|
Abstract:
I will show how ideals in algebra can represent geometric objects, and
how manipulations of the geometric objects correspondingly manipulate
the ideals. Many of these manipulations of ideals can be done on the
computer (even when the object lives in a high-dimensional space and
so the geometric picture cannot be given). I will demonstrate some
such calculations on the computer. Some calculations are fast, and
some are slow. Even before there were computers, mathematicians were
studying the number of steps needed to compute certain manipulations.
The most relevant of these is the work of Grete Hermann, a student of
Emmy Noether, who gave upper bounds on many computations. The upper
bounds she gave were horrendous, and for more than 60 years people
believed that Hermann was much too generous with her upper bounds.
But then in 1964, Mayr and Meyer produced a class of ideals which
prove that Hermann was right. I will talk about the Mayr-Meyer ideals
and some other computationally beastly ideals.
|