Monodromy in Topology and Geometric Group Theory

拓扑学和代数几何是当代数学中的两门重要学科。拓扑学研究空间结构(例如，甜甜圈的表面、宇宙的大规模形状、缠结的DNA链、海量数据集中的一簇点)，而代数几何研究无处不在的多项式方程的数学。这些学科有许多不同的接触点；首席研究员将特别研究一个：黎曼曲面族。曲面是一个拓扑对象，就像咖啡杯的二维曲面，可能有很多额外的把手。黎曼曲面提供了一种特殊的方法来描述这样一个物体，作为多项式方程的解，就像我们在高中代数中学到的那样，一些方程描述圆，另一些方程描述椭圆，以及其他更复杂的形状。通过改变用来定义黎曼曲面的方程，就产生了一族黎曼曲面--人们可以想象“转动旋钮”来拉伸和扭曲曲面的形状。黎曼曲面族存在于数学的许多领域，在理论物理中也起着重要的作用。主要研究人员将应用拓扑学和几何群论的相关学科的工具，以便更好地理解当今数学家感兴趣的一些最重要的黎曼曲面族。更广泛的影响包括建立国家定向阅读项目网络的新篇章。该项目由两个部分组成。首先将研究阿贝尔微分层(平移面)的拓扑结构。动态学家和几何学家已经对这些家族进行了深入的研究，但仍有一些基本的拓扑问题仍然存在。特别是，(奥比福尔德)地层的基本群体仍然非常神秘。这可以通过单行表示、映射到映射类组中的方式有效地探测到。首席研究员将继续他的工作，描述这些单元性的表示，并将开发新的工具来理解单元性核心，从而进一步发展基本地层群的理论。这项工作的核心将是阐明阿尔丁集团和基本地层集团之间的确切关系。该项目的第二个组成部分涉及通过分支保护层构建的族。这些族已成为拓扑学、代数几何和群论中的重要例子来源。首席研究员将调查这些家族的单一性，目的是进一步将拓扑学方面(辫子群、映射类群)与代数(算术群和推广、Burau类表示、本原同调)相结合。该奖项反映了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 and algebraic geometry are two subjects of central importance in contemporary mathematics. Topology is the study of spatial structures (e.g. the surface of a donut, the large-scale shape of the cosmos, a tangled strand of DNA, a cluster of points in a massive data set), while algebraic geometry studies the mathematics of the ubiquitous polynomial equation. These disciplines have many and diverse points of contact; the principal investigator will study one in particular: families of Riemann surfaces. A surface is a topological object like the two-dimensional surface of a coffee cup, possibly one with lots of extra handles. A Riemann surface gives a special way of describing such an object as the solution to a polynomial equation, much as we learn in high school algebra that some equations describe circles, others ellipses, and others far more complicated shapes. A family of Riemann surfaces arises by varying the equations used to define the Riemann surface - one can imagine "turning a knob" to stretch and distort the shapes of the surfaces. Families of Riemann surfaces arise in many parts of mathematics and also play an important role in theoretical physics. The principal investigator will apply tools from topology and the related discipline of geometric group theory in order to better understand some of the most important families of Riemann surfaces that mathematicians today are interested in. Broader impacts include the establishment of a new chapter of the national Directed Reading Project network.The project has two components. The first will investigate the topology of strata of Abelian differentials (translation surfaces). These families have been intensively studied by dynamicists and geometers, but there are foundational topological questions that still remain. In particular, the (orbifold) fundamental groups of strata are still highly mysterious. This can be effectively probed by way of the monodromy representation, a map into the mapping class group. The principal investigator will continue his work describing these monodromy representations, and will develop new tools to understand the monodromy kernel and so further develop the theory of fundamental groups of strata. Central to this endeavor will be an elucidation of the precise relationship between Artin groups and fundamental groups of strata. The second component of this project concerns families constructed via branched covers. These families have served as an important source of examples in topology, algebraic geometry, and group theory. The principal investigator will investigate the monodromy of these families with the objective of further synthesizing the topological aspects (braid groups, mapping class groups) with the algebraic (arithmetic groups and generalizations, Burau-like representations, primitive homology).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.