Goals: This year, the class fill focus on algebraic geometry. Simply put, algebraic geometry studies exactly the same questions as linear algebra using polynomial equations in place of linear equations. Because of this, there is a wealth of applications. But because polynomials can be more intricate than linear functions, the work we must do is harder. Here are some examples of the differences, and why the study of algebraic geometry is so interesting:

• In linear algebra, linear equations give rise to planes, pehaps of a higher dimension. In algebraic geometry, polynomial equations give rise to surfaces. Some examples are below. Some may be old friends from multivariate calculus.

Images courtesy of the Virtual Math Museum.
• In linear algebra, the intersection of any number of planes is another plane, usually of a smaller dimension. The intersection of a number of surfaces can look like almost anything!
• In linear algebra, a basis of a subspace can be found using a fairly simple algorithm and every basis has the same dimension. In algebraic geometry, two bases for the same object can have wildly different number of elements.

There is a huge number of applications, including machine learning (where data often lies on one or multiple surfaces), robotics (intersections of polynomials limit the motion of robotic arms), cryptography (polynomial equations called elliptic curves form the basis of modern encryption schemes), phylogenetics and computational linguistics(polynomials determine evolutionary trees), and automated theorem proving (if you set things up correctly, whether or not a family of polynomials has a non-empty intersection will tell you whether a theorem is true or not.)

Prerequisites: The formal prerequisite for the class is Math 2702 although that is really much too restrictive. Quoting from the book I will be using for the class: "The mathematical prerequisites of the book are modest: the students should have had a course in linear algebra and a course where they learned how to do proofs." There will be parts of the class where having had a class in either groups or rings will help a little bit, but the class is really about polynomials: something that we are all quite familiar with. If you would like to take this class and do not have the Math 2702 prerequisite, please send me an email. I will have to give you permission to register.

Course structure: The class will be lecture an laboratory based. Homework will be weekly and will consist of proof-based and computer-based problems. Prior experience in programming is not required; the software we will use is quite friendly. There will be a mid-term exam. A final project, that may be completed as part of a group or individually, will explore an application of the material we convered during the semester.

Projects:
I have begun a list of possible final project topics.

Robotic motion planning	Computational linguistics
Data analysis	Automated theorem verficiation

Course Objectives: A student completing the class should be able to:

• enunciate the motivating questions of algebraic geometry;
• define and prove elementary results about real algebraic varieties and ring ideals;
• describe the The Algebra-Geometry Dictionary with a focus on how problems in geometry can be solved by using abstract algebra as well as the reverse; and
• apply methods of algebraic geometry to a selection of problems in robotics, statistics and data science, biology, computational linguisitcs, automated theorem proving, among others.

Book and Materials: We will closely follow the fourth edition of Ideals, Varieties, and Algorithms by Cox et al. Make sure to secure your own copy as soon as practical. The very recent volume Applications of Polynomial Systems by Cox et al. will be extremely useful as a source of final projects. It details a number of recent applications and open questions in the field.

Weekly schedule: Every Sunday night, if not earlier, I will post a complete class schedule for the upcoming week on this web page. It will contain all the details for that week, including homework, due dates, and links to video lectures as well as labs.

Reading and watching mathematics: Learning mathematics is not a spectator sport. Reading mathematics is not like reading a novel; watching mathematics is not like watching an action thriller. Some paragraphs are easy to digest, but you may find yourself looking at one line of text for five or more minutes trying to understand what the author is trying to say.  Use the pause button when watching a video. As you read or watch, take notes, just as you do in class. This is crucial! If questions arise, write them down and ask during office hours. In each module, I will let you know how long you should expect to spend reading and watching the material.  Some weeks, this will be a substantial commitment of time even before you start the homework.

Office hours: I will reseve my Monday mornings for office hours for this class. To schedule a meeting, either individually or as a group, please send me an email with a couple of times that will work. I will reply with a link and a time we can all meet. Further, I will hold an official office hour for this course on Mondays at 11:30am. The link will be available on each week's web page. You will need a password to join.

Homework: Homework problems will be assigned after every class, and will be due once a week. Each submission will consist of three parts:

• Cover sheet: In addition to your name, I will also ask you to recognize individuals you worked with and sources you used to complete your work. There will also be a space to briefly discuss your group meetings that week.
• Class notes: As you watch class videos and lectures, I expect you to actively take notes. You will submit them as part of the homework each week. They will be graded generously, but should be complete and legible.
• Homework problems: Most of the homework will come from our text. Please leave ample room for grader comments.

Your homework will be graded using [GradeScope]. To begin, you will need to set up an account.  I will send you the code for our class by email.  Each homework assignment will need to be submitted as a .pdf file.  If you edit your homework electronically, make sure you can save or export your work in this format.  If you write-up your homework the old-fashioned way using pencil and paper, use a scanner or a phone scanner app.  See the GradeScope help document for a list of suggestions [PDF]. Please let me know if this does not work for you; I will come up with an alternative.

Collaboration and groups: Throughout the course of the class, you will be a part of a group. While your work will be written-up individually, I would like all of you to check in with each other and discuss the course material and homework at least once a week outside of class.

Exams: There will be one exam during the semester, tentatively scheduled for Friday, March 26. It will be a flexibly-timed take-home affair. Details will be announced in class and on the web page.

Final Project: Your final course evaluation will consist of a project where you will use algebraic geometry to describe and solve a problem. A number of possible examples are listed in the prior section, but I will provide many resources. The project can be completed individually or as a group, although everyone will submit an individual report. You will also be required to meet with me a few weeks before the due date to check in and iron out the details.

Grades: Your course grade will be based on the homework (40%), exam (20%), final project (20%), and class engagement (20%).

Computation will be a part of the class. There are a number of software packages that are can be used for algebraic geometry, but my favorite is SageMath. It is Python-based, so any programming skills you accidentally acquire will apply broadly. You can install SageMath on your own machine, sign up for a CoCalc account, or use Bowdoin's High Performance Computing server. But for a quick one-off computation, nothing beats SageCell. In the example below, you can graph an algebraic variety defined by a polynomial:

Bowdoin's HPC cluster: If you would like to use Bodwoin's cluster to run your computations, Dj Merrill wrote a quick set of instructions. Note that if you are off-campus, you will need to VPN to campus first. Here is the link to Bowdoin's Jupyter hub:

jupyter.bowdoin.edu

One advantage of using the cluster for your computations is the ability to save your work for future reference.

Our scheduled meeting times are Wednesdays at 3pm and Fridays 10:30am. Also, the official office hour for this class is 11:30am on Monday. Both can be accessed via the following link, but you will need a password:

[Meeting Link]

If you would like to meet with me at any other point, please send me an email; I hold Monday mornings reserved especially for meetings with students.

Lecture recordings: The live lectures for this week will be recorded for on-campus use. Once they are available, you can find them at the following links:

[Wednesday Lecture]

[Friday Lecture]

[Handout 1]

Discussion boards: There is a Slack channel for the class. Use it to post questions and comments, or answer those posted by others; I will monitor it and respond daily.

[Slack]

Homework due Tuesday at 5pm: The homework this week consists of four parts:

• Make sure that you have completed all the items in my "welcome" course email.
• Complete the following problems:
  • Practice: §1.2 1, 4d.
  • Official: §1.2 6, 12, 13, 15 §1.3 6
  • Challenge: Read §1.1 Proposition 5 until the end. Then consider §1.2 8, 10.
  At some point, solutions will be available: [solutions] .
• a [cover sheet] with your name, the assignment number, a bibliography recognizing the contributions of others in your work as well as any inanimate sources you may have used, as well as a brief description of your group meeting this week,
• a copy of your notes from this week's lectures and videos,

You should submit your homework solutions as a single .pdf file; a nice way to create this document is to use a [phone scanner app]. Submit it via [GradeScope].

Our scheduled meeting times are Wednesdays at 3pm and Fridays 10:30am. Also, the official office hour for this class is 11:30am on Monday. Both can be accessed via the following link, but you will need a password:

[Meeting Link]

If you would like to meet with me at any other point, please send me an email; I hold Monday mornings reserved especially for meetings with students.

Lecture recordings: The live lectures for this week will be recorded for on-campus use. Once they are available, you can find them at the following links:

[Wednesday Lecture]

[Friday Lecture]

[Handout 2]

Discussion boards: There is a Slack channel for the class. Use it to post questions and comments, or answer those posted by others; I will monitor it and respond daily.

[Slack]

Homework due Tuesday at 5pm: The homework this week consists of two parts:

• Complete the following problems:
  • Practice: §1.4 3(a), 3(b)
  • Official: §1.4 2, 3, 7, 8 §1.5 2
  • Challenge: §1.4 15
  You will need a [cover sheet] . Eventually, I will post [solutions] .
• a copy of your notes from this week's lectures and videos,

You should submit your homework solutions as a single .pdf file via [GradeScope].

Our scheduled meeting times are Wednesdays at 3pm and Fridays 10:30am. Also, the official office hour for this class is 11:30am on Monday. Both can be accessed via the following link, but you will need a password:

[Meeting Link]

If you would like to meet with me at any other point, please send me an email; I hold Monday mornings reserved especially for meetings with students.

Lecture recordings: The live lectures for this week will be recorded for on-campus use. Once they are available, you can find them at the following links:

[Wednesday Lecture]

[Friday Lecture]

[Handout 3]

Discussion boards: There is a Slack channel for the class. Use it to post questions and comments, or answer those posted by others; I will monitor it and respond daily.

[Slack]

Homework due Tuesday at 5pm: The homework this week consists of two parts:

• Complete the following problems:
  • Practice: §1.5 8, 9
  • Official: §1.5 3, 5 §2.1 Read Examples 2 and 3. Do problems 1, 2, 3. §2.2 9
  • Challenge: §1.5 Read exercises #11-16 and write up a solution to #17
  You will need a [cover sheet] . Eventually, I will post [solutions] .
• a copy of your notes from this week's lectures and videos,

You should submit your homework solutions as a single .pdf file via [GradeScope].

Our scheduled meeting times are Wednesdays at 3pm and Fridays 10:30am. Also, the official office hour for this class is 11:30am on Monday. Both can be accessed via the following link, but you will need a password:

[Meeting Link]

If you would like to meet with me at any other point, please send me an email; I hold Monday mornings reserved especially for meetings with students.

Lecture recordings: The live lectures for this week will be recorded for on-campus use. Once they are available, you can find them at the following links:

[Wednesday Lecture]

[Dickson's Lemma]

[Friday Lecture]

[Handout 4]

Discussion boards: There is a Slack channel for the class. Use it to post questions and comments, or answer those posted by others; I will monitor it and respond daily.

[Slack]

Homework due Tuesday at 5pm: The homework this week consists of two parts:

• Complete the following problems:
  • Practice: §2.3 6, 7
  • Official: §2.3 8 §2.4 8, 10ab §2.5 3ab, 7
  • Challenge: §2.4 9
  You will need a [cover sheet] . Eventually, I will post [solutions] .
• a copy of your notes from this week's lectures and videos,

You should submit your homework solutions as a single .pdf file via [GradeScope].

Our scheduled meeting times are Wednesdays at 3pm and Fridays 10:30am. Also, the official office hour for this class is 11:30am on Monday. Both can be accessed via the following link, but you will need a password:

[Meeting Link]

If you would like to meet with me at any other point, please send me an email; I hold Monday mornings reserved especially for meetings with students.

Lecture recordings: The live lectures for this week will be recorded for on-campus use. Once they are available, you can find them at the following links:

[Wednesday Lecture]

[Friday Lecture]

Discussion boards: There is a Slack channel for the class. Use it to post questions and comments, or answer those posted by others; I will monitor it and respond daily.

[Slack]

For your computational enjoyment, here is a code snippet that will compute a Gröbner basis, should you find it necessary either now or in the future. By the way, if there is an error in my HTML, just cut and paste the code below into the SageMath box at the top of the page; that one should work.

Homework due Tuesday at 5pm: The homework this week consists of two parts:

• Complete the following problems:
  • Practice: §2.6 2
  • Official: §2.5 10, 16, 17, 18 §2.6 4
    Read examples 2 and 3 in §2.8 and do problems 3 and 4 therein.
  • Challenge: §2.6 12

Eventually, I will post [solutions] .

Our scheduled meeting times are Wednesdays at 3pm and Fridays 10:30am. Also, the official office hour for this class is 11:30am on Monday. Both can be accessed via the following link, but you will need a password:

[Meeting Link]

If you would like to meet with me at any other point, please send me an email; I hold Monday mornings reserved especially for meetings with students.

Lecture recordings: The live lectures for this week will be recorded for on-campus use. Once they are available, you can find them at the following links:

[Wednesday Lecture]

[Friday Lecture]

Discussion boards: There is a Slack channel for the class. Use it to post questions and comments, or answer those posted by others; I will monitor it and respond daily.

[Slack]

Homework due Tuesday at 5pm: None!

Our scheduled meeting times are Wednesdays at 3pm and Fridays 10:30am. Also, the official office hour for this class is 11:30am on Monday. Both can be accessed via the following link, but you will need a password:

[Meeting Link]

If you would like to meet with me at any other point, please send me an email; I hold Monday mornings reserved especially for meetings with students.

Lecture recordings: The live lectures for this week will be recorded for on-campus use. Once they are available, you can find them at the following links:

[Friday Lecture]

Discussion boards: There is a Slack channel for the class. Use it to post questions and comments, or answer those posted by others; I will monitor it and respond daily.

[Slack]

Our scheduled meeting times are Wednesdays at 3pm and Fridays 10:30am. Also, the official office hour for this class is 11:30am on Monday. Both can be accessed via the following link, but you will need a password:

[Meeting Link]

If you would like to meet with me at any other point, please send me an email; I hold Monday mornings reserved especially for meetings with students.

Lecture recordings: The live lectures for this week will be recorded for on-campus use. Once they are available, you can find them at the following links:

[Wednesday Lecture]

[Friday Lecture]

Discussion boards: There is a Slack channel for the class. Use it to post questions and comments, or answer those posted by others; I will monitor it and respond daily.

[Slack]

Homework due Tuesday at 5pm: The homework this week consists of two parts:

• Complete the following problems:
  • Official: §3.3 14 §3.4 8, 20
  • Challenge: §3.3 11
  You will need a [cover sheet] . Eventually, I will post [solutions] .
• a copy of your notes from this week's lectures and videos,

You should submit your homework solutions as a single .pdf file via [GradeScope].

Our scheduled meeting times are Wednesdays at 3pm and Fridays 10:30am. Also, the official office hour for this class is 11:30am on Monday. Both can be accessed via the following link, but you will need a password:

[Meeting Link]

If you would like to meet with me at any other point, please send me an email; I hold Monday mornings reserved especially for meetings with students.

Lecture recordings: The live lectures for this week will be recorded for on-campus use. Once they are available, you can find them at the following links:

[Wednesday Lecture]

[Friday Lecture]

Discussion boards: There is a Slack channel for the class. Use it to post questions and comments, or answer those posted by others; I will monitor it and respond daily.

[Slack]

Homework due Tuesday at 5pm: The homework this week consists of two parts:

• Complete the following problems:
  • Official: §4.1 1, 2 §4.2 6b, 7b, 14
  • Challenge: §4.1 8, §4.2 16
  You will need a [cover sheet] . Eventually, I will post [solutions] .
• a copy of your notes from this week's lectures and videos,

You should submit your homework solutions as a single .pdf file via [GradeScope].

Our scheduled meeting times are Wednesdays at 3pm and Fridays 10:30am. Also, the official office hour for this class is 11:30am on Monday. Both can be accessed via the following link, but you will need a password:

[Meeting Link]

If you would like to meet with me at any other point, please send me an email; I hold Monday mornings reserved especially for meetings with students.

Lecture recordings: The live lectures for this week will be recorded for on-campus use. Once they are available, you can find them at the following links:

[Wednesday Lecture]

[Friday Lecture]

Discussion boards: There is a Slack channel for the class. Use it to post questions and comments, or answer those posted by others; I will monitor it and respond daily.

[Slack]

Homework due Tuesday at 5pm: The homework this week consists of two parts:

• Complete the following problems:
  • Official: §4.3 1, 8, 9
  • Challenge: §4.3 11
  You will need a [cover sheet] . Eventually, I will post [solutions] .
• a copy of your notes from this week's lectures and videos,

You should submit your homework solutions as a single .pdf file via [GradeScope].

Our scheduled meeting times are Wednesdays at 3pm and Fridays 10:30am. Also, the official office hour for this class is 11:30am on Monday. Both can be accessed via the following link, but you will need a password:

[Meeting Link]

If you would like to meet with me at any other point, please send me an email; I hold Monday mornings reserved especially for meetings with students.

Lecture recordings: The live lectures for this week will be recorded for on-campus use. Once they are available, you can find them at the following links:

[Wednesday Lecture]

[Friday Lecture]

Discussion boards: There is a Slack channel for the class. Use it to post questions and comments, or answer those posted by others; I will monitor it and respond daily.

[Slack]

Homework due Tuesday at 5pm: The homework this week consists of two parts:

• Complete the following problems:
  • Official: §4.4 1bc, 3, 16b
  You will need a [cover sheet] . Eventually, I will post [solutions] .
• a copy of your notes from this week's lectures and videos,

You should submit your homework solutions as a single .pdf file via [GradeScope].

Our scheduled meeting times are Wednesdays at 3pm and Fridays 10:30am. Also, the official office hour for this class is 11:30am on Monday. Both can be accessed via the following link, but you will need a password:

[Meeting Link]

If you would like to meet with me at any other point, please send me an email; I hold Monday mornings reserved especially for meetings with students.

Lecture recordings: The live lectures for this week will be recorded for on-campus use. Once they are available, you can find them at the following links:

[Wednesday Lecture]

[Friday Lecture]

Discussion boards: There is a Slack channel for the class. Use it to post questions and comments, or answer those posted by others; I will monitor it and respond daily.

[Slack]

Homework due Tuesday at 5pm: The homework this week consists of two parts:

• Complete the following problems:
  • Official:§4.4 4 (ignore last equation), §4.5 2, 6
  You will need a [cover sheet] . Eventually, I will post [solutions] .
• a copy of your notes from this week's lectures and videos,

You should submit your homework solutions as a single .pdf file via [GradeScope].

Our scheduled meeting times are Wednesdays at 3pm and Fridays 10:30am. Also, the official office hour for this class is 11:30am on Monday. Both can be accessed via the following link, but you will need a password:

[Meeting Link]

If you would like to meet with me at any other point, please send me an email; I hold Monday mornings reserved especially for meetings with students.

Lecture recordings: The live lectures for this week will be recorded for on-campus use. Once they are available, you can find them at the following links:

[Wednesday Lecture]

[Friday Lecture]

Discussion boards: There is a Slack channel for the class. Use it to post questions and comments, or answer those posted by others; I will monitor it and respond daily.

[Slack]

Homework due Tuesday at 5pm: The homework this week consists of two parts:

• Complete the following problems:
  • Official:§4.4 4 (ignore last equation), §4.5 2, 6, 11 §4.6 10
  You will need a [cover sheet] . Eventually, I will post [solutions] .
• a copy of your notes from this week's lectures and videos,

You should submit your homework solutions as a single .pdf file via [GradeScope].

Our scheduled meeting times are Wednesdays at 3pm and Fridays 10:30am. Also, the official office hour for this class is 11:30am on Monday. Both can be accessed via the following link, but you will need a password:

[Meeting Link]

If you would like to meet with me at any other point, please send me an email; I hold Monday mornings reserved especially for meetings with students.

Lecture recordings: The live lectures for this week will be recorded for on-campus use. Once they are available, you can find them at the following links:

[Wednesday Lecture]

[Friday Lecture]

As part of your final evaluation for the course, I would like you to complete a project that shows your mastery of some of the concepts covered in the course. You will hand in a written report of your work, the goal of which is to show

• your mastery of knowledge of some component of algebraic geometry that we have studied this semester, as well as
• your ability to build upon what we have learned, either by learning additional material beyond what we have covered or by applying techniques we have learned to a tangible problem.

The following tabs suggest some projects you can pursue. However, you should not feel limited to choose only from the projects presented here, nor should you view the project descriptions as instructions that must be followed to the letter. This is your chance to follow an interest in algebraic geometry!

Instructions: The projects will be due on May 24th.  Although you should work on these projects as part of a larger group, your final submission must be completed individually.  You are encouraged to use any external resources that you deem appropriate (both animate as well as inanimate), but you must make sure to cite them. A reasonable written project is expected to be in the neighborhood of ten single-spaced pages.  Written work should demonstrate your mastery of the theoretical material as well as your ability to apply it to solve a problem.  In particular, a significant fraction of your written work should include an exposition of the relevant theory that we have learned in this course, and in certain cases, that you have learned from other sources during your research. As a general rule of thumb, you should aim to write a paper that could be read and understood by an outstanding undergraduate student who is unfortulately not familiar with algebraic geometry.

Topics: I have begun a list of possible final project topics. Click on the images for the details.

Robotic motion planning	Phylogenetics
Stable points of dynamical systems	Automated theorem verficiation
Rigidity	Rational equivalence and variety isomorphism