MA 726: Algebraic Geometry, NCSU, Spring 2022

Instructor:

Email: mhelmer at ncsu.edu
Office:
Office Hours:

Moodle:

Course notes and assignments will be posted on the Moodle page (click here).

Books:

The primary course text will be Invitation to Nonlinear Algebra below. We will refer to this book as MS, the book by Cox, Little and O'Shea as CLO.

Invitation to Nonlinear Algebra by Mateusz Michałek and Bernd Sturmfels.

We will also reference the following books for certain portions of the course:

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.

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.

A major component of this study will be the theory of Gröbner basis. At the end of the course students will be able to answer such questions as: Does a given system of polynomials have finitely many solutions? If so, what are they? If there are infinitely many solutions, how can can these be described and understood?

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.

Suggested Topics:
• Any of the the topics listed in Appendix D §2 of CLO.
• 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.
Time line :
• Febuary 17: Proposal Submission
• April 7: Draft Submission
• April 18: Peer Review Submission
• April 26: Project Submission
Mark break down for project:
• 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.
• 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