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Lecture course by Don Zagier

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Starting May 10, Don Zagier will give a joint IGAP*/MPIM lecture course entitled “From 3-manifold invariants to number theory”, intended to be accessible to mathematicians in all fields and at all levels. The course will take place on

Mondays 4pm–6pm and Fridays 2pm–4pm,

starting on Monday, May 10, and ending on July 16, 2021. All lectures will be streamed online on Zoom, from MPIM in May and from Trieste in June and July. The course is available to everybody (including mathematicians not at the MPI, SISSA, or ICTP), but one must register in order to participate. The meeting details are given below.

From 3-manifold invariants to number theory

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 a development from the last few years has 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. It is based on 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.

Meeting details:
Meeting ID: 969 5251 6566
Passcode: 307018

There were serious problems with the video and audio quality of the last 
few lectures of the course, as was reported by various of the listeners.
We apologize for this and have taken the following steps to remedy the 
1. Starting Friday, June 18, all of the remaining lectures of the course
will be streamed from the ICTP rather than SISSA, since they have rooms
with larger blackboards and that are completely covered by the cameras.
2. Recordings of all of the lectures up to now (and also of all of the 
subsequent ones) are now publically available on the link
3. A copy of the handwritten notes of one of the participants (Muhammad
Sohaib Khalid) of the course are being made publically available, with 
his kind permission but of course with no guarantee of completeness or 
correctness since they were made for private use and were not originally 
intended for distribution.


An announcement of the course can also be found on the following websites. or

* The Institute for Geometry and Physics (IGAP) is a new joint venture between SISSA and ICTP in Trieste devoted to the exchange of ideas, techniques and experiences, and the training of young researchers interested in this fascinating research area. Don Zagier is affiliated with both SISSA and ICTP and this course is intended to be the first of a series of annual IGAP courses.


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