Traces in Algebraic K-theory and Topological Fixed Point Invariants

回答许多数学问题的障碍之一就是有太多的信息——太多的例子和太多的可能性。有许多方法可以尝试处理这个问题，但其中许多方法都是基于不记住我们示例之间的一些差异。一种方法是定义一个函数，一个不变量，给我们的每个例子分配一些简单的信息，比如一个数字。这本身并不能帮助进步，因为函数可能没有特殊的模式，而且它可能会区分太多的例子。我们可以通过对这些函数施加条件使它更好地工作。最重要的选项之一是请求一个函数，该函数可以由其在较小块上的值确定。这种观点非常强大，并为许多拓扑不变量的研究提供了信息，但与不动点理论相关的重要不变量却没有得到它的支持。这个项目的目标是纠正这一点，并使用优先考虑这种可加性的工具来进一步开发不动点不变量。该项目还支持PI与肯塔基大学不同社会经济背景的研究生群体的合作。PI致力于建立一个社区，学生可以在这里发现困难，轻松地提出问题，并学会有效地为自己辩护。欧拉特征的可加性是这个非常重要的不变量的最重要的性质之一。捕捉这种可加性的一种方法是观察欧拉特征可以被理解为代数k理论中一个类的映像。经典地，与拓扑不动点理论有关的欧拉特征的推广忽略了可加性的意义。本项目的工作旨在纠正这一疏漏。先前的研究结果提出了实现这一目标的两种方法。第一种认识到Reidemeister迹是欧拉特征的一种改进，它给出了与Lefschetz不动点定理的逆，在拓扑Hochschild同调中取值。拓扑Hochschild同调从拓扑限制同调中得到一个映射，Reidemeister迹很可能通过这个映射提升到拓扑限制同调中的一个拓扑有意义的类。另一种方法是从理解e -无穷环谱上模的自同态的k理论开始。从这里开始，我们的目标是描述双范畴和对称单范畴中切环轨迹和轨迹之间的联系。在所有这些工作中，将结果建立在拓扑意义上也很重要——验证在这些不同组中构造的类是与有趣的问题相关的不变量。例如，这些构造应该从给出周期点的不变量开始，并可能扩展到动力系统的不变量。这种观点也符合一个更长远的目标，即使用不动点理论作为产生由错误路径映射或转移引起的加性不变量的新方法的测试用例。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。

One of the barriers to answering many math questions is that there is just too much information - too many examples and too many possibilities. There are many ways to attempt to manage this problem but many of them are based on not remembering some of the differences between our examples. One way to do this is to define a function, an invariant, that assigns some simpler information, say a number, to each of our examples. This by itself doesn't help make progress since there maybe no special pattern to the function and it may distinguish between too many examples. We can make this work much better by imposing conditions on these functions. One of the most important options is to ask that for a function that can determined by its values on smaller pieces. This perspective is very powerful and informs approaches to many topological invariants, but important invariants associated to fixed point theory have escaped its reach. The goal of this project is to rectify this and use the tools that prioritize this additivity to further develop fixed point invariants. The project also supports the PI's work with the socioeconomically diverse graduate student population at the University of Kentucky. The PI has worked to build a community where students can identify difficulties, feel comfortable asking questions, and learn to effectively advocate for themselves. The additivity of the Euler characteristic is one of the most important properties of this very important invariant. One way to capture this additivity is to observe that the Euler characteristic can be understood as the image of a class in algebraic K-theory. Classically, the generalizations of the Euler characteristic that relate to topological fixed point theory ignored the significance of additivity. The work in this project seeks to rectify this omission. Previous results suggest two to approaches to this goal. The first recognizes that the Reidemeister trace, a refinement of the Euler characteristic that gives a converse to the Lefschetz fixed point theorem, takes values in topological Hochschild homology. Topological Hochschild homology receives a map from topological restriction homology and it seems likely that the Reidemeister trace will lift through this map to a topologically meaningful class in topological restriction homology. An alternative approach is to start with the question of understanding K-theory of endomorphisms of modules over E-infinity ring spectra. From here the goal is to describe connections between the cyclotomic trace and trace in bicategories and symmetric monodial categories. In all of this work it also important to ground the results in topological meaning - to verify that classes constructed in these various groups are invariants associated to interesting questions. For example, these constructions should start by giving interesting invariants for periodic points and possibly extend to invariants for dynamical systems. This perspective also fits within a longer term goal of use fixed point theory as a test case for new methods of producing additive invariants that arise from wrong-way-maps or transfers.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.