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Resurgence and asymptotic expansions of quantum invariants – understanding the connection between Chern-Simons theory and topological quantum field theory

PhD defence, Friday 20 September 2019. William Elbæk Mistegård.

2019.09.20 | Maia Høyer Monod

William Elbæk Mistegård

It is a fundamental problem in mathematical physics to obtain a rigorous mathematical understanding of the Feynman path integrals used with formidable succes in quantum field theory. In Chern-Simons theory, which is a mathematical example of a three-dimensional field theory, it is widely believed that the path integrals are modelled by the quantum invariants of three-manifolds. These invariants are parts of a rich mathematical structure known as a topological quantum field theory (TQFT). During his PhD studies, William Elbæk Mistegård studied the connection between TQFT and Chern-Simons theory and used a new promising theory from mathematical physics known as resurgence.

During his PhD studies William Elbæk Mistegård collaborated with his supervisor to prove the following four results. 1) In the case where the manifold is a mapping torus the quantum invariants admits an asymptotic expansion in terms of Chern-Simons invariants. 2) In the case the manifold is Seifert fibered the Borel transform of the quantum invariant determines the Chern-Simons invariants, and the Borel transform determines the quantum invariant. 3) All quantum invariants can be constructed from an analytic function that conjecturally determines the Chern-Simons invariants. 4) The Reshetikhin-Turaev TQFT admits a duality and unitarity structure satisfying strict compatibility.

The PhD degree was completed at the Centre for Quantum Geometry of Moduli Spaces, Science and Technology, Aarhus University.

This résumé was prepared by the PhD student.

Time: Friday 20 September 2019 at 11.00
Place: Building 1534, room 125, Auditorium F, Department of Mathematics, Aarhus University, Ny Munkegade 118, 8000 Aarhus C
Title of PhD thesis: Quantum Invariants and Chern-Simons theory
Contact information: William Elbæk Mistegård, e-mail: william@qgm.au.dk, Phone: +45 87155817
Members of the assessment committee:
Professor Martin Schlichenmaier, Mathematics Research Unit, University of Luxembourg, Belgium
External Scientific Member Stavros Garoufalidis, Max Planck Institute for Mathematics, Germany
Professor Marcel Böckstedt, (chair), Department of Mathematics, Aarhus University
Main supervisor:
Professor Jørgen Ellegaard Andersen, Department of Mathematics, Aarhus University
Language: The PhD dissertation will be defended in English

The defence is public.
The PhD thesis is available for reading at the Graduate School of Science and Technology/GSST, Ny Munkegade 120, building 1521, 8000 Aarhus C.    

PhD defence
12183 / i43