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

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.

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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

situation:

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

https://nextcloud.mpim-bonn.mpg.de/s/4XG3xSJG7AwBmde

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.