Embeddings, Intersections and Symmetries

这一建议涉及流形同伦理论的各个方面，所有这些方面都以这样或那样的方式与嵌入理论有联系。本文主要分为四个部分：1)嵌入理论。PI将使用由庞加莱复合体的同伦理论产生的技术来研究嵌入问题。2)小组行动。利用同伦理论对等变流形分类问题的研究提供了新的途径。3)字符串拓扑。这是由Chas和sullivan创建的新兴领域，研究与流形自由环空间相关的代数结构。PI提出研究弦拓扑的同伦理论方面。4)增厚。空间的“增厚”是其同伦型的流形模型。提出了分析有限复形增厚的模空间的同伦类型。“流形”是局部欧几里得的空间。流形自然地出现在物理、化学和生物学中，作为一组适当的“好”代数方程的解的空间，它们模拟了研究的科学对象（时空、原子、动力系统等）。流形在数学中起着核心作用。通常情况下，流形的数学问题可以用流形相关空间间的参数化函数族来表示。同伦理论是一门旨在解决这类函数族问题的学科。PI提出研究一类有希望从同伦理论工具箱中分析出来的流形问题。这些问题涉及到与相交、嵌入和对称有关的问题。

This proposal involves various aspects of the homotopy theory ofmanifolds, all of which, in one way or another, have connections tothe theory of embeddings. The proposalhas four parts: 1) Embedding theory. The PI will use techniques arising from thehomotopy theory of Poincare complexes to study embedding questions. 2) Group Actions. The study of equivariant manifold classificationproblems will be approached in a new way using homotopy theory. 3) String Topology. This is emerging field, created by Chas andSullivan, studies algebraic structures associated with the free loopspace of a manifold. The PI proposesto study homotopy theoretic aspects of string topology. 4) Thickenings. A "thickening" of a space is a manifold model forits homotopy type. The PI proposes to analyse the homotopy type of the moduli space of thickenings of a finite complex.A "manifold" is a space which is locally Euclidean. Manifolds arisenaturally in physics, chemistry and biology as spaces of solutions ofa suitably "nice" set of algebraic equations modeling the scientificobject of study (space-time, atoms, dynamical systems, etc.) .Manifolds play a central role in mathematics. It is often the casethat mathematical questions about manifolds can be formulated interms of parametrized families of functions between spaces associatedwith manifolds. Homotopy theory is a subject designed to tacklequestions about such families of functions. The PI proposes to studycertain kinds of manifold questions which can be hopefully be analyzedfrom the homotopy theoretic toolbox. These involve problems relatedwith intersections, embeddings and symmetries.