*What is infinity factorial (and why might we
care)?*

Euler, in the 18th century, computed the values for the sums of
(divergent) series such as 1+2+3+ … + n + (n+1) + … (which
is equal to -1/12). His computations can be interpreted using the
Riemann zeta function, (actually first considered by Euler himself).
I shall discuss these and other divergent sums and products, including
how we can give a value to ∞!=1.2.3. … n.(n+1). …
. I shall give some indication of how these kinds of computations
are related to results in number theory and geometry. This lecture
should be accessible to students who have a good knowledge of calculus,
and have some knowledge of complex numbers.

The Clay Mathematics Institute Senior Scholars program aim is to
foster mathematical research and the exchange of ideas by providing
support for senior mathematicians who will play a central role in
a topical program at an institute or university. Senior Scholars
will be in residence for a substantial fraction of the program and
are expected to interact extensively with the other participants.