FRG: Collaborative Research: Homotopical Methods in Algebraic Geometry

主要研究人员将加入一个合作的努力，利用现代同调技术研究代数几何的基本问题；这些问题的一个统一线索是各种不变量的重要性，从纯代数几何到纯拓扑。首先，研究了实代数变体之间的态射空间的结构，特别是“实代数”态射空间的不稳定同伦型和稳定同伦型。其次，pi将研究各种离散和算术群的上同调，包括上同调的同伦不变性的代数版本和相关的Friedlander-Milnor猜想。第三，pi建议研究由涉及cdh拓扑的方法产生的奇点不变量，继续最近在该主题中的活动。最后，在代数几何不变量和拓扑不变量之间的比较的激励下，pi将研究代数变体的半拓扑或形态不变量，它们位于代数几何和拓扑世界之间。代数几何是数学最古老的分支之一，其核心目标是研究多项式方程组的解的结构；这些解的集合称为代数变种。同伦理论，有时被称为橡胶板几何，试图研究几何物体的那些与它们被拉动或扭曲的方式无关的方面；一种方法是为这些对象附加“不变量”，例如数字（或更一般的代数结构）。实系数或复系数方程的代数变异可以用同伦理论来研究，其不变量必然受到一定的限制。本课题的目的是利用由同伦理论产生的代数变异的不变量来研究代数几何中的经典问题。该项目的一个主要目的是将主要研究人员的一些热情、技术和数学目标传达给以研究生和博士后为代表的下一代数学家。招募和吸引早期职业数学家的方法包括组织一次大型国际会议，举办几次研讨会，分享旅行资金，以及邀请其他机构的访问者参加活动。

The Principal Investigators will join in a collaborative effort to investigate fundamental questions in algebraic geometry using modern homotopical techniques; a unifying thread in these questions is the importance of various classes of invariants ranging from purely algebro-geometric to purely topological. First, the PIs propose to investigate the structure of morphism spaces between real algebraic varieties, especially unstable and stable homotopy types of spaces of "real algebraic" morphisms. Second, the PIs will examine the cohomology of various discrete and arithmetic groups, including algebraic versions of homotopy invariance for cohomology and the related Friedlander-Milnor conjecture. Third, the PIs propose to investigate invariants of singularities arising from methods involving the cdh-topology, continuing the recent flurry of activity in this subject. Finally, motivated by comparisons between algebro-geometric and topological invariants, the PIs will investigate semi-topological or morphic invariants of algebraic varieties, which lie partway between the worlds of algebraic geometry and topology.Algebraic geometry, one of the oldest branches of mathematics, has at its heart the goal of studying the structure of solutions to systems of polynomial equations; these collections of solutions are called algebraic varieties. Homotopy theory, sometimes called rubber sheet geometry, attempts to study those aspects of geometric objects that are independent of the way they are pulled or twisted; one way to do this is to attach "invariants," e.g., numbers (or more general algebraic structures), to these objects. Algebraic varieties arising from equations with real or complex coefficients can be studied by means of homotopy theory, and the invariants that arise are necessarily somewhat restricted. The goal of this project is to study classical questions in algebraic geometry using invariants of algebraic varieties arising from homotopy theory. A major aim of this project is to convey some of the enthusiasm, techniques, and mathematical goals of the principal investigators to the next generation of mathematicians represented by graduate students and postdoctoral fellows. Methods to recruit and involve early career mathematicians will include the organization of a large international conference, the running of several workshops, the sharing of travel funds, and activities involving visitors from other institutions.