Geometry of Groups and Complexes

PI对群和复合体的大规模几何感兴趣。一个项目涉及分析群的填充不变量的渐近行为。在与Martin Bridson， Max Forester和Krishnan Shankar的合作下，PI正在研究群的一阶和二阶Dehn函数。在与Max Forester和Krishnan Shankar的合作中，PI正在探索CAT(0)群和双曲群的子群的一阶或二阶Dehn函数的类型。PI和John Crisp， Anton Kaul和Jon McCammond的另一个项目涉及到Artin群的广义Garside结构和相关复合物。另一个项目由PI与John Crisp和Jon McCammond的单独合作组成，重点关注群论中的非正曲率和维度，以及大尺度概念和局部非正曲率概念之间的联系。其他一些项目包括研究广义baumslag -孤群的共轭和同构，银汞合金的末端，双曲群的子群的畸变，以及双曲群的表面子群的存在性。群被数学家用来研究对称性。群就是一个物体对称性的集合。例子包括墙纸图案的几何对称性，或晶体结构的几何对称性，或埃舍尔绘画的几何对称性，或多项式根的代数对称性。自19世纪以来，数学家们就把群体作为抽象的代数对象进行了深入研究。20世纪80年代M. Gromov提出把群看作几何对象，并开始推导群的几何性质和代数性质之间的深刻联系。从Gromov的工作中出现的一个主题是，具有深刻代数后果的几何性质不是局部性质，而是粗糙的或“大规模的”。例如，无限阶梯和无限直线不是局部相似的，而是大尺度相似的，它们的对称群会有许多代数上的相似。PI研究群体中的大尺度等周问题（面积与周长问题），以及粗糙负弯曲群体的几何。这些研究有助于加深我们对对称本质的理解。

The PI is interested in the large scale geometry of groups and complexes.One project involves analyzing the asymptotic behavior of filling invariants of groups. In collaboration with Martin Bridson, Max Forester and Krishnan Shankar, the PI is investigating the first and second order Dehn functions of groups. In joint work with Max Forester and Krishnan Shankar, the PI is exploring what types of functions can arise as first or second order Dehn functions of subgroups of CAT(0) groups, and of hyperbolic groups. Another project of the PI and John Crisp, Anton Kaul and Jon McCammond involves generalized Garside structures and associated complexes for Artin groups. Another project, comprised of separate collaborations of the PI with John Crisp and with Jon McCammond, focuses on non-positive curvature and dimension in group theory, and on the connections between large scale notions and local notions of non-positive curvature. A number of other projects include investigations into conjugacy and isomorphism of generalized Baumslag-Solitargroups, ends of amalgams, distortion of subgroups of hyperbolicgroups, and the existence of surface subgroups of hyperbolic groups.Groups are used by mathematicians to study symmetry. A group is just a collection of symmetries of an object. Examples include the geometric symmetries of a wallpaper pattern, or of a crystal structure, or of an Escher painting, or the algebraic symmetries associated to roots of polynomials. Mathematicians have studied groups intensively as abstract algebraic objects since the 19th century. In the 1980's M. Gromov proposed that we consider groups as geometric objects, and began to derive deep connections between the geometric and the algebraic properties of groups. One theme which emerged from Gromov's work is that the geometric properties which have deep algebraic consequences are not local properties, but rather coarse or "large scale". For example, and infinite ladder and an infinite straight line are not locally alike, but are large scale alike, and their symmetry groups will have many algebraic similarities. The PI investigates large scale versions of isoperimetric problems (area versus perimeter lengthproblems) in groups, and also the geometry of coarsely negativelycurved groups. These investigations help deepen our understanding ofthe nature of symmetry.