RUI: Motivic, Operadic, and Combinatorial Homotopy Theory

近年来，当代同伦理论--形状和变形的数学--的范围急剧扩大，需要新的工具和视角来理解它的特点和潜力。PI将通过组合学(列举结构)、运算符(参数化运算)和代数几何(由多项式方程描述的形状)的镜头来探索同伦理论。督导人员亦会透过成立协作数学研究小组，支持不同组别的本科生参与研究。CMRG参与者将产生新的研究成果，并让他们的当地社区参与到数学推广工作中。PI将研究同伦理论的运算、组合和理据方面，目的是产生结构、计数和计算结果。他们的工作包括：(1)进一步探索有限范畴上模型结构的组合学，特别是着眼于任意类型的加泰罗尼亚组合学(在簇代数意义下)。(2)通过利用与其étalal变体的回忆，揭示稳定动机同伦范畴的Balmer谱中的额外结构。(3)通过计算Bachmann-Hopkins连通厄米特像J谱的切片谱序列，扩展我们对稳定动机同伦理论的计算理解。(4)为E_n-空间构建一个刚性模型，作为特定图范畴中的交换么半群。该奖项反映了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。

The scope of contemporary homotopy theory — the mathematics of shape and deformation — has expanded dramatically in recent years, and new tools and perspectives are needed to understand its features and potential. The PIs will explore homotopy theory through the lenses of combinatorics (enumerating structures), operads (parametrizing operations), and algebraic geometry (shapes described by polynomial equations). The PIs will also support the participation of a diverse group of undergraduate students in their research by organizing the Collaborative Mathematics Research Group (CMRG). CMRG participants will produce novel research results and also engage their local community in mathematics outreach efforts.The PIs will investigate operadic, combinatorial, and motivic aspects of homotopy theory with the aim of producing structural, enumerative, and computational results. Their projects include the following:(1) Further explore the combinatorics of model structures on finite categories, especially with an eye towards Catalan combinatorics of arbitrary type (in the sense of cluster algebras).(2) Uncover additional structure in the Balmer spectrum of the stable motivic homotopy category by leveraging a recollement with its étale variant.(3) Extend our computational understanding of stable motivic homotopy theory by computing the slice spectral sequence for the Bachmann-Hopkins connective Hermitian image-of-J spectrum.(4) Construct a rigid model for E_n-spaces as commutative monoids in a certain diagram category.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.