Homological tools in low-dimensional topology

Danny Calegari将研究几何、群论和动力学之间的关系，在许多情况下，共同的主题将是同调工具的使用。低维拓扑空间可以通过曲面映射非常有效地探测。该研究的线性化导致了同调及其变体（有理同伦，代数k理论等）；与几何的相互作用导致了进一步的改进（有界上同调、拟同态），这些改进是万向圆、瑟斯顿范数、11/8猜想等现象的核心。有界上同调与数论的相互作用导致了新的和很大程度上未被研究的现象，与动力学中的不动点问题和简单环猜想等拓扑问题有关。通过从一个抽象工具（即稳定换向子长度）的角度来探索这些现象，Calegari打算开发一种机制来解释它们的统一性，并解决几个开放的猜想。主要研究词双曲（和CAT(0)）群的谱隙和稳定换易子长度的合理性，以及任意群上拟同态的中心极限定理。拓扑学是几何学的定性研究。经过几个世纪的努力，近年来取得了一些惊人的进展（特别是佩雷尔曼对庞加莱猜想的证明，以及布罗克-加那利-明斯基对末层叠猜想的证明），一些非常基本的问题仍然非常神秘。如果你把一圈绳子拉紧在一个表面上，它会如何交叉？在计算机程序中对数据结构进行编码的最有效（抽象）的方法是什么？液体中的漩涡或细胞中的DNA链是如何打结的，这种打结是如何影响它们的结构稳定性的？有时候最好的“指标”是定性的；本项目在最简单的拓扑环境中研究这种大小和效率的定性度量，因为我们获得的结果将普遍适用于其他更复杂的环境。

Danny Calegari will study the relationship between geometry, group theory and dynamics in a number of contexts where the common theme will be the use of homological tools. Low dimensional topological spaces can be very effectively probed by maps of surfaces. The linearization of this study leads to homology and its variants (rational homotopy, algebraic K-theory, etc.); the interaction with geometry leads to further refinements (bounded cohomology, quasimorphisms) which are at the heart of phenomena such as universal circles, Thurston norm, the 11/8 conjecture, etc. The interaction of bounded cohomology with number theory leads to new and largely unstudied phenomena, with connections to fixed point problems in dynamics, and topological problems like the simple loop conjecture. By exploring these phenomena in terms of a single underlying abstract tool (namely stable commutatorlength) Calegari intends to develop machinery with which to explain their unity, and to settle several open conjectures. The main objects of study are spectral gaps and rationality of stable commutator length in word-hyperbolic (and CAT(0)) groups, and central limit theorems for quasimorphisms on arbitrary groups.Topology is the qualitative study of geometry. After centuries of work, and some spectacular progress in recent years (especially Perelman's proof of the Poincare Conjecture, and Brock-Canary-Minsky's proof of the ending lamination conjecture) some very fundamental questions are still very mysterious. If you pull a loop of string tight on a surface, how will it cross itself? What is the most efficient (abstract) way to encode a data structure in a computer program? How do vortices in fluids or strands of DNA in a cell become knotted, and how does this knottiness affect their structural stability? Sometimes the best "metrics" are qualitative; this project studies such qualitative metrics of size and efficiency in the simplest possible topological contexts, since the results we obtain will be universal in their applicability to other, more complicated contexts.