Geometric objects can often be described as the solution sets of algebraic equations. Simple examples in three-dimensional space are curves like

and surfaces 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 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 amenable to treatment by a computer.

However, only fairly recently (since the 1970s) have algorithms and computers 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, cryptography, computational biology, geometric modeling, statistics) can be stated as problems in algebraic geometry. We will emphasize these applications of algebraic geometry throughout the course. We will also introduce some Methods of Research. Specifically, each student will develop an independent project in pure algebraic geometry or applied algebraic geometry as described below.

Prerequisites

A good foundation in linear algebra (at the level of MTH 377) and the ability to formulate mathematical proofs. Some knowledge of abstract algebra MTH 477 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.

There will be a problem set assigned on a semi-regular basis, handed out in class, and also posted on this website.
The problems will range in difficulty from routine to more challenging. Completed solutions are to be handed in at the beginning of class on the due date specified on the respective homework set. No late homework will be accepted. However, your lowest homework score will be dropped when computing your grade. You are encouraged to work together on the exercises, but any graded assignment should represent your own work.

Some of the homework problems (and the midterm exam) will involve the use of computer algebra systems. No previous experience with computer programming is assumed, but I expect that you are able and willing to familiarize yourself with the use of the program of your choice. For overall user-friendliness, I recommend the general-purpose programs Maple and Mathematica (which can do algebra, calculus, graphics, and so on). If you prefer, you may also use Singular, Macaulay 2, CoCoA or Sage. These free software systems are explicitly designed to support computations in algebraic geometry and commutative algebra. All these systems are available for most platforms (Unix, Linux, Mac OS X, Window$, etc.).

Exams and Paper

There will be a take-home Midterm examination.
There will be no Final examination. However, students are required to work on an independent project throughout the semester. The project will involve studying a class-related topic, and writing a short summary paper on this subject, which will go through several stages of revision. Your paper should be self-contained and accessible to the other participants in the class. Achieving this should take approximately 10 pages. At the end of the course, you will read a referee report written by another student in the class, and you will also write such a report about the paper of another student. Here is a list of possible topics (in no particular order):

1. Theoretical:
   
   1. Gröbner bases over principal ideal domains.
   2. Gröbner bases for modules.
   3. Gröbner bases in power series rings.
   4. Gröbner bases of ideals with finitely many zeroes.
   5. Modules, free resolutions, and the Hilbert Syzygy Theorem.
   6. Gröbner fan of an ideal and the state polytope.
   7. Generic initial ideals.
2. Computational:
   1. Universal Gröbner bases.
   2. Complexity of computing Gröbner bases.
   3. Buchberger's criterion and improvments to Buchberger's algorithm
   4. The FGLM Gröbner basis conversion algorithm
   5. Matrix term orderings
3. Applications:
   1. Gröbner bases for toric ideals and integer linear programming
   2. Automatic Theorem Proving.
   3. Primitive partition identities
   4. Gröbner bases and numerical analysis
   5. Gröbner bases and computational chemistry: Conformation of cyclic molecules
   6. Gröbner bases and computational biology: Reverse-engineering of biochemical networks
   7. Gröbner bases and computational biology: Phylogenetic inference
   8. Gröbner bases and computational statistics: Contingency tables

You may also suggest your own project topic.

You are strongly encouraged to type your assignments. In the Computer Lab you may access some implementations of TeX (also written as ), a mathematical text processing system written by Donald Knuth. The use of TeX is simplified by LaTeX (), written by Leslie Lamport. If you wish to learn LaTeX, there are many online guides available, for example here and here. A good reference book is Math into LaTeX by George Grätzer.

If you know how to download and install new software on your computer, you might also consider using the what-you-see-is-what-you-get text editor TeXmacs written by Joris van der Hoeven. It makes it unnecessary for you to learn the LaTeX typesetting language while producing output of comparable quality. The program is freely downloadable, available for various platforms, able to import and export LaTeX files, and offers a plugin for Macaulay 2.

More detailed instructions about the project, including references for the projects listed above, will be announced in the first week of the semester.

Grading policy: Homework: 30%. Midterm Exam: 30%. Paper: 40%.

Historical Information

Click below for biographical information about some of the mathematicians whose work we will encounter in this course:

Bruno Buchberger
L. E. Dickson
Wolfgang Gröbner
Grete Hermann
David Hilbert
Heisuke Hironaka
Emanuel Lasker
Francis Sowerby Macaulay
Emmy Noether
Oscar Zariski

Interesting Links

Click below for some web pages containing beautiful pictures of work related to this course or the specific projects:

Geometrical Animations
Algebraic Surfaces Gallery
Algebraic Surfaces
Geometry of Algebraic Curves

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Back to my home page.      Last modified February 13, 2008.