SUMS: Menzies -- Complex elliptic curves

Speaker: Max Menzies (Cambridge)

COMPLEX ELLIPTIC CURVES

You might have heard of elliptic curves before, from other
mathematicians or even the media. There are two reasons for their
fame: first, they were important in the proof of Fermat’s Last
Theorem, the most famous mathematical problem in the world; secondly,
they are awesome. Perhaps the prettiest family of objects in all of
maths.

I will not prove Fermat’s Last Theorem, because it is... hard. Yeah,
it’s quite difficult. But I will introduce elliptic curves and discuss
their nicest properties. In general, it’s quite hard to define and
understand elliptic curves, because they are objects of projective
geometry, rather than plane geometry. However over the complex
numbers, they have a surprising and beautiful geometric equivalence
with something you all know and love (to eat). This is the simplest
example of an deep relationship between algebraic geometry (objects
defined by polynomial equations) and analytic geometry (complex
manifolds).