**abstract:**
Connections between topology and number theory have been
known and studied for many years, a classical example being
the relation between Bernoulli numbers and exotic spheres
discovered in the 1960s. But in the last few years it has been
discovered that also much more sophisticated number-theoretical
objects like algebraic K-groups and new types of modular forms
arise naturally from 3-dimensional topology and more explicitly
from the study of so-called quantum invariants of knots and
3-manifolds. I will try to give as gentle an introduction as
possible to these new ideas, culminating in the discovery
of a new type of modular forms. No knowledge of any of
the ingredients of the title will be assumed, and in particular
I will give a feeling for both classical modular forms and their
young quantum cousins. Most of the material presented is
joint work with Stavros Garoufalidis.

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