Starts 18 Jun 2021 14:00
Ends 16 Jul 2021 16:00
Central European Time
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IGAP/MPIM lecture course “From 3-manifold invariants to number theory”
Online/Presence
This will be a hybrid lecture course
Research Group: Geometry and Mathematical Physics
Course Type: PhD Course
Questions from topology have led to interesting number theory for many years, a famous example being the occurrence of Bernoulli numbers in connection with stable homotopy groups and exotic spheres, but some developments from the last few years have led to much deeper relationships and to highly non-trivial ideas in number theory. The course will attempt to describe some of these new interrelationships, which arise from the study of quantum invariants of knot complements and other 3-dimensional manifolds. [Joint work with Stavros Garoufalidis]
Topics to be studied include:
* The dilogarithm function, the 5-term relation, and triangulations of 3-manifolds
* Quantum invariants of 3-folds (Witten-Reshetikhin-Turaev and Kashaev invariant) definitions and first properties
* The Habiro ring (this is a really beautiful algebraic object that should be much better known and in which both of the above-named quantum invariants live)
* Perturbative series (formal power series in h) associated to knots
* Turning divergent power series into actual functions (this has connections with resurgence theory and involves some quite fun analytic considerations)
* Numerical methods (the ones needed are surprisingly subtle)
* Holomorphic functions in the upper half-plane (q-series) associated to knots
* Modular properties of both the Habiro-like and of the holomorphic invariants
These topics are all interconnected in a very beautiful way, formally summarized at the end by a single matrix invariant having different realizations in the Habiro world, the formal power series world, and the q-series world.
Although some quite advanced topics will be reached or touched upon, the course assumes no prerequisites beyond standard basic definitions from either topology, number theory, or analysis.
To access the course use the following Zoom coordinates:
Zoom meeting ID 969 5251 6566
Password 307018
Course Type: PhD Course
Questions from topology have led to interesting number theory for many years, a famous example being the occurrence of Bernoulli numbers in connection with stable homotopy groups and exotic spheres, but some developments from the last few years have led to much deeper relationships and to highly non-trivial ideas in number theory. The course will attempt to describe some of these new interrelationships, which arise from the study of quantum invariants of knot complements and other 3-dimensional manifolds. [Joint work with Stavros Garoufalidis]
Topics to be studied include:
* The dilogarithm function, the 5-term relation, and triangulations of 3-manifolds
* Quantum invariants of 3-folds (Witten-Reshetikhin-Turaev and Kashaev invariant) definitions and first properties
* The Habiro ring (this is a really beautiful algebraic object that should be much better known and in which both of the above-named quantum invariants live)
* Perturbative series (formal power series in h) associated to knots
* Turning divergent power series into actual functions (this has connections with resurgence theory and involves some quite fun analytic considerations)
* Numerical methods (the ones needed are surprisingly subtle)
* Holomorphic functions in the upper half-plane (q-series) associated to knots
* Modular properties of both the Habiro-like and of the holomorphic invariants
These topics are all interconnected in a very beautiful way, formally summarized at the end by a single matrix invariant having different realizations in the Habiro world, the formal power series world, and the q-series world.
Although some quite advanced topics will be reached or touched upon, the course assumes no prerequisites beyond standard basic definitions from either topology, number theory, or analysis.
To access the course use the following Zoom coordinates:
Zoom meeting ID 969 5251 6566
Password 307018
Next Lectures:
Monday, July 12, 2021 - 16:00 - 18:00 (Budinich Lecture Hall)
Friday, July 16, 2021 - 14:00 - 16:00 Budinich Lecture Hall)
Monday, July 12, 2021 - 16:00 - 18:00 (Budinich Lecture Hall)
Friday, July 16, 2021 - 14:00 - 16:00 Budinich Lecture Hall)