Geometric
objects can often be described as the solution sets of algebraic
equations. Simple examples in three-dimensional space are curves
like
In this course, we will investigate questions such as: How can one
compute the equations for the intersection or union of two such
objects? How can one determine whether two systems of algebraic
equations
describe the same geometric object?
These are basic questions at the foundations of
algebraic geometry.
This course is intended as an introduction to this subject, which
occupies a central place in modern mathematics. We will learn
techniques for translating (certain) geometric problems into algebraic
ones. Once they are reformulated in algebraic language, one may unleash
the power of (commutative) algebra on them. Sometimes they even become
(at least in principle) amenable to treatment by a computer.
However, only fairly recently (since the 1970s) have algorithms (and
the computers powerful enough to run them!) become available to
actually carry out the necessary computations. The engine behind these
is
Buchberger's algorithm,
which is based on the notion of
Gröbner
basis. (If you are curious about Gröbner bases already,
watch the
movie!)
The advent of these programs has enabled mathematicians to study
complicated examples which previously couldn't be investigated by hand,
in this way inspiring a wealth of new mathematics. It has also made the
subject interesting for computer scientists and
engineers, since many practical questions (e.g., in robotics) can be
stated as problems in algebraic geometry.
Prerequisities
A good foundation in linear algebra (at the level of
Math
115A) and the ability to formulate mathematical proofs. Some
knowledge of
abstract
algebra
would be useful, but is
not
strictly necessary. You should also be able to
use (though
not necessarily to program) a
computer. Please feel free to
contact me if you'd like to take this course, but are unsure whether
you have the right preparation.
Click
below for biographical information about some of the mathematicians
whose work we will encounter in this course: