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Title:Equivariant dynamics and categories of equivariant vector fields
Author(s):Klajbor-Goderich, Stefanie
Director of Research:Lerman, Eugene
Doctoral Committee Chair(s):Loja Fernandes, Rui A
Doctoral Committee Member(s):Kerman, Ely; Zharnitsky, Vadim
Department / Program:Mathematics
Discipline:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Degree:Ph.D.
Genre:Dissertation
Subject(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.
Issue Date:2020-04-22
Type:Thesis
URI:http://hdl.handle.net/2142/107880
Rights Information:Copyright 2020 Stefanie Klajbor-Goderich
Date Available in IDEALS:2020-08-26
Date Deposited:2020-05


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