MTH 470W/560: Algorithms for Elementary Algebraic Geometry

General Information

Time and Place: MWF 11:00 - 11:50, LDB 201

Instructor: Luis David Garcia-Puente

Course Homepage: http://www.shsu.edu/~ldg005/AEAG

Office:
Lee Drain Building 417B
Office Phone: (936) 294-1581
Office Hours: MW 12:00 - 2:00 and by appointment.

Description

Geometric objects can often be described as the solution sets of algebraic equations. Simple examples in three-dimensional space are curves like
 (the set of triples (x,y,z) of real numbers satisfying the equations y=x², z=x³)
and surfaces like
 (the set of triples (x,y,z) of real numbers solving the equation x²-y²z²+z³=0).
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.

Course Text

 In this course we will discuss systems of polynomial equations (ideals), their solution sets (varieties), and how these objects can be effectively manipulated (algorithms). We will try to cover at least the first four chapters of the book Ideals, Varieties, and Algorithms, An Introduction to Computational Algebraic Geometry and Commutative Algebra, Third Edition, by David Cox, John Little, and Donal O'Shea, Springer, New York, 2007. The authors of the textbook maintain a web page with errata and software.

Other textbooks of interest (that can be checked out at the SHSU Library):
• William W. Adams and Philippe Loustaunau, An Introduction to Gröbner Bases, Graduate Studies in Mathematics 3, American Mathematical Society, Providence, RI, 1994. Copies on order.

• David Cox, John Little, and Donal O'Shea, Using Algebraic Geometry, Graduate Texts in Mathematics 185, Springer, New York, 2005. (QA564 .C68 1998 -- First Edition)

• David Eisenbud, Commutative Algebra: with a View Toward Algebraic Geometry, Corrected Third Printing, Graduate Texts in Mathematics 150, Springer, New York, 2004. (QA251.3 .E38 2004)

• Ralf Fröberg, An Introduction to Gröbner Bases [electronic resource], Wiley, New York, 1997.

• Gert-Martin Greuel and Gerhard Pfister, A Singular Introduction to Commutative Algebra, Second Edition, Springer, New York 2008. (QA251.3 .G745 2002 -- First Edition)

• Brendan Hassett, Introduction to Algebraic Geometry, Cambridge University Press, Cambridge, 2007. QA564 .H256 2007

• Klaus Hulek, Elementary Algebraic Geometry, American Mathematical Society, Providence, RI, 2003. Copies on order.

• Niels Lauritzen Concrete Abstract Algebra: From Numbers to Gröbner Bases, Cambridge University Press, Cambridge, 2003. QA162 .L43 2003

• Hal Schenck, Computational Algebraic Geometry, London Mathematical Society Student Texts 85, Cambridge University Press, Cambridge, 2003. QA564 .S29 2003

• Karen E. Smith, Lauri Kahanpää, Pekka Kekäläinen and William Traves, An Invitation to Algebraic Geometry, Undergraduate Texts in Mathematics, Springer, New York, 2000. Not in library.

• Bernd Sturmfels, Solving Systems of Polynomial Equations, CBMS Regional Conference Series in Mathematics 97, American Mathematical Society, Providence, RI, 2002. Copies on order.

Homework

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.

Software

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.).

Information on how to use Maple for computations with Gröbner bases may be found in Appendix C of the textbook. The packages for Maple discussed there (which may be downloaded from the website of the authors) tend to be rather slow in comparison with a dedicated system as Macaulay 2. If you decide to use Macaulay 2, you might want to consult a chapter by Bernd Sturmfels from a book on Macaulay 2.

However the software of choice for this class will be Sage. There is a lot of information in the Sage website on how to use it. This choice will also force us to learn Singular since most algebraic computations in Sage are parsed to it. Singular has a comprehensive online manual, but you can also learn more from the Singular book which you can check out at the library.

Exams and Paper

There will be a take-home Midterm examination, due on Friday, October 17, at the beginning of class.

Students with conflicts with the Midterm Exam in this course are responsible for discussing makeup examinations with me no later than two weeks prior to the exam.

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 of ideals with finitely many zeroes.
4. Modules, free resolutions, and the Hilbert Syzygy Theorem.
5. Gröbner fan of an ideal and the state polytope.
6. Generic initial ideals.
7. Gröbner bases of binomial 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. Primary decomposition of monomial ideals.
6. Gröbner bases over Z2.
7. The F4 and F5 algorithms.
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.

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

Other information

Attendance policy: Regular and punctual attendance for this course is mandatory and will be recorded throughout the semester. If class must be missed, the student is expected to get the notes from a classmate, and to check the web-page for announcements and updated assignments. For all documented and university-approved absences (e.g. hospitalization, court appearances, university athletic conferences, etc) students must immediately contact and inform the instructor of the situation and present proper documentation before re-entering the classroom.

Scholastic dishonesty: All students are expected to engage in all academic pursuits in a manner that is above reproach. Students are expected to maintain complete honesty and integrity in the academic experiences both in and out of the classroom. Any student found guilty of dishonesty in any phase of academic work will be subject to disciplinary action at the discretion of the instructor. The University and its official representatives may initiate disciplinary proceedings against a student accused of any form of academic dishonesty including, but not limited to, cheating on an examination or other academic work which is to be submitted, plagiarism, collusion and the abuse of resource materials.

Classroom Rules of Conduct: Students will refrain from behavior in the classroom that intentionally or unintentionally disrupts the learning process and, thus, impedes the mission of the university. Cellular telephones and pagers must be turned off before class begins. Students are prohibited from eating in class, using tobacco products, making offensive remarks, reading newspapers, sleeping, talking at inappropriate times, wearing inappropriate clothing, or engaging in any other form of distraction. Inappropriate behavior in the classroom shall result in a directive to leave class. Students who are especially disruptive also may be reported to the Dean of Students for disciplinary action in accordance with university policy.

Visitors in the Classroom: Unannounced visitors to class must present a current, official SHSU identification card to be permitted in the classroom. They must not present a disruption to the class by their attendance. If the visitor is not a registered student, it is at the instructor's discretion whether or not the visitor will be allowed to remain in the classroom.

Student Absences on Religious Holy Days Policy: Section 51.911(b) of the Texas Education Code requires that an institution of higher education excuse a student from attending classes or other required activities, including examinations, for the observance of a religious holy day, including travel for that purpose. A student whose absence is excused under this subsection may not be penalized for that absence and shall be allowed to take an examination or complete an assignment from which the student is excused within a reasonable time after the absence.
University policy 861001 provides the procedures to be followed by the student and instructor. A student desiring to absent himself/herself from a scheduled class in order to observe (a) religious holy day(s) shall present to each instructor involved a written statement concerning the religious holy day(s). This request must be made in the first fifteen days of the semester or the first seven days of a summer session in which the absence(s) will occur. The instructor will complete a form notifying the student of a reasonable timeframe in which the missed assignments and/or examinations are to be completed.

Disabled Student Policy: It is the policy of Sam Houston State University that no otherwise qualified disabled individual shall, solely by reason of his/her handicap, be excluded from the participation in, be denied the benefits of, or be subjected to discrimination under any academic or Student Life program or activity. Disabled students may request assistance with academically related problems stemming from individual disabilities by contacting the Director of the Counseling Center in the Lee Drain Annex or by calling (936) 294-1720. Please bring all the necessary paperwork to the instructor before the end of the first week of classes in order to proceed with the requested accommodations. All disclosures of disabilities will be kept strictly confidential. NOTE: no accommodation can be made until the student registers with the Counseling Center.

Copyright policy: All printed materials disseminated in class or on the web are protected by Copyright laws. One xerox copy (or download from the web) is allowed for personal use. Multiple copies or sale of any of these materials is strictly prohibited.

Additional Information: All information on this syllabus is subject to change. Any changes will be announced in class.

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
Francis Sowerby Macaulay
Emmy Noether
Oscar Zariski