Thirteen??

David Mumford

A large part of what a mathematician does may be described as exploring
"things" which they discover through reasoning, which become as real to them as
the house they live in but which have no material existence at all. Some easy
examples are the higher-dimensional analogs of the Platonic solids. One of the
spires on Gaudi's *Sagrada Familia* is expected to have an icosahedron on it --
the twenty-sided regular polyhedron. This is a real tangible thing. But
mathematicians all feel space can have more than three dimensions and in the
19^{th} century Schläfli discovered some marvelous four-dimensional versions of
Plato's list that we can only imagine. My own passion for a long time focused on
the things written as *M _{g}* in my formula, known as the
"moduli space of curves of genus g". Even by a mathematician's standards for
reality, when I was a student these spaces seemed shrouded in fog, in a limbo
between full-fledged mathematical existence and fantasy. I wanted to change that.

At that time, Alexander Grothendieck burst on the scene. A truly unique
person, he had the skill of passing to higher levels of abstraction than anyone
before and using these abstractions to shine a brilliant light on relatively
concrete things we didn't yet understand. It turned out that one of his deep
results could be applied to my still dimly perceived space. I didn't know then
what to make of this. The power of the result only became apparent some twenty
years later working with Joe Harris, when we knew enough to be able to treat
*M _{g}* as a real object (technically, a quasi-projective
algebraic variety).

So what is the formula about and why is it beautiful? It states that two
things ("line bundles") are essentially the same ("isomorphic"). The one on
the left is the key object that determines the geometry of any space like
*M _{g}*. Going back to Gauss, we have known that in crudest
terms there are three kinds of spaces: flat ones like a plane, positively
curved ones like the surface of the earth and negatively curved ones like the
surface of a saddle (where the sum of the angles of a triangle is less than
180°). The object on the left controls where our space sits in this
tripartite categorization. The object on the right is what Raoul Bott called
a "tautological" structure: a basic decoration produced by the very definition
of the space

*M*. Their isomorphism turned out to be the key that showed these spaces belong to the negative category if

_{g}*g*is large enough.

The most startling thing about the formula is the number thirteen. You can look in many math journals and find that the only numbers greater than two are the page numbers. This thirteen has a pedigree in a long tradition of counting things, e.g. that a cubic surface has exactly 27 straight lines on it (including complex ones!) but, frankly, it still feels to me like a strange joke the creator is playing on us.