**SISSA, Via Bonomea 265, 34136 Trieste Italy**

**10 May 2021 – 16 Jul 2020**

**Lecturer:** Don Zagier

**Course Description:** 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 invariantsThese 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.

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