Outer automorphisms of free groups, outer space, and related deformation spaces
自由群、外层空间和相关变形空间的外自同构
基本信息
- 批准号:RGPIN-2019-04318
- 负责人:
- 金额:$ 1.17万
- 依托单位:
- 依托单位国家:加拿大
- 项目类别:Discovery Grants Program - Individual
- 财政年份:2020
- 资助国家:加拿大
- 起止时间:2020-01-01 至 2021-12-31
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
One of the most beautiful interplays in mathematics is between a deformation space (encoding all metrics on an object) and its symmetry group. This proposal describes a study of two such deformation spaces: Culler-Vogtmann outer space and Teichmuller space.
Outer space is a simplicial complex (minus some faces) encoding all weighted graphs of a fixed fundamental group. I study outer space with a focus on two interconnected themes: (1) how one efficiently deforms such weighted graphs, via studying geodesics in outer space, and (2) what happens when one repeatedly applies an automorphism to a free group element, i.e. the asymptotic invariants.
Eigenvectors and eigenvalues are classical examples of asymptotic invariants in a matrix group setting. And, just as in that setting, these invariants play a crucial role in understanding both what happens as one repeatedly applies the automorphism and the efficient deformation of metrics, again encoded in the geodesics of outer space.
I describe a plan for overcoming substantial obstacles to build and understand a useful dynamical system of certain geodesics in outer space. In conjunction, I give answers to the question asking which asymptotic invariants are most common for free group automorphisms, both in random walk and entropy senses.
The novelty of our techniques, which we greatly expand in the program, are in their discretely codifying geodesics, in a manner utilizing and illuminating their relationships to the invariants.
While more broad applications of understanding outer space are just now beginning to be explored, they are clearly extensive. Weighted graphs arise in such diverse settings as biology, artificial intelligence, and more general computer science. Outer space even relates to algebraic geometry, specifically tropical geometry.
Instead of encoding weighted graphs, Teichmuller space encodes hyperbolic metrics on a fixed finite surface. Our work on Teichmuller space is in giving an alternate proof of its Thurston compactification. The benefit of our approach is two-fold. First, we construct foliations closely approximating hyperbolic metrics. Second, our methods may be used in other settings, such as in compactifiying the space of convex projective structures on a surface.
I plan to train 11 HQPs, with focus on teaching (highly transferrable) skills in an extensively active research area and overall preparing HQPs for successful mathematics careers by helping them learn to communicate and contextualize their mathematics, while meeting new collaborators. The graduate students are given outer space projects, as the skills they learn will not only allow further study of outer space, but will prepare them to participate in recent trends of mimicking methods used to study outer space and the automorphism group of the free group in studying other groups, and in studying graphs modeling natural systems, such as phylogenetic trees, or technological systems, such as neural networks.
数学中最美丽的相互作用之一是变形空间(对对象上的所有度量进行编码)与其对称群之间的相互作用。本文描述了两个这样的变形空间:Culler-Vogtmann外空间和Teichmuller空间。
项目成果
期刊论文数量(0)
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科研奖励数量(0)
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Pfaff, Catherine其他文献
Counting conjugacy classes of fully irreducibles: double exponential growth
计算完全不可约的共轭类:双指数增长
- DOI:
10.1007/s10711-024-00885-4 - 发表时间:
2024 - 期刊:
- 影响因子:0.5
- 作者:
Kapovich, Ilya;Pfaff, Catherine - 通讯作者:
Pfaff, Catherine
Random outer automorphisms of free groups: Attracting trees and their singularity structures
自由群的随机外自同构:吸引树及其奇点结构
- DOI:
10.1090/tran/8472 - 发表时间:
2022 - 期刊:
- 影响因子:1.3
- 作者:
Kapovich, Ilya;Maher, Joseph;Pfaff, Catherine;Taylor, Samuel J. - 通讯作者:
Taylor, Samuel J.
Random trees in the boundary of outer space
外太空边界的随机树
- DOI:
10.2140/gt.2022.26.127 - 发表时间:
2022 - 期刊:
- 影响因子:2
- 作者:
Kapovich, Ilya;Maher, Joseph;Pfaff, Catherine;Taylor, Samuel J - 通讯作者:
Taylor, Samuel J
A train track directed random walk on Out (F r )
Out (Fr ) 上的火车轨道定向随机游走
- DOI:
10.1142/s0218196715500186 - 发表时间:
2015 - 期刊:
- 影响因子:0.8
- 作者:
Kapovich, Ilya;Pfaff, Catherine - 通讯作者:
Pfaff, Catherine
Stable Strata of Geodesics in Outer Space
外层空间测地线稳定地层
- DOI:
10.1093/imrn/rnx269 - 发表时间:
2018 - 期刊:
- 影响因子:1
- 作者:
Algom-Kfir, Yael;Kapovich, Ilya;Pfaff, Catherine - 通讯作者:
Pfaff, Catherine
Pfaff, Catherine的其他文献
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{{ truncateString('Pfaff, Catherine', 18)}}的其他基金
Outer automorphisms of free groups, outer space, and related deformation spaces
自由群、外层空间和相关变形空间的外自同构
- 批准号:
RGPIN-2019-04318 - 财政年份:2022
- 资助金额:
$ 1.17万 - 项目类别:
Discovery Grants Program - Individual
Outer automorphisms of free groups, outer space, and related deformation spaces
自由群、外层空间和相关变形空间的外自同构
- 批准号:
RGPIN-2019-04318 - 财政年份:2021
- 资助金额:
$ 1.17万 - 项目类别:
Discovery Grants Program - Individual
Outer automorphisms of free groups, outer space, and related deformation spaces
自由群、外层空间和相关变形空间的外自同构
- 批准号:
RGPIN-2019-04318 - 财政年份:2019
- 资助金额:
$ 1.17万 - 项目类别:
Discovery Grants Program - Individual
Outer automorphisms of free groups, outer space, and related deformation spaces
自由群、外层空间和相关变形空间的外自同构
- 批准号:
DGECR-2019-00346 - 财政年份:2019
- 资助金额:
$ 1.17万 - 项目类别:
Discovery Launch Supplement
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Outer automorphisms of free groups, outer space, and related deformation spaces
自由群、外层空间和相关变形空间的外自同构
- 批准号:
RGPIN-2019-04318 - 财政年份:2022
- 资助金额:
$ 1.17万 - 项目类别:
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Outer automorphisms of free groups, outer space, and related deformation spaces
自由群、外层空间和相关变形空间的外自同构
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- 资助金额:
$ 1.17万 - 项目类别:
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Outer automorphisms of free groups, outer space, and related deformation spaces
自由群、外层空间和相关变形空间的外自同构
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Geometry of Free Group Automorphisms: Beyond the Frontier
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