a Jordan curve J on the Riemann sphere S^{2}, then S^{2} \J = Ω∪Ω^{∗} where Ω and Ω^{∗}aretwosimply−connectedregions.T hereareconformalmapsf,gfromtheinterirandexterioroftheunitdisctotheregionsΩ, Ω^{∗} which extend as boundary homeomorphisms. Therefore ont the unit circle we define the orientation=preserving hoomeomorphism h = g^{−1} ∘ f, the welding. The welding problem consists of the collection of questions relating h to J.

With N.G. Makarov, Gavin Jones has developed an operator-based approach, relating properties of the Jordan curve welded by h with properties of the linear map V_{h} : f ↦ f ∘h, and the associated Grunsky operator Γ. His work has led to the following.

Theorem [To appear in Mich. Math.J.]

Let f be a conformal map from the unit disc to a Jordan curve with welding h. The following are equivalent, where γ_{p} denotes the p’the Schatten class of operators on the Dirichelt space p ∈ [1, ∞).

1. Γ ∈ γ_{p}.

2. Let H denote the Hilbert transform. Then the commmutator [V_{h}, H] ∈ γ_{p}.

3. log f′ ∈ B_{p}, where B_{p} is the p’th Besov space (p ∈ [1, ∞)).

4. f extends to a quasiconformal map F of the plane μ_{F} ∈ L^{p}(dλ), where dλ denotes nonEuclidean area measure on U^{∗}.

5. S_{f} ∈ L^{p}(dm_{p}(z)), where dm_{p}(z) = (1 − |z|)^{2p−2}, and S_{f} denotes the Schwarzian derivative.

Theorem (submitted to Ann. Acad. Sci. Fenn.)

Let D denote Dirichlet space and let D_{⊒} denote a weighted Dirichlet space. Then for certain w, if V_{h} is bounded from D_{⊒} to D, then h welds a Jordan curve unique to Möbius transformation.

For further details contact: G.L.Jones, 67 Manor Way, Blackheath, London SE3 9XG.