Stein structures: existence and flexibility

This survey on the topology of Stein manifolds is an extract from our recent joint book. It is compiled from two short lecture series given by the first author in 2012 at the Institute for Advanced Study, Princeton, and the Alfred Renyi Institute of Mathematics, Budapest.Comment: 29 pages, 11 figure

Stein structures and holomorphic mappings

We prove that every continuous map from a Stein manifold X to a complex manifold Y can be made holomorphic by a homotopic deformation of both the map and the Stein structure on X. In the absence of topological obstructions the holomorphic map may be chosen to have pointwise maximal rank. The analogous result holds for any compact Hausdorff family of maps, but it fails in general for a noncompact family. Our main results are actually proved for smooth almost complex source manifolds (X,J) with the correct handlebody structure. The paper contains another proof of Eliashberg's (Int J Math 1:29--46, 1990) homotopy characterization of Stein manifolds and a slightly different explanation of the construction of exotic Stein surfaces due to Gompf (Ann Math 148 (2):619--693, 1998; J Symplectic Geom 3:565--587, 2005). (See also the related preprint math/0509419).Comment: The original publication is available at http://www.springerlink.co

OMEGA: A NEW COLD X-RAY SIMULATION FACILITY FOR THE EVALUATION OF OPTICAL COATINGS

We report on recent progress for the development of a new cold X-ray optical test capability using the Omega Facility located at the Laboratory for Laser Energetics (LLE) at the University of Rochester. These tests were done on the 30 kJ OMEGA laser at the Laboratory for Laser Energetics (LLE) at the University of Rochester, Rochester, NY. We conducted a six-shot series called OMEGA II on 14 July 2006 in one eight-hour day (supported by the Defense Threat Reduction Agency). The initial testing was performed using simple protected gold optical coatings on fused silica substrates. PUFFTFT analyses were completed and the specimen's thermal lateral stress and transverse stress conditions were calculated and interpreted. No major anomalies were detected. Comparison of the pre- and posttest reflective measurements coupled with the TFCALC analyses proved invaluable in guiding the analyses and interpreting the observed damage. The Omega facility is a high quality facility for performing evaluation of optical coatings and coupons and provides experience for the development of future National Ignition Facility (NIF) testing

Differential and holomorphic differential operators on noncommutative algebras

Abstract This paper deals with sheaves of differential operators on noncommutative algebras, in a manner related to the classical theory of D-modules. The sheaves are defined by quotienting the tensor algebra of vector fields (suitably deformed by a covariant derivative). As an example we can obtain enveloping algebra like relations for Hopf algebras with differential structures which are not bicovariant. Symbols of differential operators are defined, but not studied. These sheaves are shown to be in the center of a category of bimodules with flat bimodule covariant derivatives. Also holomorphic differential operators are considered

Vitamin D Status and Bone and Connective Tissue Turnover in Brown Bears (Ursus arctos) during Hibernation and the Active State

BACKGROUND: Extended physical inactivity causes disuse osteoporosis in humans. In contrast, brown bears (Ursus arctos) are highly immobilised for half of the year during hibernation without signs of bone loss and therefore may serve as a model for prevention of osteoporosis. AIM: To study 25-hydroxy-vitamin D (25OHD) levels and bone turnover markers in brown bears during the hibernating state in winter and during the active state in summer. We measured vitamin D subtypes (D₂ and D₃), calcitropic hormones (parathyroid hormone [PTH], 1,25-dihydroxy-vitamin D [1,25(OH)₂D]) and bone turnover parameters (osteocalcin, ICTP, CTX-I), PTH, serum calcium and PIIINP. MATERIAL AND METHODS: We drew blood from seven immobilised wild brown bears during hibernation in February and in the same bears while active in June. RESULTS: Serum 25-hydroxy-cholecalciferol (25OHD₃) was significantly higher in the summer than in the winter (22.8±4.6 vs. 8.8±2.1 nmol/l, two tailed p-2p = 0.02), whereas 25-hydroxy-ergocalciferol (25OHD₂) was higher in winter (54.2±8.3 vs. 18.7±1.7 nmol/l, 2p<0.01). Total serum calcium and PTH levels did not differ between winter and summer. Activated 1,25(OH)₂D demonstrated a statistically insignificant trend towards higher summer levels. Osteocalcin levels were higher in summer than winter, whereas other markers of bone turnover (ICTP and CTX-I) were unchanged. Serum PIIINP, which is a marker of connective tissue and to some degree muscle turnover, was significantly higher during summer than during winter. CONCLUSIONS: Dramatic changes were documented in the vitamin D₃/D₂ ratio and in markers of bone and connective tissue turnover in brown bears between hibernation and the active state. Because hibernating brown bears do not develop disuse osteoporosis, despite extensive physical inactivity we suggest that they may serve as a model for the prevention of this disease

The Cohomologies of the Iwasawa Manifold and of Its Small Deformations

We prove that, for some classes of complex nilmanifolds, the Bott–Chern cohomology is completely determined by the Lie algebra associated with the nilmanifold with the induced complex structure. We use these tools to compute the Bott–Chern and Aeppli cohomologies of the Iwasawa manifold and of its small deformations, completing the computations by M. Schweitzer (arXiv:0709.3528v1 [math.AG])