Parity of the spin structure defined by a quadratic differential

According to the work of Kontsevich-Zorich, the invariant that classifies non-hyperelliptic connected components of the moduli spaces of Abelian differentials with prescribed singularities,is the parity of the spin structure. We show that for the moduli space of quadratic differentials, the spin structure is constant on every stratum where it is defined. In particular this disproves the conjecture that it classifies the non-hyperelliptic connected components of the strata of quadratic differentials with prescribed singularities. An explicit formula for the parity of the spin structure is given.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol8/paper12.abs.htm

Measuring real value and inflation

The most important economic measures are monetary. They have many different names, are derived in different theories and employ different formulas. Yet, they all attempt to do basically the same thing: to separate a change in nominal value into a ‘real part’ due to the changes in quantities and an inflation due to the changes of prices. Examples are: real national product and its components, the GNP deflator, the CPI, various measures related to consumer surplus, as well as the large number of formulas for price and quantity indexes that have been proposed. The theories that have been developed to derive these measures are largely unsatisfactory. The axiomatic theory of indexes does not make clear which economic problem a particular formula can be used to solve. The economic theories are for the most part based on unrealistic assumption. For example, the theory of the CPI is usually developed for a single consumer with homothetic preferences and then applied to a large aggregate of diverse consumers with non-homothetic preferences. In this paper I develop a unitary theory that can be used in all situations in which monetary measures have been used. The theory implies a uniquely optimal measure which turns out to be the Törnqvist index. I review, and partly re-interpret the derivations of this index in the literature and provide several new derivations. The paper also covers several related topics, particularly the presently unsatisfactory determination of the components of real GDP

Balanced Topological Field Theories

We describe a class of topological field theories called ``balanced topological field theories.'' These theories are associated to moduli problems with vanishing virtual dimension and calculate the Euler character of various moduli spaces. We show that these theories are closely related to the geometry and equivariant cohomology of ``iterated superspaces'' that carry two differentials. We find the most general action for these theories, which turns out to define Morse theory on field space. We illustrate the constructions with numerous examples. Finally, we relate these theories to topological sigma-models twisted using an isometry of the target space.Comment: 40 pages, harvmac, references added, to appear in Commun. Math. Phy