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

不动点理论研究动力系统中的固定或定常状态，这些状态对于拓扑学、分析学甚至经济学中的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意义。这个项目关注的是这两个领域之间重要的和意想不到的联系。更准确地说，它们可以用代数K-理论联系起来，这是一个深刻而抽象但无处不在的学科，其覆盖范围已经涵盖了几个数学学科，包括同伦论、微分拓扑学、代数几何和数论。这些研究项目将发展这些联系，给我们新的方式来思考经典问题，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、研讨会组织和编写基础工作，以降低进入这些学科的门槛。将进行四组项目。首先证明了Fuller迹是真G-谱中的一种非对易迹，它为通过同伦消除映射中的n个周期点提供了一个完全不变量。第二部分利用K-自同态理论到拓扑限制同调的迹，证明了这种Fuller迹是线性代数特征多项式的拓扑化、非对易推广。第三个项目证明了Waldhausen的参数化h-Coobordism定理的等变精化，这是高维流形理论中的一个基本和著名的结果。第四个项目使用非对易迹来计算代数K理论中的传递映射，导致了识别不能稳定平滑的奇异丛和纤丝的扭转计算。换句话说，定点理论技术可以用来理解光滑的结构。这个奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.