Applications of Homotopy Theory

摘要/ abstract摘要：dms -0707014首席研究员：James E. mcclure首席研究员计划在四个完全不同的领域开展工作，这些领域可以应用同伦理论方法。第一个领域是字符串拓扑。项目的这一部分是与ralph Kaufmann；Kaufmann和McClure将结合他们之前在弦拓扑上的工作，以获得任意属的弦拓扑到链水平的提升。第二个领域是手术光谱。这是和Gerd Laures合作的；Laures和mcclure将完成他们的证明，证明某些对称l谱具有e -无穷环结构，并且对称签名是e -无穷环映射。他们还将扩展他们的工作，以证明从代数k理论到由Weiss和Williams定义的对称l理论的某个映射是一个e -无穷环映射。第三个领域是交同源性。这是格雷格·弗里德曼和斯科特·威尔逊的合作；Friedman，McClure和Wilson将致力于发展交集链上的同伦理论结构，这将（例如）导致分层空间的理性同伦理论的发展。第四个领域是C*代数。这是和marius Dadarlat合作的；Dadarlat和McClure将证明Dixmier和Douady关于纤维为紧算子代数的束的经典工作可以推广到纤维为Kirchberg代数的束。同伦理论的最初目标是发展可以应用于几何问题（包括高于3维的问题）的方法。自20世纪初它的起源以来，同伦理论已经发展成为一个独立的学科，但与几何的联系一直是一个中心问题。大约从1960年开始，同伦理论已经明显地应用于除几何之外的其他领域，例如分析、代数，以及最近的物理学。首席研究员和他的合作者们的工作是将同伦理论的最新发展（例如，对称谱理论）应用到其他领域，包括几何问题（高维形状的分类）和数学物理问题（弦拓扑是许多人正在进行的项目的一部分，目的是对弦理论的思想进行更清晰的数学理解）。

AbstractAward: DMS-0707014Principal Investigator: James E. McClureThe principal investigator plans to work in four quite differentareas where homotopy-theoretic methods can be applied. The firstarea is string topology. This part of the project is joint withRalph Kaufmann; Kaufmann and McClure will combine their previouswork on string topology in order to obtain a lifting of stringtopology to the chain level for arbitrary genus. The second areais surgery spectra. This is joint with Gerd Laures; Laures andMcClure will complete their proof that certain symmetricL-spectra have E-infinity ring structures and that the symmetricsignature is an E-infinity ring map. They will also extend theirwork in order to show that a certain map from algebraic K-theoryto symmetric L-theory defined by Weiss and Williams is anE-infinity ring map. The third area is intersection homology.This is joint with Greg Friedman and Scott Wilson; Friedman,McClure and Wilson will work to develop homotopy theoreticstructure on intersection chains which should (for example) leadto the development of a rational homotopy theory of stratifiedspaces. The fourth area is C* algebras. This is joint withMarius Dadarlat; Dadarlat and McClure will work to show that theclassical work of Dixmier and Douady on bundles whose fibre isthe algebra of compact operators can be extended to bundles whosefibre is a Kirchberg algebra.The original goal of homotopy theory was to develop methods whichcould be applied to geometric problems (including problems indimensions higher than three). Since its origin in the earlytwentieth century, homotopy theory has developed into anautonomous discipline, but the connection with geometry hasalways been a central concern. Since about 1960, it has becomeapparent that homotopy theory has applications to other thingsbesides geometry, for example to analysis, algebra, and morerecently to physics. The work of the principal investigator andhis coauthors is intended to take the latest developments inhomotopy theory (for example, the theory of symmetric spectra)and apply them to other areas, including geometric problems (theclassification of higher-dimensional shapes), and problems inmathematical physics (string topology is part of the programbeing carried on by many people to develop a clearer mathematicalunderstanding of ideas from string theory).