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From 3-manifold invariants to number theory

Course Type: 
PhD Course
Academic Year: 
Spring 2021
40 h

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. 

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.

ICTP: 18 June - Euler Lecture Hall; 21 June - L. Stasi Seminar Room; 25, 28 June, 2 July - Euler Lecture Hall; 5, 9, 12, 16 July - Budinich Lecture Hall
Next Lectures: 

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