Cohomology Jumping Loci

这个项目从不同的角度探索了空间的拓扑和几何之间的深刻而复杂的联系，它的基本群的代数结构，以及相关上同调跳跃轨迹的代数几何。这项研究是在这样一个背景下进行的，它抽象出了超平面排列补的拓扑学和代数工作之间的两个关键特征：形式性(它导致了切锥定理，将特征和共振簇联系在一起)和拟投射性(它确保特征簇由可能翻译的子环组成)。这两个性质一起对共振簇的几何构成了很强的限制，导致有限表示群可以实现为光滑的(准)射影复变集的基本群。最近，出现了新的联系，将空间的上同调跳跃轨迹与它的自由阿贝尔覆盖的同调有限性质联系起来，从而与Dwyer-Fred和Bieri-Neumann-Strebel-Renz不变量的结构有关。经典跳跃轨迹的推广--从秩单局部系统到非阿贝尔环境，从上同调环到微分分次代数-扩大了研究的范围在这个项目中使用的拓扑学和群论方法揭示了发生在各种背景下的物体的组合和几何结构，允许不同领域之间的交叉授粉，来自给定领域的想法在新的背景下得到了有效的应用。上同调跳跃轨迹理论影响了一系列广泛的空间和群的研究，包括环面复形和矩角复形；直角Artin和Coxeter群；实和复准环面流形；Kaehler流形和拟Kaehler流形；超平面排列的Milnor纤维；以及配置空间和模空间的紧致。对这些对象的研究，以及它们与奇点理论、图论和低维拓扑的多重必要性，提供了代数、拓扑和组合学之间的广泛相互作用，产生了从拓扑机器人到理论计算机科学的广泛应用。这项调查是与几位合作者以及研究生和本科生一起进行的。参加密集研究阶段和研讨会的目的是将新一代的学生和年轻的研究人员介绍到一个非常活跃的跨学科研究领域。

This project explores from various angles the deep andintricate connections between the topology and geometryof a space, the algebraic structure of its fundamentalgroup, and the algebraic geometry of the associatedcohomology jumping loci. The study is done in a contextwhich abstracts two of the crucial features that makethe dictionary between topology and algebra work sowell for complements of hyperplane arrangements:formality (which leads to the Tangent Cone Theorem,relating the characteristic and resonance varieties), andquasi-projectivity (which insures that the characteristicvarieties consist of possibly translated subtori). Together,these two properties put strong constraints on the geometryof the resonance varieties, leading to powerful obstructionsto finitely presented groups being realizable as fundamentalgroups of smooth, (quasi-) projective complex varieties.Recently, new connections have emerged, relating the cohomologyjumping loci of a space to the homological finiteness propertiesof its free abelian covers, and thereby to the structure of theDwyer-Fried and Bieri-Neumann-Strebel-Renz invariants.Generalizations of the classical jump loci---from rank onelocal systems to the non-abelian setting, and from cohomologyrings to differential graded algebras---extend the scope ofthe investigation, and broaden the range of its applicability.The topological and group-theoretic methods utilizedin this project shed new light on the combinatorial andgeometric structure of objects occurring in a varietyof contexts, allowing for cross-pollination betweendifferent fields, with ideas originating in a given areabeing fruitfully applied in new settings. The theory ofcohomology jumping loci impacts the study of a widearray of spaces and groups, including toric complexesand moment-angle complexes; right-angled Artin andCoxeter groups; real and complex quasi-toric manifolds;Kaehler and quasi-Kaehler manifolds; Milnor fibrationsof hyperplane arrangements; as well as configurationspaces and compactifications of moduli spaces.The study of these objects, with their multipleconnections to the theory of singularities, graphtheory, and low-dimensional topology, provides arich interplay between algebra, topology, andcombinatorics, yielding applications to areasranging from topological robotics to theoreticalcomputer science. The investigation is being conductedtogether with several collaborators, as well as graduateand undergraduate students. Participation in intensiveresearch periods and workshops is meant to introduce anew generation of students and young researchers to avery active, interdisciplinary area of study.