Homological tools in low-dimensional topology

丹尼·卡莱加里将在一些背景下研究几何、群论和动力学之间的关系，在这些背景下，共同的主题将是使用同源工具。利用曲面映射可以非常有效地探索低维拓扑空间。这项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精化(有界上同调，拟同态)，这是万有圆，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大部分未被研究的现象，与动力学中的不动点问题有关，以及与简单回路猜想等拓扑问题的联系。通过用一个单一的抽象工具(即稳定的换位长度)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，并解决几个公开的猜想。主要研究对象是字双曲(和CAT(0))群的谱间隙和稳定交换子长度的合理性，以及任意群上拟态射的中心极限定理。拓扑学是对几何学的定性研究。经过几个世纪的工作，以及近年来一些引人注目的进展(特别是佩雷尔曼对庞加莱猜想的证明，以及布罗克-金丝雀-明斯基对结束分层猜想的证明)，一些非常基本的问题仍然非常神秘。如果你在一个表面上拉紧一圈绳子，它将如何穿过它自己？在计算机程序中对数据结构进行编码的最有效(抽象)方法是什么？细胞中的流体或DNA链中的漩涡是如何打结的，这种打结如何影响它们的结构稳定性？有时最好的“度量”是定性的；这个项目在尽可能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.