842 Applied ALGEbra: Varieties, Applications, and polynomial system Solving

• Lectures: There are two types of lectures in this course. A Reading Supplement Lecture (RS-Lecture) is designed to help digest the material from the textbook and work through examples. On the other hand, a Varieties in Action Lecture (VA-Lecture) will be relatively self-contained and discuss how a variety appears in the world, i.e., how to model a problem using algebra. The format of a VA-lecture will vary depending on the reference material. (E.g., instructor/student presenting slides versus moderated discussion on a video lecture.)
  Here is a sample of possible topics. 

  • Algebraic Singular Value Decomposition

  • Method of moments

  • Polynomial systems in economics

  • Polynomial systems in phylogenetics 

  • Likelihood geometry for discrete statistical models

  • Unbalanced Procrustes Problem

  • Quantum optics

  • Multiview varieties

  • Multiparameter eigenvalue problems

  • Algebraic geometry and singularity theory in kinematics

  • Open to suggestions.

• The Textbook Material is the first seven chapters of Using Algebraic Geometry (online reference):

  • Polynomials and Ideals; Monomial Orders and Polynomial Division; Grobner Bases; Affine Varieties

  • Solving Polynomial Systems by Elimination; Finite-Dimensional Algebras; Grobner Basis Conversion; Solving Equations via Eigenvalues and Eigenvectors ; Real Root Location and Isolation

  • Resultants

  • Computation in Local Rings (4.5 Optional)

  • Modules

  • Free Resolutions

  • Polytopes, Resultants, and Equations

• Mathematical Tidbit Presentation: Starting in Week 2, every lecture day has an assigned person who has up to five minutes at the end of class to share a tidbit. This can be a proof of a theorem from the textbook, a neat example, or a solution to an exercise. Since you aren't required to present, there is a backup speaker for that day who can informally say what they found interesting about the course or something about their research interest (like an elevator pitch). 

• Learning outcomes: The development of  a common language amongst the class to communicate across wide ranging disciplines such as statistics, engineering, and the sciences. Learning outcomes by week.

• Participation will account for 30% of the course grade. 

  • About every three weeks there will be a participation survey. 

  • Here is a Canvas link to the first one.

• Final projects will account for 70% of the course grade. I am open to people working in a group of two, but a collaboration agreement form needs to be filled out. 

For the final project you are tasked with designing a minisymposium that has a connection to Applied Algebra. The components of the final project are below (subject to minor changes). The peer review components have strict deadlines. I can likely do extensions for the other components, but a request needs to be made through Canvas before the deadline (submit the request instead of the work).

• • 10% Acknowledgement: this is a confirmation that you know about the project and an opportunity to express concerns and ask questions, e.g. if it is okay to work with a partner. 

  • 10% Title and first draft of the minisymposium abstract (3 - 12 sentences, due February 17)

    • • Here is a website of a minisymposium organized by Thomas Yahl with an abstract after the organizers' names.

      • A long list of minisymposium for SIAM AG23 is found here and by clicking on a link you will be taken to a page with a list of speakers and minisumposium abstract. 

  • 10% Speaker details: List of four to six speakers, their websites (if available), titles of their talks (this can be the title of the relevant article), and pdfs of the article(s) you want the speaker to cover (due February 29 )

  • 10% Peer feedback of your current progress (Due between March 1 and March 14)

    • • Peer feedback can be obtained from students in the class or other math/stat/engineering graduate students who attend the University of Wisconsin --- Madison.

      • A student in the class can provide feedback to at most two other classmates. 

      • You only have to submit the name of the peer and how you got the feedback (was it over Zoom, email, or in-person). 

      • I am open minded on how you obtain the feedback. Here are two ideas to be concrete.  

        •  Idea 1: Send a peer your minisymposium materials (title, abstract, and speaker details) and ask (1) if they can identify the target audience for your minisymposium, (2) do they think the target audience would come to the session based on the minisymposium materials, and (3) if they have comments/suggestions for additional speakers or replacements. 

        • Idea 2: Send a peer your minisymposium title and abstract to have a conversation about who the speakers are and why you chose them. 

  • 10% Revised title, abstract and speaker information (Update according to peer feedback, due March 17)

  • 10% No more than 500 words explaining your choice of speakers and their work --- think of this as justifying travel support for these speakers to a funding agency (due March 24)

  • 10% A logo and less than 250 words explaining the logo: a logo is some type of graphic here are examples from MXM (due April 1)

  • 10% Second peer feedback of your current progress (Due between April 9 and  April 22th)

  • 10% Reflection: State some open problems and new directions that would be discussed at the minisymposium. This may also involve the student working out some preliminary examples to put these into context. (A bulleted list is acceptable, but a 1-2 page summary with some examples or key ideas is preferred) (Due by the last day of class)

    • • Code can be included in the 1-2 page summary

  • 10% Presentation: A short presentation to introduce your minisymposium or a related result (If a student doesn't like public speaking there are alternatives to this component such as recording the talk or submitting slides with notes. (Please mention any concerns about this in the Acknowledgement component at the start of the semester) (Due by the last day of class)

    • • Sign-up dates for presentations are in the tidbit google sheet.

      • Presentations can be 15 -25 minutes long 

      • Slide talks and chalk talks are good

      • Here are two types of presentations if you are looking for defaults:

        • Focus on one result of a speaker

        • Focus on a broad problem and how the session pulls together people in this context

        • Give an introduction to the  themes of your sympsosium

      • Please upload slides/notes to Canvas before or after the presentation.

•  Many of the topics in this course will be related to the long program Algebraic statistics and our changing world at IMSI (Chicago) Fall 2023.

• Check out the learning outcomes (updated continuously) for this information