Chromatic homotopy theory, algebraic K-theory, and L-functions

代数拓扑学是通过相关的代数实体研究拓扑空间的一门学科。其中最重要的实体或不变量之一是拓扑空间的同伦群，它衡量空间是否可以连续变形为一个点。球面同伦群的计算一直是代数拓扑学发展的动力，在数学的许多其他领域都有广泛的应用，包括数论。这个项目的首要目标是进一步探索数论和同伦论之间的深刻联系。这个项目的更广泛的影响集中在色同伦理论、学生指导和会议组织方面的研究生课程发展。PI计划调查色同伦理论、代数K-理论和L函数之间的各个方面的联系。在他以前工作的基础上，PI将使用等变代数K-理论的工具，探索Quillen-Lichtenbaum猜想到与Galois表示的阿贝尔特征相关的各种L函数的推广。代数K-理论谱具有相干乘法结构。PI计划给出这种结构的显式代数公式，并研究它在Zeta函数中的含义。PI将研究的另一个问题是如何将Eisenstein级数和Hida的二元L函数与椭圆上同调的幂运算联系起来。最后，PI计划使用算术几何的工具来研究色同伦理论中的消失猜想和Picard群。这一奖项反映了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。

Algebraic topology is a subject that studies topological spaces via associated algebraic entities. One of the most important such entities, or invariants, is the homotopy groups of a topological space, which measure whether or not the space can be continuously deformed into a single point. The computation of homotopy groups of spheres has been a driving force in the development of algebraic topology with wide applications to many other fields of mathematics, including number theory. The overarching goal of this project is to further explore the profound connections between number theory and homotopy theory. Broader impacts of this project focus on graduate curriculum development in chromatic homotopy theory, student mentoring, and conference organization.The PI plans to investigate various aspects of the connections between chromatic homotopy theory, algebraic K-theory, and L-functions. Building on his previous work, the PI will explore generalizations of the Quillen-Lichtenbaum Conjecture to various L-functions associated to abelian characters of Galois representations, using tools from equivariant algebraic K-theory. Algebraic K-theory spectra have a coherent multiplicative structure. The PI plans to give explicit algebraic formulas for this structure and investigate its implications in zeta functions. Another question the PI will study is how to relate Eisenstein series and Hida’s L-functions of two variables to power operations on elliptic cohomology. Finally, the PI plans to use tools from arithmetic geometry to investigate the Vanishing Conjecture and Picard groups in chromatic homotopy theory.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.