Applications of Homotopy Theory

摘要奖：DMS-0707014主要研究者：James E.首席研究员计划在同伦理论方法可以应用的四个完全不同的领域工作。 第一个领域是弦拓扑。 这个项目的这一部分是与Ralph Kaufmann联合进行的; Kaufmann和麦克卢尔将联合收割机结合他们以前在弦拓扑上的工作，以获得任意亏格的链拓扑的提升。 第二个领域是手术谱。 这是与Gerd Laures联合的; Laures和McClure将完成他们的证明，即某些非线性L-谱具有E-无限环结构，并且非线性签名是E-无限环映射。 为了证明韦斯和威廉姆斯定义的从代数K-理论到对称L-理论的映射是E-无限环映射，他们也将推广他们的工作。 第三个领域是相交同调。这是与格雷格·弗里德曼（Greg Friedman）和斯科特·威尔逊（Scott Wilson）的联合研究;弗里德曼、麦克卢尔和威尔逊将致力于发展相交链上的同伦理论结构，这应该（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。 这是联合与马里乌斯达达拉特;达达拉特和麦克卢尔将工作表明，古典工作的Dixlane和Douady丛的纤维是代数的紧凑运营商可以扩展到丛whosefibre是一个基希贝格代数。原来的目标同伦理论是制定的方法，可适用于几何问题（包括问题的维数高于三）。 自世纪早期起源以来，同伦理论已经发展成为一门独立的学科，但它与几何学的联系一直是人们关注的焦点。 自1960年以来，同伦理论在几何学之外的其他领域也有应用，例如分析、代数，以及最近的物理学。 首席研究员和他的合著者的工作是打算采取同伦理论的最新发展（例如，对称谱理论），并将其应用于其他领域，包括几何问题（高维形状的分类），以及数学物理问题（弦拓扑是许多人进行的计划的一部分，以发展对弦理论思想的更清晰的理论理解）。

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).