Equivariant dynamics and categories of equivariant vector fields

Klajbor-Goderich, Stefanie

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https://hdl.handle.net/2142/107880 Copy

Description

Title

Equivariant dynamics and categories of equivariant vector fields

Author(s)

Klajbor-Goderich, Stefanie

Issue Date

2020-04-22

Director of Research (if dissertation) or Advisor (if thesis)

Lerman, Eugene

Doctoral Committee Chair(s)

Loja Fernandes, Rui A

Committee Member(s)

• Kerman, Ely
• Zharnitsky, Vadim

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• equivariant dynamics
• relative equilibria
• bifurcation

Abstract

We present a framework for studying the dynamics of equivariant vector fields near relative equilibria. To overcome the lack of linearization at a relative equilibrium or the possible non-smoothness of the orbit space, we categorify the space of equivariant vector fields. A category where the objects are equivariant vector fields was first introduced by Hepworth in the context of smooth stacks. Central to our approach is the ensuing notion of isomorphic equivariant vector fields. The idea is that considering equivariant vector fields, and their corresponding dynamics, up to isomorphism is a way to take into account the symmetries of the group action without passing to the orbit space. In particular, the category of equivariant vector fields near a relative equilibrium is equivalent to the category of equivariant vector fields on the slice representation of the relative equilibrium. We apply this to the stability and motion of relative equilibria, bifurcations to and from relative equilibria, and to the genericity of conditions for equivariant bifurcation from relative equilibria.

Graduation Semester

2020-05

Type of Resource

Thesis

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http://hdl.handle.net/2142/107880

Copyright and License Information

Copyright 2020 Stefanie Klajbor-Goderich

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