Geometry of Groups and Complexes

PI对群和复形的大规模几何学感兴趣。其中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的合作中，PI正在研究群的一阶和二阶Dehn函数。在与Max Forester和Krishnan Shankar的联合工作中，PI正在探索什么类型的函数可以作为CAT（0）群和双曲群的子群的一阶或二阶Dehn函数出现。PI和John Crisp，Anton Kaul和Jon McCammond的另一个项目涉及Artin群的广义Garside结构和相关复合物。 另一个项目，包括独立的合作PI与约翰克里斯普和乔恩McCammond，侧重于非积极的曲率和尺寸在群论，并在大规模的概念和非积极的曲率局部概念之间的连接。 一些其他的项目包括研究广义Baumslag-Solitargroups的共轭性和同构性，汞齐的末端，双曲群子群的变形，以及双曲群曲面子群的存在性。一个群只是一个物体的对称性的集合。例子包括墙纸图案的几何对称性，或晶体结构，或绘画，或与多项式根相关的代数对称性。 自19世纪以来，数学家们就把群作为抽象的代数对象进行了深入的研究。在20世纪80年代，M。格罗莫夫提出我们将群视为几何对象，并开始推导群的几何性质和代数性质之间的深层联系。 一个主题出现了从格罗莫夫的工作是，几何性质有深刻的代数后果不是当地的财产，而是粗糙或“大规模”。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，它们的对称群将有许多代数相似性。 PI研究大规模版本的等周问题（面积与周长问题）在组中，也是几何的粗糙negativelycurved组。这些研究有助于加深我们对对称性本质的理解。

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.