Monodromy in Topology and Geometric Group Theory

拓扑学和几何群论中的单向性

• 批准号: 2153879
• 负责人: Nicholas Salter
• 金额: $ 16.18万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-10-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2153879&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.

拓扑学和代数几何学是当代数学中两个具有中心重要性的学科。拓扑学是空间结构的研究（例如甜甜圈的表面，宇宙的大尺度形状，DNA的缠结链，大量数据集中的点簇），而代数几何学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联系点;首席研究员将特别研究其中之一：Riemann曲面族。曲面是一个拓扑对象，就像咖啡杯的二维曲面，可能有很多额外的控制柄。黎曼曲面提供了一种特殊的方式来描述多项式方程的解，就像我们在高中代数中学习的那样，有些方程描述圆，有些方程描述椭圆，还有一些方程描述更复杂的形状。黎曼曲面族是通过改变用于定义黎曼曲面的方程而产生的-人们可以想象“转动旋钮”来拉伸和扭曲曲面的形状。黎曼曲面族出现在数学的许多领域，在理论物理中也扮演着重要的角色。首席研究员将应用拓扑学和几何群论的相关学科的工具，以更好地理解一些最重要的黎曼曲面，今天的数学家感兴趣的家庭。更广泛的影响包括建立了国家指导阅读项目网络的一个新的分会。第一部分将研究阿贝尔微分层（平移曲面）的拓扑结构。这些族已经被动力学家和几何学家深入研究，但仍然存在一些基本的拓扑问题。特别是，地层的（轨道）基本群仍然是高度神秘的。这可以通过单值表示（monodromy representation），即映射到映射类组中的映射来有效地探测。首席研究员将继续他的工作描述这些monodromy表示，并将开发新的工具来理解monodromy内核，从而进一步发展理论的基本群体的地层。这一奋进的核心将是阐明阿廷集团和基本集团的阶层之间的精确关系。该项目的第二个组成部分涉及通过分支盖建造的家庭。这些家庭已担任一个重要来源的例子在拓扑学，代数几何，群论。首席研究员将调查这些家庭的monodromy与进一步综合的拓扑方面（辫子群，映射类组）与代数（算术群和推广，Burau样表示，原始同源）的目标。这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。