FRG: Collaborative Research: Homotopical Methods in Algebraic Geometry

首席研究人员将共同努力，使用现代同伦技术研究代数几何中的基本问题；这些问题中的一个统一线索是从纯粹的代数几何到纯粹的拓扑学的各种不变量的重要性。首先，PI建议研究实代数簇之间的态射空间的结构，特别是“实代数”态射的不稳定和稳定同伦类型的空间。其次，PI将检查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建议研究涉及cdh-拓扑的方法产生的奇点的不变量，继续最近在这一学科中的一系列活动。最后，受代数几何和拓扑不变量比较的启发，PI将研究代数簇的半拓扑不变量或形态不变量，它介于代数几何和拓扑学的世界之间。代数几何是数学中最古老的分支之一，其核心目标是研究多项式方程组解的结构；这些解的集合被称为代数簇。同伦理论，有时被称为橡皮片几何，试图研究几何对象的那些方面，这些方面与它们被拉或扭曲的方式无关；实现这一点的一种方法是将数字(或更一般的代数结构)的“不变量”附加到这些对象上。由实系数或复系数方程产生的代数变体可以用同伦理论来研究，而产生的不变量必然受到一定的限制。这个项目的目标是利用同伦理论中产生的代数簇的不变量来研究代数几何中的经典问题。这个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研究员为代表的下一代数学家。招募和吸收早期职业数学家的方法将包括组织一次大型国际会议，举办几个讲习班，分享旅费，以及有其他机构的参观者参加的活动。

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.