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。这种灵活性很重要，因为它在其他领域扮演着重要的角色，能够处理有些嘈杂或不准确的数据，并对此进行严格的推理。然而，为了许多目的，人们想知道拓扑结构有多大或多复杂，以及对扰动的解有多稳定。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，而且还要研究将测量(如导数的大小)限制在函数上的空间。这一混合分析拓扑研究将是该项目的核心，应用于几何复杂性，应用于具有奇点的空间，这些研究受益于此类研究，但也需要额外的几何和代数工具。这个项目将更详细地研究光滑和PL环境下同伦、嵌入、浸入和(体积)上的同伦的Lipschitz常数和双Lipschitz常数所度量的复杂性。在某种意义上，这应该与通常的几何拓扑学具有相同的关系，就像理论计算机科学与逻辑之间的关系一样。事实上，使用不可判定的结果，可以证明较低的复杂性界(如在Nabutovsky的ICM谈话中)，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Brown)。这些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，表明它们在直径上有一些界限，与对其体积(也称为熵，或覆盖数)的更大估计(欠逼近理论和一些学习理论)相比，这些界限是相当惊人的。除了对这些问题的内在兴趣外，它们还涉及变分问题，也许还涉及计算(例如，在Blum-Shub-Smear模型中)。这些定量问题的早期变体已经出现在平方可积上同调和受控(和有界)拓扑的研究中。这些理论将继续被研究，有望揭示Borel/Baum-Connes猜想在特殊情况下(如虚拟可解群)的更微妙的改进，并应用于理解非球面流形上的群作用。有趣的是，虽然目前还没有关于一个流形在余维至少为三的另一个流形中嵌入的数量的已知的量化界限，但有限是已知的(对于任何双Lipschitz界限)，有一种自然的策略结合了受控拓扑和关于函数空间的新信息来给出这些有限数目的精确估计。这个奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.