Announcements

• Online Instruction Announcement (Updated March 20):
  This announcement will summarize how we will go forward with online instruction
  and will be pinned at the the top of the page until further notice.
  The current plan is to have online instruction for the remainder of the year.
  All quizzes and homework will be submitted through Canvas during this time.
  Class will be done online through GoogleMeet which can be accessed here: https://google.wisc.edu using your NetID.
  We will have an online class through GoogleMeet that will meet at the usual time and days of class.
  There is no passcode---you will receive an invitation from me by email at the time of the meeting.
  Office hours can be scheduled with me by appointment over email, and these will be done through GoogleMeet.
  In addition, you are expected to discuss with a subset of your group members at least once a week via some online platform. Since the DLN and Kinematics group is small, I hope that you will discuss much more than this.
  There are several discussion threads in Canvas to facilitate communication. Please take advantage of these.
  Colin, will post software resources at the usual place in Canvas and monitor the discussion threads.
  On Tuesday's and Thursday's (starting March 17) Colin will be watching the discussion threads from 10:30 to 12, to actively engage with you.


• Apr 9: Because of the switch to online instruction. Emails right before class will be the primary source of communication of updates.
• Apr 8: You have homework due tonight please check canvas.
• Apr 3: (Syllabus update) Lecture materials and recordings for this class are protected intellectual property at UW-Madison. Students in this course may use the materials and recordings for their personal use related to participation in this class. Students may also take notes solely for their personal use. If a lecture is not already recorded, you are not authorized to record my lectures without my permission unless you are considered by the university to be a qualified student with a disability requiring accommodation. [Regent Policy Document 4-1] Students may not copy or share lecture materials and recordings outside of class, including posting on internet sites or selling to commercial entities. Students are also prohibited from providing or selling their personal notes to anyone else or being paid for taking notes by any person or commercial firm without the instructor’s express written permission. Unauthorized use of these copyrighted lecture materials and recordings constitutes copyright infringement and may be addressed under the university's policies, UWS Chapters 14 and 17, governing student academic and non-academic misconduct.


• Mar 20: I look forward to seeing you all Monday.


• Mar 29: This is a message. to students about the transition to online learning.
• Mar 16-b: If you have become homeless because of the current situation, then feel free to reach out. There are some university resources available. I am happy to advocate on your behalf.
• Mar 16-a: This is a reminder that the NASO (see Mar 15-a) starts tomorrow at 11am Madison time.
• Mar 15-b: Two quizzes have been posted to Canvas.


• Mar 15-a: The NASO (Nonlinear Algebra Seminar Online), to be organized by Alexander Heaton, Miruna-Stefana Sorea, and Bernd Sturmfels, is a new MPI Leipzig seminar series is entirely online and everyone in the world is invited to participate. The meeting is Tuesdays and Thursdays, 17:00-19:00h, local time in Germany. Each session has three talks of 30 minutes, followed by questions. Details are posted at here.
  No registration or sign up is needed. To participate, simply visit this page at the scheduled time.
• Mar 12-d: I have discussed with the MLE group and the kinematics group about possible direction to take their respective projects. I am happy with this. I didn't get a chance to talk with the DLN group yet as Friday is your focus day, but I will send you an email tomorrow to check in.
• Mar 12-c: After spring break, we will switch to online instruction until at least April 10. During the online instruction period, I will post slides on mathematical concepts that I would like you to know. I will also post code to help you with programming questions that you will have. Your quizzes will be online.
• Mar 12-b: Over spring break, you are encouraged to work on your projects with your groups. I am hoping to set up online office hours and maybe Piazza so that you have multiple forums to ask questions.
• Mar 12-a: Assuming the university sticks to its current recommendations, I plan to have lecture tomorrow. I understand if you are uneasy about attending. If you plan not to come, then please check in with your group by email and send me a message.
• Mar 7: At the beginning of each class day next week, you will have a pop-quiz on your group project.
• Feb 21-a: For the those that are curious. For more information about the underlying method for PHCpack see the Wikipedia entry on Puiseux series and Chapters 1.4 and 3 in Sturmfels' solving polynomial systems book.
• Feb 18-b: Due this Friday, you have an open note Canvas Quiz on one of the guides to writing.
• Feb 18-a: If you are feeling stuck with the applications, then revisit the references for mathematical writing in the Feb 15 announcements. Use what you learn to improve the illustrative example submission that is due Wednesday.
• Feb 17 (DLN Group): Your focus day is on Friday, but don't hesitate to ask questions before then.
• Feb 17 (Kinematics Group): Congratulations on deriving your polynomial systems. Please try to solve these systems and see if your results agree with the literature. If you have any questions that you would like us to work on your focus day, then please email me (and cc your other group members).
• Feb 17 (MLE Group): Please continue working on finding many real solutions to the polynomial system you solved for the Bernoulli Mixture Model we discussed today. For more information on local maxima see page 12 of this.
• Feb 17-d: Your attendance is part of your participation grade.
• Feb 17-c: For information on how to make functions in M2 see this.
• Feb 17-b: For information on how to make functions in M2 see this.
• Feb 17-a: For information on how to run for loops in M2 see this.
• Feb 16: This is a reminder that announcements on this page should not be ignored. This class has begun to move at a quicker pace and we will no longer go over each message on this website at the beginning of every class.
• Feb 15-c: Please check out these guides to mathematical writing: Kevin P. Lee Kevin Houston.
• Feb 15-b: By February 19, 11pm please submit a written summary of an illustrative example you've worked on with your group.
• Feb 15-a: In the next weeks, you will work primarily in your groups. As a default, I will focus on the MLE group on Monday, kinematics on Wednesday, and DLN on Friday. Come prepared with questions.
• Feb 11-b: By next Wednesday (2/19) I would like to see a written summary of an illustrative example complete with reference. By next next Monday (2/24), I would like to see a list of questions/computations that you aim to address (I will elaborate more on this during class).
• Feb 11-a: Your computer lab grade is primarily based off of attendance and this will be checked going forward. You are expected to attend one computer lab session a week.
• Feb 6-b: Homework submission is now open.
• Feb 6-a: From Colin: We will be having help session every week on Thursday at the same time in the same place. If you can, come by either 11am or at 4pm and I'll explain how to determine degree of polynomial systems using macaulay2.
• Feb 4-b: By Friday morning, for HW LaTeX your solutions (or partial solutions) to Feb 3rd's problems.
• Feb 4-a: From Colin: Jose and I wrote down a short explanation of what degree and dimension are as part of a previous project. If you need a reminder of what these concepts are about, you can check out theorem 4.2. There's a lot of other information in this document that you won't need, so feel free to read past it.
• Feb 3: Please work on the following problems. 1a. Determine the degree of this system xz-y=z-y^2=0 . 1b. Try to make sense of the system's multidegree. 2a. Determine the degree of this system xy-1=xz-y=z-y^2=0. 2b. Try to make sense of this system's multidegree. 3. Find a polynomial system you like in the Jan 31st references and determine it's degree or multidegree.
• Jan 31-b: For more information regarding WURC, see here,
• Jan 31-a: Take a look at these references and pick your favorite one: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11.
• Jan 30: Additional office hours will be on Monday's at 11:30am.
• Jan 29-c: From Colin: The first computer help session is Thursday 11:00-12:00 in VAN HISE 0119, and the second help session is Thursday 4:00-5:00pm in VAN VLECK B333. Please either bring a laptop, or you can check one out at the library: https://ecs.library.wisc.edu// Again, show up to either one.
• Jan 29-b: Friday's HW assignment is to write up a solution to Problem 4 using LaTeX. Submit your tex, bib, and pdf file on Canvas. This document should include at least one reference. A description of your experiments and how you came about your answer would also be great to include.
• Jan 29-a: Friday's quiz will involve generalized eigenvalue problems and discriminants.
• Jan 27-d: The computer sessions will be 11-noon and 4-5 on Thursdays. Show up to either one, but at least. The location is TBD but will be announced on canvas and then on this website. To start Colin will go over how to log into wurc and use macaulay2 and latex.
• Jan 27-c: If you are off campus, then you have to use WiscVPN when sshing to campus. Just another level of security. See here for more information.
• Jan 27-b: Please LaTeX your solution to Problem 1.4. This solution should be understandable without reading the problem in the book.
• Jan 27-a: Using the terminal, try to ssh into wurc using the command ssh YourNetID@wurc.math.wisc.edu . If you try this command and are unsuccessful, then please email me.
• Jan 23-d: For the computational problems you are encouraged to use your favorite software. If you do not have a favorite software I suggest pairing with someone to work on the problems and/or checking out Macaulay2 or Julia. Next week I expect you to get access to the cluster and this should make things easier with the help of Colin.
• Jan 23-c: Here is a link to an article on Polynomial matrix spectral factorization. One of you asked me about this today after class. A very important concept in linear algebra that appears in this article is Cholesky decomposition.
• Jan 23-b: From Chapter 1.6 in Solving Systems of Polynomial Equations by Bernd Sturmfels try to do Problems 3, 4, 5. If you successfully complete Problem 3, then try it again for GEP and PEP with k=2, 3, 4. If you get stuck on Problem 4, then work it out for some small examples. If you get stuck on Problem 5, then try it for polynomials with degree 5 instead of 15.
• Jan 23-a: For quick guide to SVD see here.
• Jan 22-b: Look up the Sylvester matrix to prepare for Friday's class.
• Jan 22-a: Review Jordan decomposition and singular value decomposition for Friday's class.

An introduction to mathematical research: Solving polynomial systems (numerical nonlinear algebra)

• Algebraic kinematics
• Nash equilibria
• Reaction networks
• Deep linear networks and critical points
• Maximum likelihood estimation
• Monodromy, Galois groups, braid groups
• Numerical implicitization of determinantal representations
• Nonlinear eigenvalue problems, polynomial eigenvalue problem, generalized eigenvalue problem, multiparameter eigenvalue problem

Goals of the course

• learn some basic background about numerical methods in algebraic geometry
• learn how to translate problems into the language of algebraic geometry
• learn the process of producing mathematical data and exploring it for patterns
• develop mathematical computer programming skills
• develop presentation skills
• develop mathematical writing skills
• develop LaTeX proficiency

Prerequisites

• No computational experience is strictly necessary, but familiarity with a modern computer programming language will be useful. Interest in exploring mathematics through computer generated examples is a must!
• Linear algebra will be necessary.
• A knowledge of numerical algorithms and/or abstract algebra is a plus, but one can also learn what is needed along with the course material.

Some recommended reading

• Algebra -- group, ring, field, homomorphism, ideal, polynomial ring
• Ideal, Varieties, and Algorithms by David Cox for an introduction to computational algebraic geometry
• Numerically solving polynomial systems for an introduction to Bertini
• Solving Systems of Polynomial Equations by Bernd Sturmfels
• Invitation to Nonlinear Algebra by Mateusz Michalek and Bernd Sturmfels
• Algebraic Statistics by Seth Sullivant
• Abstract Algebra by Dummit and Foote
• Contemporary Abstract Algebra by Joseph Gallian

Course requirements

Caveat : We will be doing mathematical research, and by its very nature, we can't predict the outcome and so it is possible that the course requirements may change to better reach our research goals, e.g. perhaps groups will be merged or split if it turns out two questions have a common thread or one question splits into two, or perhaps presentations will be longer and less frequent if we need more time to devote to each presentation.

Tentative Course Outline

Preparation: First 5 weeks (Jan 22-Feb 21)

• Show up on time and bring paper and writing implement to every class Not only will your notes be a crucial reference (as there is no textbook), but you will also need these for quizzes (see below).
• Quizzes (10%) There will be 5 minute quizzes at the start of many classes in the first 5 weeks, which will require basic recall of definitions from previous classes, so you will need to review your notes before each class.
• Homework (10%) Homework will be given during each class period, and due the following Friday in class. Homework must be Typeset in LaTeX (see below), and a printed copy of your pdf or dvi is due in class. The final homework assignment is due March 2. Students may work together on problems but must each produce their own latex file of written solutions, and should include in their write-up the names of each student they worked with.
• Computer lab office hours (10%) Each student should attend computer lab office hours with Colin Crowley once each week to keep on track with the LaTeX and computational requirements.
• Project Work The projects will be introduced during weeks 4 and 5 and groups should start working on them as soon as their project is introduced.

Projects:

• Project Work The project work should include from each group member at least 5-8 hours of work per week.
• Presentations (0%) Each group will give a weekly project report. This report should include a handout or projected computer presentation which gives relevant charts or tables of data, and summarizes (e.g. in clearly written bullet points) the main updates of the report, as well as future questions to be considered. The group may decide how to split the presentation duties, but it is recommendation that they rotate in a reasonable way. The presentations will be graded based on the expectations for project work given above.
• Presentation Participation (0%) Students will be expected to given constructive feedback, ask questions, and make suggestions on the other groups' presentations.
• Write up progress (40%) Each group will give make updates to their final writeup on a weekly basis.

Final Summary and Poster Presentation:

• Outline (10%) An outline of the final summary and poster for each group is due to be emailed to the class email list by TBD.
• Poster drafts (0%) (No posters because of the coronavirus).
• Final Paper (20%) The .tex and all necessary files, along with a pdf, is due emailed to the instructor by noon TBD.

Grades

LaTex Resources

• Windows: Download MikTeX.
• Mac: You can download either of the following: TeXShop or MacTeX.
• Linux: If you use Linux, you might already have a TeX editor installed somewhere. If not, I think TeXLive might work.
• Any platform: One option is to use Auctex, which runs under emacs, and Skim as a viewer (this one is Mac only). One nice emacs client for a mac is Aquamacs, which, I think, comes with AucTex preinstalled.
• Yvonne Nagel has a pretty comprehensive list of resources here.
• Finally, here is a sample LaTeX file to get you started.

• The Not So Short Introduction to LaTeX (pdf)
• Googling a specific TeX question usually works out.
• Detexify -- draw a symbol and detexify tries to figure out how to produce it with LaTeX.
• If Googling doesn't work, go to the stackexchange Q&A site to have your questions quickly answered by experts.
• Beamer quickstart -- a guide to making slides in LaTeX.

Computers

I like this Sage reference manual

Math department linux machines you can work on are listed here.