Homotopical Methods in Fixed Point Theory

该奖项支持参加2022年7月11日至15日在科罗拉多大学博尔德分校举办的暑期班“不动点理论中的同伦方法”。这个工作坊将为来自不同机构和背景的参与者提供一个学习代数拓扑学高级主题的机会，代数拓扑学是一个数学领域，使用代数中的工具来研究几何对象的某些不变性质。研讨会的好处是，这个领域的问题和问题很容易陈述和激发，但他们的研究自然会引导人们考虑代数拓扑学领域的现代发展。积极学习的形式，以及对在社区中彼此共同创造数学的重视，将允许参与者之间进行重大合作，并与科学委员会进行互动。专题的可接近性和讲习班的形式将吸引来自具有较小拓扑学研究小组的机构的与会者，以及那些身份处于边缘化或代表性不足的机构的与会者。科学委员会本身由职业生涯早期的研究人员组成，组织和科学委员会中有一半以上是女性。这种表现形式和它为暑期班设计提供的不同视角将对与会者产生积极的影响。暑期班的科学目标是通过不动点理论的镜头向学员介绍代数拓扑和同伦理论的工具和思想。工作坊将围绕以主动学习方式教授的迷你课程进行组织，课程主题将从经典的不动点理论到现代工具，如代数K-理论中的对偶、谱和迹方法。这一范围反映了同伦理论和高范畴理论的最新进展和表面上不相关的进展，这些进展为拓扑不动点理论提供了新的方法。这些方法使用通常难以概括的单纯结构定义了精炼的经典不变量。这种视角的转变也说明了有用的不动点不变量的可加性的中心性。由于代数K-理论是加性不变量的普遍家园，寻找不动点理论和代数K-理论之间的联系是很自然的。最近K群计算方面的重要进展进一步支持了这种联系。科学委员会的一些成员的工作已经确定，这些计算方法可以使用新形式的不动点不变量来有效地建模。这为这些重要的发展提供了一种非常不同的方法，并提供了一个新的直觉来源。Https://sites.google.com/colorado.edu/fixedpointtheory2020/This奖反映了国家科学基金会的法定使命，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。

This award supports participation in the summer school “Homotopical Methods in Fixed Point Theory,” taking place July 11 to 15, 2022 at the University of Colorado, Boulder. This workshop will be an opportunity for participants from a wide variety of institutions and backgrounds to learn advanced topics in algebraic topology, an area of mathematics that uses tools from algebra to study certain invariant properties of geometric objects. The advantage of the workshop is that the questions and problems in this field are easy to state and motivate, yet their study naturally leads one to consider modern developments in the field of algebraic topology. The active-learning format, and emphasis on co-creating mathematics in community with one another, will allow for significant collaboration between participants and for interactions with the scientific committee. The approachability of the topic and the format of the workshop will attract participants coming from institutions with smaller topology research groups, as well as those with marginalized or underrepresented aspects of their identity. The scientific committee itself is made up of early-career researchers, and more than half of the organizing and scientific committee are women. This representation and the diverse perspectives it offers for the design of the summer school will have a positive impact on attendees.The scientific goal of the summer school is to introduce participants to tools and ideas from algebraic topology and homotopy theory through the lens of fixed point theory. The workshop will be structured around mini-courses taught in an active-learning style and course topics will range from classical fixed point theory to modern tools such as duality, spectra, and trace methods in algebraic K-theory. This range reflects recent and superficially unrelated advances in homotopy theory and higher category theory that have provided new approaches to topological fixed point theory. These approaches have refined classical invariants defined using simplicial structures that were often difficult to generalize. This shift in perspective also illuminated the centrality of additivity for useful fixed point invariants. Since algebraic K-theory is the universal home for additive invariants, it is natural to look for connections between fixed point theory and algebraic K-theory. This connection is further supported by the recent important developments in computations of K-groups. Work of some members of the scientific committee has established that these computational approaches can be effectively modeled using the new forms of fixed point invariants. This provides a very different approach to these important developments and a new source of intuition. https://sites.google.com/colorado.edu/fixedpointtheory2020/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.