Stable commutator length

在以前的工作中，PI开发了一些强大而精细的工具，通过稳定的交换子长度(相对二维的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类似于双曲三维流形拓扑中体积谱的自由基团的SCL谱的结构，并从手术的角度进行了部分解释，以及与高维连分式理论、计数组合学、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范数的多面体结构中隐含的抽象刚性，这表现在表面群的辛表示的刚性，以及某些辛同构群的Ruelle不变量及其推广的动态刚性。PI将致力于统一和解释这些现象，并给予它们严格的证明。两千多年来，几何一直被定义和研究在一维术语中：为了测量两点之间的距离，找到从一个点到另一个点的最短直线(即最小的一维对象)。自16世纪以来，数学家们已经认识到，一维“实”数只是二维“复”数的影子。最近，几何学以类似的方式“复杂化”，数学家通过用二维表面探测空间几何的不同方面来探索它。这样的表面可以是金属丝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链之间的假想表面。PI将使用代数、群论、拓扑学和泛函分析中的工具来研究解决“极端”几何问题(在许多意义上)的曲面。

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.