Research in Geometric and Quantitative Topology

拓扑学通常被认为是几何学的一种定性形式。这种灵活性对于它在其他领域中的作用是很重要的，因为它能够处理有些嘈杂或不精确的数据，并对其进行严格的推理。然而，对于许多目的，人们想要知道拓扑结构有多大或复杂，以及扰动的解有多稳定。 从技术上讲，这要求人们不仅要研究从一个空间到另一个空间的非线性函数，还要研究约束函数的测量（例如导数的大小）的空间。 这种混合的解析拓扑研究将是该项目的核心，应用于几何复杂性，奇异空间，从这种研究中受益，但也需要额外的几何和代数工具。本计画将研究光滑与PL设定下同伦、嵌入、浸入与协边的Lipschitz常数与双Lipschitz常数之复杂性。 在某种意义上，这应该与通常的几何拓扑学有着相同的关系，就像理论计算机科学与逻辑学的关系一样。事实上，使用不可判定性的结果，人们可以证明较低的复杂性界限（如Nabutovsky的ICM演讲），但许多同伦理论，特别是稳定同伦理论是可判定的，但不是有效的（跟随布朗）。新的信息可以被认为是提供了关于Lipschitz映射的函数空间的几何信息，表明它们有一些直径上的界限，与它们的体积（也称为熵或覆盖数）的更大估计（作为近似理论和一些学习理论的基础）相比，这些界限非常引人注目。 除了对这些问题的内在兴趣之外，它们还涉及变分问题，也许还涉及计算（例如Blum-Shub-Smale模型）。 这些定量问题的早期变体已经出现在平方可积上同调和控制（和有界）拓扑的研究中。 这些理论将继续被研究，希望能揭示在特殊情况下（如虚拟可解群）的Borel/Baum-Connes拓扑的更微妙的改进，以及理解非球面流形上的群作用的应用。 有趣的是，虽然目前还没有一个已知的数量限制，在一个流形嵌入另一个流形的余维至少为3，有限性是已知的（对于任何Bi-Lipschitz界），有一种自然的策略，将控制拓扑与关于函数空间的新信息相结合，该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Topology is usually thought of as a qualitative form of geometry. This flexibility is important for the role it plays in other areas in being able to deal with somewhat noisy or imprecise data, and reason rigorously about it. For many purposes, however, one wants to know how large or complex topological constructions are, and how stable solutions are to perturbations. This requires, technically, that one studies not just nonlinear functions from one space to another, but also spaces that bound measurements (such as the size of derivatives) on the functions. This mixed analytic topological study will be at the core of the project, with applications to geometric complexity, to spaces with singularities, which benefit from such study, but also require additional geometric and algebraic tools. Application to problems of numerical computation with natural resource bounds is anticipatedIn more detail, this project will study the complexity measured in terms of Lipschitz constants and bi-Lipschitz constants of homotopies, embeddings, immersions and (volumes for) cobordisms in smooth and PL settings. In some sense this should have the same relation to usual geometric topology as theoretical computer science has to logic. Indeed, using undecidability results, one can prove lower complexity bounds (as in Nabutovsky's ICM talk), but much homotopy theory, especially stable homotopy theory are decidable, but not effectively so (following Brown). The new information can be thought of as providing geometric information about function spaces of Lipschitz maps, showing that they have some bounds on diameter that are quite striking in comparison to the much larger estimates (that underly approximation theory, and some learning theory) for their volumes (also known as entropy, or covering numbers). Besides the intrinsic interest in these questions, they also bear on variational problems and perhaps on computation (as in, for example, the Blum-Shub-Smale model). Earlier variants of these quantitative concerns have already arisen in the study of square integrable cohomology and controlled (and bounded) topology. These theories will continue to be studied, hopefully revealing more subtle refinements of the Borel/Baum-Connes conjectures in special cases (such as virtually solvable groups) and applications to understanding group actions on aspherical manifolds. Interestingly, although there is currently no known quantitative bound at all on the number of embeddings of one manifold in another in codimension at least three, finiteness is known (for any Bi-Lipschitz bound), there is a natural strategy combining controlled topology with new information about function spaces to give, conjecturally, sharp estimates for these finite numbers.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.