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Algebraic K-Theory in Fixed-Point Theory and Smooth Manifolds

定点理论和光滑流形中的代数 K 理论

基本信息

批准号：
2005524
负责人：
Cary Malkiewich
金额：
$ 15.4万
依托单位：
SUNY at Binghamton
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-08-01 至 2024-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2005524&HistoricalAwards=false
关键词：
Algebraic Theory Fixed Point Smooth

项目摘要

Fixed-point theory studies the fixed or stationary states in dynamical systems, which are critical to questions in topology, analysis, and even economics. Smooth manifolds are a special class of high-dimensional shapes that are of fundamental importance in geometry and physics. This project is concerned with important and unexpected connections between these two fields. To be more precise, they can be related using algebraic K-theory, a deep and abstract but pervasive subject whose reach already encompasses several mathematical disciplines, including homotopy theory, differential topology, algebraic geometry, and number theory. These research projects will develop these connections, giving us new ways to think about classical problems and to use techniques from fixed-point theory in unexpected places. Broader impacts include mentorship, workshop organization, and the writing of foundational works to lower the entry barrier into these disciplines.Four groups of projects will be pursued. The first shows that the Fuller trace, a certain noncommutative trace in genuine G-spectra, provides a complete invariant for removing n-periodic points from a map by a homotopy. The second uses the trace from K-theory of endomorphisms to topological restriction homology (TR) to show that this Fuller trace is a topological, non-commutative generalization of the characteristic polynomial from linear algebra. The third project proves the equivariant refinement of Waldhausen's parametrized h-cobordism theorem, a fundamental and celebrated result in high-dimensional manifold theory. The fourth project uses noncommutative traces to compute transfer maps in algebraic K-theory, leading to torsion calculations that identify exotic bundles and fibrations that cannot be stably smoothed. In other words, fixed-point theory techniques can be used to understand smooth structures.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

不动点理论研究动力系统中的固定或定态，这对拓扑、分析甚至经济学问题都是至关重要的。光滑流形是一类特殊的高维形状，在几何和物理中具有重要的基础意义。这个项目关注的是这两个领域之间的重要和意想不到的联系。更准确地说，它们可以用代数k理论联系起来，代数k理论是一个深刻、抽象但普遍的学科，其范围已经涵盖了几个数学学科，包括同伦理论、微分拓扑、代数几何和数论。这些研究项目将发展这些联系，为我们提供思考经典问题的新方法，并在意想不到的地方使用不动点理论的技术。更广泛的影响包括指导、研讨会组织和基础作品的写作，以降低进入这些学科的门槛。将进行四组项目。首先证明了真g谱中的一个非交换迹——富勒迹，为用同伦从映射中去除n个周期点提供了一个完全不变量。第二，利用自同态k理论到拓扑限制同调（TR）的迹来证明富勒迹是线性代数特征多项式的拓扑非交换推广。第三个项目证明了Waldhausen的参数化h-协定理的等变细化，这是高维流形理论中的一个基本和著名的结果。第四个项目使用非交换轨迹来计算代数k理论中的转移映射，从而导致扭转计算，从而识别无法稳定平滑的奇异束和纤维。换句话说，不动点理论技术可以用来理解光滑结构。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。