Dixmier traces and some applications to noncommutative geometry

This is a survey of some recent advances in the theory of singular traces in which the authors have played some part and which were inspired by questions raised by the book of Alain Connes (Noncommutative Geometry, Academic Press 1994). There are some original proofs and ideas but most of the results have appeared elsewhere. Detailed information on the contents is contained in the Introduction.Comment: To appear in Russian Mathematical Surveys (in Russian). New version corrects Latex problems, minor errors and reference

The spectral shift function and spectral flow

This paper extends Krein's spectral shift function theory to the setting of semifinite spectral triples. We define the spectral shift function under these hypotheses via Birman-Solomyak spectral averaging formula and show that it computes spectral flow.Comment: 47 page

Spitzer's mid-infrared view on an outer Galaxy Infrared Dark Cloud candidate toward NGC 7538

Infrared Dark Clouds (IRDCs) represent the earliest observed stages of clustered star formation, characterized by large column densities of cold and dense molecular material observed in silhouette against a bright background of mid-IR emission. Up to now, IRDCs were predominantly known toward the inner Galaxy where background infrared emission levels are high. We present Spitzer observations with the Infrared Camera Array toward object G111.80+0.58 (G111) in the outer Galactic Plane, located at a distance of ~3 kpc from us and ~10 kpc from the Galactic center. Earlier results show that G111 is a massive, cold molecular clump very similar to IRDCs. The mid-IR Spitzer observations unambiguously detect object G111 in absorption. We have identified for the first time an IRDC in the outer Galaxy, which confirms the suggestion that cluster-forming clumps are present throughout the Galactic Plane. However, against a low mid-IR back ground such as the outer Galaxy it takes some effort to find them.Comment: Accepted for publication in ApJL -- 11 pages, 2 figures (1 colour