Metrics, Measures, and Identities on Moduli Spaces

PI研究几何，物理，动力学，统计和数论领域之间的相互联系。空间上的几何是测量空间中距离的一种手段，可以被认为是给空间一个形状。研究空间几何的一种方法是考虑测地线流;这是一个将直线概念推广到弯曲空间的对象。通过研究这个物体的性质，可以发现许多关于几何本身的东西。例如，有多少流动路径关闭的问题与素数的分布有关。通常，不同的几何形状可以放置在一个空间上，从而获得形状的空间。一个自然的问题是，这个形状空间本身是否可以被赋予一个漂亮的形状。PI建议在几何空间上研究这些几何并研究它们的性质。PI将继续致力于本科和研究生教育。PI将 指导研究生和博士后助理教授进行与项目相关的研究。PI还将就与提案相关的材料进行研究讲座、临时讲座、小型课程和系列讲座，并组织会议。PI的研究计划围绕使用某些几何测量来定义模空间和表示变体上的结构。这类测度包括克莱因群极限集上的Hausdorff测度、测地线流、克莱因群的Patterson-Sullivan测度、使用与双曲群表示相关联的热力学定义的平衡测度，以及由某些几何定义的函数对体积测度的推进。其中一个研究领域是高等Teichmuller理论，这是研究的代表性空间的双曲群成半简单的李群。这些是经典Teichmuller空间的推广。使用热力学，PI和合作者在这个更高的Teichmuller空间上定义了一个压力几何。空间PI建议研究这个度量的几何性质，包括它的曲率、度量完备化和等距群。建议研究的另一个领域是几何恒等式;这些方程在几何模空间上成立。PI和合作者通过研究双曲流形上测地线流的统计特性来推导出这样的恒等式。PI建议研究这些身份及其与其他已知身份的关系。

The PI studies the interconnections between the fields of geometry, physics, dynamics, statistics and number theory. A geometry on a space is a means of measuring distance in the space and can be thought of as giving the space a shape. One approach to studying the geometry on a space is to consider the geodesic flow; this is an object that generalizes the notion of a straight line to spaces that are curved. By studying the properties of this object much can be discovered about the geometry itself. For example, the question of how many flow paths close up is related to the distribution of prime numbers. Often different geometries can be placed on a space and thus one obtains a space of shapes. A natural question to ask is if this space of shapes can be given a nice shape itself. The PI proposes to study these geometries on the space of geometries and investigate their properties. The PI will continue his commitment to both undergraduate and graduate education. The PI will mentor graduate students and postdoctoral assistant professors on research related to the project. The PI will also give research talks, expository talks, minicourses and lecture series on material related to the proposal as well as organize conferences. The research plan of the PI centers around the use of certain geometric measures to define structures on moduli spaces and representation varieties. Such measures include the Hausdorff measure on the limit set of a Kleinian group, geodesic currents, the Patterson-Sullivan measure of a Kleinian group, equilibrium measures defined using Themodynamics associated with representations of hyperbolic groups, and push-forwards of volume measures by certain geometrically defined functions. One area of study is Higher Teichmuller Theory which is the study of representation spaces of hyperbolic groups into semi-simple Lie groups. These are generalizations of the classical Teichmuller space. Using Thermodynamics, the PI and collaborators define a Pressure geometry on this Higher Teichmuller space. space. The PI proposes to study the geometric property of this metric including its curvature, metric completion, and isometry group. Another area of proposed study is geometric identities; these are equations that hold on a moduli space of geometries. The PI and collaborators derive such identities by studying the statistical properties of the geodesic flow on a hyperbolic manifold. The PI proposes to study these identities and their relation to other known identities.