Stable commutator length

在之前的工作中，PI开发了一些强大而精细的工具，用于通过稳定的换易子长度（相对的二维Gromov范数）在二维有界上同调中进行精确计算，特别是在自由群中。实验和一些理论工作揭示了自由群的scl谱的结构类似于双曲3流形拓扑中的体积谱，并给出了部分外科解释，以及与高维连分式理论、枚举组合学、耦合非线性振子的锁相理论和DNA热力学的联系。其他理论工作揭示了scl范数多面体结构中隐含的抽象刚性，这种刚性表现为曲面群的辛表示的刚性，以及某些辛形态群的Ruelle不变量及其推广的动态刚性。PI将试图统一和解释这些现象，并给他们一个严格的理由。两千多年来，几何学一直是用一维术语来定义和研究的：为了测量两点之间的距离，找到从一点到另一点的最短线（即最小的一维物体）。自16世纪以来，数学家们已经明白，一维的“实数”只是二维“复数”的影子。最近，几何学以类似的方式被“复杂化”了，数学家们通过用二维曲面探测空间来探索空间几何的不同方面。这样的表面可以是一个在线框上有边界的肥皂泡，或者是湍流和非湍流之间的界面，或者是DNA链之间的想象表面。PI将研究解决“极端”几何问题（在许多意义上）的曲面，使用工具来解决起源于代数、群论、拓扑和功能分析的问题。

In previous work, the PI developed some powerful and delicate tools for making exact calculations in 2-dimensional bounded cohomology, especially in free groups, via the stable commutator length (a relative 2-dimensional Gromov norm). Experiments and some theoretical work reveals structure in the scl spectrum of a free group analogous to the volume spectrum in hyperbolic 3-manifold topology, together with a partial explanation in terms of surgery, and connections to the theory of higher-dimensional continued fractions, enumerative combinatorics, phase locking for coupled nonlinear oscillators, and the thermodynamics of DNA. Other theoretical work reveals an abstract rigidity implicit in the polyhedral structure of the scl norm, which manifests itself in rigidity for symplectic representations of surface groups, and dynamical rigidity of Ruelle invariants and their generalizations for certain groups of symplectomorphisms. The PI will atempt to unify and explain these phenomena, and to give them a rigorous justification.For more than two thousand years, geometry has been defined and studied in 1-dimensional terms: to measure the distance between two points, find the shortest line (i.e. the smallest 1-dimensional object) that goes from one to the other. Since the 16th century, mathematicians have understood that the 1-dimensional "real" numbers are just a shadow of the 2-dimensional "complex" numbers. Recently, geometry has been "complexified" in a similar way, and mathematicians explore different aspects of the geometry of a space by probing it with 2-dimensional surfaces. Such a surface could be a soap bubble with boundary on a wire frame, or an interface between turbulent and non-turbulent flow, or an imaginary surface interpolating between strands of DNA. The PI will study the surfaces that solve "extremal" geometric problems (in many senses of the word) using tools from - and in order to solve problems that originate in - algebra, group theory, topology, and functional analysis.