Differential invariants of Lie pseudogroups

Schneider, Eivind

dc.contributor.advisor	Kruglikov, Boris
dc.contributor.author	Schneider, Eivind
dc.date.accessioned	2019-06-25T08:11:29Z
dc.date.available	2019-06-25T08:11:29Z
dc.date.issued	2019-05-10
dc.description.abstract	We compute differential invariants for several Lie pseudogroups, and use them for solving the equivalence and classification problem for a variety of mathematical structures appearing in geometry and mathematical physics. We demonstrate utility of the algebra of rational scalar differential invariants for solving these two problems, also in cases where the structures are given as solutions to a nonlinear partial differential equation and the Lie pseudogroup is infinite-dimensional. In addition to contributing to the study of important mathematical structures, our work brings new insight into the general theory of differential invariants.	en_US
dc.description.doctoraltype	ph.d.	en_US
dc.description.popularabstract	Many interesting mathematical structures and physical phenomena are described as solutions to differential equations, and solving such equations is a task of great importance both in mathematics and physics. However, it may happen that different solutions describe the same mathematical structure, or the same physical phenomenon. Mathematically, this fact manifests itself as a group of transformations acting on the space of solutions, and we may say that two solutions are equivalent if there exists a transformation taking one solution to the other. By relying on the most recent developments in the theory of differential invariants, we study several differential equations with two main goals: determining when two solutions are equivalent and describing the set of equivalence classes. Relying on the mathematical theory and computer algebra systems, we demonstrate utility of the algebra of rational scalar differential invariants for solving these two problems. Our work contributes to the study of several important equations of mathematical physics, and it brings new insight into the general theory of differential invariants.	en_US
dc.identifier.isbn	978-82-8236-346-4 (trykt) 978-82-8236-347-1 (pdf)
dc.identifier.uri	https://hdl.handle.net/10037/15600
dc.language.iso	eng	en_US
dc.publisher	UiT Norges arktiske universitet	en_US
dc.publisher	UiT The Arctic University of Norway	en_US
dc.relation.haspart	<p>Paper I: Schneider, E. (2018). Projectable Lie algebras of vector fields in 3D. <i>Journal of Geometry and Physics, 132</i>, 222-229. Also available in Munin at <a href=https://hdl.handle.net/10037/14569> https://hdl.handle.net/10037/14569</a>. <p>Paper II: Schneider, E. Differential invariants of surfaces. (Manuscript). Available in the file “thesis_entire.pdf”. <p>Paper III: Kruglikov, B. & Schneider, E. (2017). Differential invariants of self-dual conformal structures. <i>Journal of Geometry and Physics, 113</i>, 176-187. Also available at <a href=https://doi.org/10.1016/j.geomphys.2016.05.017>https://doi.org/10.1016/j.geomphys.2016.05.017</a>. Accepted manuscript version available in Munin at <a href=https://hdl.handle.net/10037/15607>https://hdl.handle.net/10037/15607</a>. <p>Paper IV: Kruglikov, B. & Schneider, E. (2018). Differential invariants of Einstein-Weyl structures in 3D. <i>Journal of Geometry and Physics, 131</i>, 160-169. Also available in Munin at <a href=https://hdl.handle.net/10037/15058> https://hdl.handle.net/10037/15058</a>. <p>Paper V: Kruglikov, B., McNutt, D. & Schneider, E. (2019). Differential invariants of Kundt waves. (Submitted manuscript). Accepted manuscript version available at <a href=https://iopscience.iop.org/article/10.1088/1361-6382/ab28c5>https://iopscience.iop.org/article/10.1088/1361-6382/ab28c5</a>. <p>Paper VI: Schneider, E. (2019). Differential invariants in thermodynamics. (Submitted manuscript). Published version in Kycia, R., Ułan, M. & Schneider, E. (Eds). <i>Nonlinear PDEs, Their Geometry, and Applications. Tutorials, Schools, and Workshops in the Mathematical Sciences.</i> Birkhäuser, Cham, available at <a href=https://doi.org/10.1007/978-3-030-17031-8_7>https://doi.org/10.1007/978-3-030-17031-8_7</a>.	en_US
dc.rights.accessRights	openAccess	en_US
dc.rights.holder	Copyright 2019 The Author(s)
dc.rights.uri	https://creativecommons.org/licenses/by-nc-sa/3.0	en_US
dc.rights	Attribution-NonCommercial-ShareAlike 3.0 Unported (CC BY-NC-SA 3.0)	en_US
dc.subject	VDP::Mathematics and natural science: 400::Mathematics: 410::Topology/geometry: 415	en_US
dc.title	Differential invariants of Lie pseudogroups	en_US
dc.type	Doctoral thesis	en_US
dc.type	Doktorgradsavhandling	en_US