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We describe a natural decomposition of a normal complex surface singularity (X, 0) into its “thick” and “thin” parts. The former is essentially metrically conical, while the latter shrinks rapidly in thickness as it approaches the origin. The thin part is empty if and only if the singularity is metrically conical; the link of the singularity is then Seifert fibered. In general the thin part will not be empty, in which case it always carries essential topology. Our decomposition has some analogy with the Margulis thick-thin decomposition for a negatively curved manifold. However, the geometric behavior is very different; for example, often most of the topology of a normal surface singularity is concentrated in the thin parts.
By refining the thick-thin decomposition, we then give a complete description of the intrinsic bilipschitz geometry of (X, 0) in terms of its topology and a finite list of numerical bilipschitz invariants.
We prove an analogue of the Madsen–Weiss theorem for high-dimensional manifolds. In particular, we explicitly describe the ring of characteristic classes of smooth fibre bundles whose fibres are connected sums of g copies of Sn×Sn, in the limit . Rationally it is a polynomial ring in certain explicit generators, giving a high-dimensional analogue of Mumford’s conjecture.
More generally, we study a moduli space of those null-bordisms of a fixed (2n–1)-dimensional manifold P which are (n–1)-connected relative to P. We determine the homology of after stabilisation using certain self-bordisms of P. The stable homology is identified with that of an infinite loop space.
Wehrl used Glauber coherent states to define a map from quantum density matrices to classical phase space densities and conjectured that for Glauber coherent states the mininimum classical entropy would occur for density matrices equal to projectors onto coherent states. This was proved by Lieb in 1978 who also extended the conjecture to Bloch SU(2) spin-coherent states for every angular momentum J. This conjecture is proved here. We also recall our 1991 extension of the Wehrl map to a quantum channel from J to with corresponding to the Wehrl map to classical densities. These channels were later recognized as the optimal quantum cloning channels. For each J and we show that the minimal output entropy for the channels occurs for a J coherent state. We also show that coherent states both Glauber and Bloch minimize any concave functional, not just entropy.