Math 669 Winter 2022
Combinatorics and Geometry of Amplitudes

Lectures: Tuesday and Thursday 1-2:30pm 4088 East Hall

Instructor: Thomas Lam,

Office Hours: Thursday after lectures.

This is a graduate level mathematics course. There are no specific prerequisites but various concepts from combinatorics, algebra, geometry, topology, etc. will be (sometimes reviewed) and used. Students are expected to chase down less familiar concepts with assistance from the instructor. No knowledge of physics is assumed.

Grading: There will be problem sets roughly once a month. Students will be asked to write a term paper, roughly 5 pages in length. There will be no exams.
Grades will be calculated from: Paper (25%) and Problem Sets (75%).

References: We will not be following a textbook.
Physics background:
H. Elvang and Y.-T. Huang, Scattering Amplitudes in Gauge Theory and Gravity, Cambridge University Press, 2015.
G. M. Ziegler, Lectures on Polytopes, Graduate Texts in Mathematics, Springer.
Positive geometries:
N. Arkani-Hamed, Y. Bai, T. Lam, Positive Geometries and Canonical Forms, arXiv link
Adjoint hypersurfaces:
K. Kohn, K. Ranestad, Projective geometry of Wachspress coordinates, arXiv link
Background on toric varieties:
W. Fulton, Introduction to Toric Varieties, Annals of Mathematics Studies, Princeton University Press.
N. Arkani-Hamed, Y. Bai, S. He, G. Yan, Scattering Forms and the Positive Geometry of Kinematics, Color and the Worldsheet, arXiv link
Non-polytope positive geometries:
K. Kohn, R. Piene, K. Ranestad, F. Rydell, B. Shapiro, R. Sionn, M.-S. Sorea, S. Telen, Adjoints and Canonical Forms of Polypols, arXiv link
Moduli space M0,n
F. Brown, Multiple zeta values and periods of moduli spaces M0,n, arXiv link
Stringy integrals
N. Arkani-Hamed, S. He, T. Lam, Stringy canonical forms, arXiv link
Scattering equations
F. Cachazo, S. He, E.Y. Yuan, Scattering of Massless Particles: Scalars, Gluons and Gravitons, arXiv link
Surveys on spinors, gauge theory amplitudes
L.J. Dixon, A brief introduction to modern amplitude methods, arXiv link
R. Britto, Constructing scattering amplitudes, link
Grassmannian integrals
N. Arkani-Hamed, F. Cachazo, C. Cheung, J. Kaplan, A Duality for the S-Matrix, arXiv link
L. Mason, D. Skinner, Dual Superconformal Invariance, Momentumn Twistors and Grassmannians, arXiv link
Positive Grassmannian
A. Postnikov, Total positivity, Grassmannians, and networks, arXiv link
T. Lam, Totally nonnegative Grassmannian and Grassmann polytopes, arXiv link
N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, A.B. Goncharov, A. Postnikov, J. Trnka, Scattering Amplitudes and the Positive Grassmannian, arXiv link
Positroid varieties
A. Knutson, D. Speyer, T. Lam, Positroid Varieties: Juggling and Geometry, arXiv link

Lecture notes:
Some facts about semialgebraic sets
Summary of polytope canonical form formulae
Uniqueness of adjoint hypersurfaces

Paper topics:
The paper is due on Tuesday April 19. Please pick a topic for your paper by April 1 and discuss your choice of topic for the term paper with me before starting!
Plan to write a paper that is roughly 5 page in length, typeset in LaTeX.
Ideas for term paper topics

Pset problems:

Submit 6 problems from any of the psets on or before each of the following dates: February 7, March 7, April 7

these files will be updated throughout the semester
Positive Geometries
phi^3 amplitudes
Stringy integrals
Spinors and Twistors
Positive Grassmannian

Homework policy: Homework must be written in LaTeX.
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, or elsewhere, you must acknowledge the source.

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

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In our classrooms all students are expected to adhere to the required safety measures and guidelines of the State of Michigan and the University of Michigan, wearing a face covering that covers the mouth and nose in all classrooms, and not coming to class when ill or in quarantine. It is important to also be thoughtful about group gatherings as well as about classroom activities and exercises that require collaboration.
Any student who is not able and willing to comply with campus safety measures for this course should contact the course instructor or their academic advisor to discuss alternate participation or course options. Students who do not adhere to these safety measures while in a face-to-face class setting, and do not have an approved exception or accommodation, may be asked to disenroll from the class.
For additional information refer to the LSA Student Commitment to the Wolverine Culture of Care and the OSCR Addendum to the Statement of Student Rights and Responsibilities on the OSCR website.

Disabilities: The University of Michigan recognizes disability as an integral part of diversity and is committed to creating an inclusive and equitable educational environment for students with disabilities. Students who are experiencing a disability-related barrier should contact Services for Students with Disabilities; 734-763-3000 or For students who are connected with SSD, accommodation requests can be made in Accommodate. If you have any questions or concerns please contact your SSD Coordinator or visit SSD’s Current Student webpage. SSD considers aspects of the course design, course learning objects and the individual academic and course barriers experienced by the student. Further conversation with SSD, instructors, and the student may be warranted to ensure an accessible course experience.

Course Recordings: I plan to have course lectures audio/video recorded and made available to other students in this course. Please be aware that as part of your participation in this course, you may be recorded. If you have concerns about this, please let me know as soon as possible.

Topics covered (tentative):
Polytopes, positive geometries, phi^3-tree amplitudes, moduli space of marked rational curves, tree-level string amplitudes, Grassmannians, spinors and twistors, super Yang-Mills tree amplitudes, amplituhedra and Grassmann polytopes

List of lectures: