Monodromy in Topology and Geometric Group Theory

拓扑学和几何群论中的单向性

• 批准号: 2003984
• 负责人: Nicholas Salter
• 金额: $ 16.18万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-07-01 至 2021-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2003984&HistoricalAwards=false
• 关键词: Monodromy; Topology; Geometric; Group; Theory

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.

拓扑和代数几何形状是当代数学中至关重要的两个主题。拓扑是对空间结构的研究（例如，甜甜圈的表面，宇宙的大规模形状，缠结的DNA链，大量数据集中的一个点簇），而代数几何学研究了无处不在的多项式方程的数学。这些学科具有许多不同的接触点；主要研究者将特别研究一个：Riemann表面家庭。表面是拓扑对象，例如咖啡杯的二维表面，可能是一个额外的手柄。 Riemann表面提供了一种特殊的方式来描述诸如对多项式方程式的解决方案，就像我们在高中代数中所了解的那样，某些方程式描述了圆圈，其他椭圆形和其他更为复杂的形状。一个黎曼表面的家族是通过改变用于定义Riemann表面的方程而产生的 - 人们可以想象“转动旋钮”以伸展和扭曲表面的形状。黎曼表面的家族在数学的许多地方都出现，并且在理论物理学中也起着重要作用。首席研究者将应用拓扑结构的工具和几何群体理论的相关学科，以便更好地了解当今数学家感兴趣的一些最重要的Riemann表面家庭。更广泛的影响包括建立国家有导性阅读项目网络的新章节。该项目具有两个组成部分。第一个将研究Abelian差异层层（翻译表面）的拓扑。这些家庭已经由动态主义者和几何学家进行了深入的研究，但是仍然存在基本的拓扑问题。特别是（Orbifold）基本阶层的基本群体仍然是高度神秘的。这可以通过单一莫术表示有效地探测，该图是映射类组的地图。首席研究者将继续他的工作描述这些单肌表示，并将开发新的工具来了解单构核，从而进一步发展基本阶层基本群体的理论。这项努力的核心将是对Artin群体与基本阶层群体之间的确切关系的阐明。该项目的第二部分涉及通过分支封面建造的家庭。这些家庭是拓扑，代数几何学和群体理论的重要例子的重要来源。主要研究者将研究这些家族的单片，目的是通过代数（算术组和概括，类似Burau的代表，原始同源性）来进一步综合拓扑方面（编织组，映射课程组）。这一奖项反映了NSF的法定任务，并反映了审查的范围，并已被认为是构成构成的范围和范围。