MA 726: Algebraic Geometry, NCSU, Spring 2022

Instructor:

Martin Helmer

Moodle:

Books:

Invitation to Nonlinear Algebra by Mateusz Michałek and Bernd Sturmfels.

Ideals, varieties, and algorithms : an introduction to computational algebraic geometry and commutative algebra (Fourth Edition) by David Cox, John Little, and Donal O'Shea.

The references below overlap with our course and interested students may enjoy consulting them on occasion. Note, however, that these references contain some material that we will not cover.

• Algebraic geometry: a first course by Joe Harris.
• Algebraic Geometry class notes by Andreas Gathmann.

Course Goals:

The goal of this course is to introduce students to algebraic geometry in a hands on manner. Students are encouraged to use computer tools such as Macaulay2 or Sage to explore examples and investigate problems.

The primary object at study will by systems of polynomial equations in $n$ variables. The solutions set of a system of polynomial equations forms a geometric object called a variety; we will see that this corresponds to a (radical) ideal in a polynomial ring. We will explore the geometry of varieties both computationally and abstractly using the algebraic structure of polynomial rings.

Term Project/Paper:

This course will involve a term project. The project will require students to independently study a class-related topic. The results of your work and the understanding that you have gained will be summarized in a short paper. Your paper should be self-contained and should be written so that to the other students in our class can understand it. The target length will be approximately 10 pages. If appropriate your project may also have a software component, in such cases the report may be somewhat shorter but should still contain the ideas behind the algorithms present in your software.

• Any of the the topics listed in Appendix D §2 of CLO.
• A topic of your suggestion (please run it by me).
• Tropical Geometry: The goal here would be to expand on what is covered in Chapter 7 of MS by consulting other references.
• Tensors: The goal here would be to expand on what is covered in Chapter 9 of MS by consulting other references.
• Representation Theory: The goal here would be to expand on what is covered in Chapter 10 of MS by consulting other references.
• Semidefinite Programming: The goal here would be to expand on what is covered in Chapter 9 of MS by consulting other references.
• Automated Geometric Theorem Proving: The goal here would be to expand on what is covered in §6.4 of CLO by consulting other references.
• Robotics and Kinematics: The goal here would be to expand on what is covered in §6.1 to §6.3 of CLO by consulting other references.
• Invariant Theory of Finite Groups. The goal here would be to summarize and expand on what is found in CLO and MS.
• Modern Gröbner Bases algorithms. The goal of this project would be to describe recent advances in algorithms to compute Gröbner Bases. A starting point for this would be Chapter 10 of CLO, and in particular §10.3 and §10.4. This project would be a good candidate to include some programming, but it would not be required.
• A brief introduction to intersection theory and the Chow ring. This is covered in Chapter 1 of the first reference. This project could, for example, focus on a proof of Corollary 2.4 of the first reference below (note that this project involves many new concepts beyond what we cover in class):
  • David Eisenbud, and Joe Harris. 3264 and all that: A second course in algebraic geometry. Cambridge University Press, 2016.
  • William Fulton. Introduction to intersection theory in algebraic geometry. No. 54. American Mathematical Soc., 1984.

• Febuary 17: Proposal Submission
• April 7: Draft Submission
• April 18: Peer Review Submission
• April 26: Project Submission

• Main report: 80% of your project grade
• This should be an approximately 10--15 page report detailing what you have studied. It should give the key definitions and build to some sort of main result, either a theorem or algorithm, and it should detail the theoretical basis for this result in either case. Your report should be written so that other students in the class can understand it.
• Peer review: 10% of your project grade
• As part of your projects you will be asked to swap a first draft with one other student in the class. You will then write feedback on the other students report, this feedback should point out any mathematical (or typographical) errors you note and should offer comments on things that could be clarified. Your mark for this will be based on your comments, i.e. on how well you critically read the other students work
• Proposal: 10% of your project grade
• A one page outline of what you propose to study and how you plan to do so.
• There may be an option to give a talk on your project, but this will depend on scheduling and level of interest.

• Your project (and ideally your proposal) should be prepared using LaTex. To get you started there is an example file below.
• The tex file
• The resulting pdf file

Algebra Software:

• Macaulay2
• Download and install
• Use online
• M2 may also be used online via the Sage Math Cloud. To use M2 this way create an account (this is free) and open a new project, then start a new Terminal (>_ icon). Once the Sage terminal opens type the command: M2. M2 will then open and you can enter M2 commands.
• Computations in polynomial rings:
• Polynomial Rings
• Quotient Rings
• Factoring Polynomials
• Computing gcd
• Division with remainder, i.e. getting a representative in a quotient ring, inverting an element in a quotient ring which is also a field etc.
• Check if an integer or ideal is prime
• Check equality of ideals (or numbers or other objects)

Installing M2 in Windows

• Download/install Ubuntu 18.04 from the Windows Store
• Follow the Ubuntu Install instructions for M2 (if you use 18.04 Ubuntu the following should do it)
• sudo echo 'deb http://www.math.uiuc.edu/Macaulay2/Repositories/Ubuntu bionic main' >/etc/apt/sources.list.d/macaulay2.list
• sudo apt-key adv --keyserver hkp://keys.gnupg.net --recv-key CD9C0E09B0C780943A1AD85553F8BD99F40DCB31
• sudo apt-get update -q
• sudo apt-get install -y -q macaulay2
Then typing 'M2' in the Ubuntu terminal should do it, optionally you can also install Xserver and run the emacs GUI.
• Install Xserver and emacs (if desired)
• Sage
• Sage Cell
• Sage Math Cloud
• Computations in polynomial rings:
• Univariate Polynomial Rings Description
• General Polynomial Rings Description

Syllabus:

The Syllabus.

Exam Dates:

• TBA.