Lectures: Tuesday and Thursday 1-2:30pm B737 East Hall
Instructor: Thomas Lam, [email protected]
Office Hours: approach me after class, or EH 2834 Tuesday, Thursday 2:30-3:50pm.
Prerequisites:
This is a graduate level mathematics course and intended to be challenging. Students will be assumed to be proficient at linear algebra. Students are expected to have taken multiple graduate level classes (e.g. 565, 566, 593, 594) in and around algebra and combinatorics. Concepts from geometry and topology may be used after being reviewed.
Grading:
There will be problem sets assigned roughly once a month. There will be no exams.
Grades will be calculated from: Problem Sets (98%) and submission of course evaluation (2%).
References:
We will not follow a single text closely.
[Pin] A. Pinkus, Totally Positive Matrices, Cambridge Tracts in Mathematics
[Kar] S. Karlin, Total positivity, Volume 1, Stanford University Press.
[FZ] S. Fomin and A. Zelevinsky, Total positivity: tests and parametrizations, The Mathematical Intelligencer, 22 (2000), 23-33.
[Pos] A. Postnikov, Total positivity, Grassmannians, and networks.
[Lam] T. Lam, Totally nonnegative Grassmannian and Grassmann polytopes, Current Developments in Mathematics, 2014.
Related courses:
Pavel Galashin
Xuhua He
Lauren Williams
David Speyer
Ten+ years ago
Alex Postnikov
Pavlo Pylyavskyy
Pset problems:
Problem set problems.
Hand in five problems by each of the following dates: Feb 17, Mar 17, Apr 14.
Homework must be typed and written in LaTeX. Homework must be submitted either in person, or emailed to [email protected]. Homework must be submitted by the start class time (i.e. 1pm on Tuesday or Thursday). Late homework will not generally be accepted.
You are encouraged to work with other students on the problem sets, but you must include the names of those you worked with when you hand in your homework. You are not allowed to post homework problems on question websites such as mathoverflow or stackexchange. If you use a solution you find in a book, online, by asking AI, or elsewhere, you must acknowledge the source.
It is assumed that you understand any solution that you submit. In particular, I may ask you to explain what you wrote on your homework to me.
Academic Misconduct: The University of Michigan community functions best when its members treat one another with honesty, fairness, respect, and trust. The college promotes the assumption of personal responsibility and integrity, and prohibits all forms of academic dishonesty and misconduct. All cases of academic misconduct will be referred to the LSA Office of the Assistant Dean for Undergraduate Education. Being found responsible for academic misconduct will usually result in a grade sanction, in addition to any sanction from the college. For more information, including examples of behaviors that are considered academic misconduct and potential sanctions, please see lsa.umich.edu/lsa/academics/academic-integrity.html.
Disabilities: If you think you need an accommodation for a disability, please let me know as soon as possible. In particular, a Verified Individualized Services and Accommodations (VISA) form must be provided to me at least two weeks prior to the need for a test/quiz accommodation. The Services for Students with Disabilities (SSD) Office (G664 Haven Hall) issues VISA forms.
Topics covered (tentative):
Total nonnegativity for GL_n and planar networks.
Totally positive Grassmannian and plabic graphs.
Positroids, Grassmann necklaces, and bounded affine permutations.
Dimer model.
Totally positive orthogonal and Lagrangian Grassmannians.
Ising model and resistor network model.
Topology of totally positive spaces?
Amplituhedra and Grassmann polytopes?
Tropical totally positive Grassmannian?
List of lectures: