Thomas Lam (University of Michigan): Positive geometries
Abstract: Positive geometry lies at the intersection of combinatorics, geometry, and theoretical physics. In this lecture series I will give an introduction to positive geometry with a focus on polytopes, hyperplane arrangements, and matroids. A general reference for the subject is the survey [arXiv:2509.25372].
Lecture 1: Positive geometry of polytopes. I will explain the statement “polytopes are positive geometries”. We will explore the canonical form of a polytope.
References: [arXiv:2208.05407, Section 1] and [arXiv:2410.21688].
Lecture 2: Positive geometry of hyperplane arrangements. I will discuss applications of canonical forms in the theory of hyperplane arrangements, with a special focus on the physically important case of the moduli space of n points on P1.
Reference: [arXiv:2502.20782].
Lecture 3: Matroids and amplitudes. I will explain a notion of scattering amplitudes for hyperplane arrangements and matroids.
Reference: [arXiv:2412.06705].
Lecture 1 (PDF)
Lecture 2 (PDF)
Lecture 3 (PDF)
Lecturer scribe: Mackenzie Bookamer
Problems 1 (link)
Problems 2 (link)
Problems 3 (link)
Pavlo Pylyavskyy (University of Minnesota): Cluster algebras, T-systems, and Zamolodchikov periodicity
Abstract: Cluster algebras were introduced around the year 2000 and have since grown into a major subject, with connections to many areas of mathematics and physics. At first, however, the theory was met with a certain amount of skepticism. A natural question was: what genuinely new results can cluster algebras prove?
One of the earliest and most compelling answers came from the Zamolodchikov periodicity conjecture, a statement about remarkable periodic behavior in certain recurrences arising from physics. In this mini-course, I will explain the T-system setting in which this phenomenon occurs, discuss Zamolodchikov periodicity and Volkov’s proof in a special case, and then describe classification results and a broader generalization of periodicity known as integrability. Parts of the course will be based on joint work with Pavel Galashin.
Lecture notes (PDF)
Lecturer scribe: Yuhan Jiang
Jessica Striker (North Dakota State University): Webs and alternating sign matrices
Abstract: Alternating sign matrices are intriguing combinatorial objects that simultaneously generalize both permutations and Catalan objects. They have a nice enumeration and deep connections to algebra, geometry, and statistical physics. Webs are combinatorial diagrams which encode polynomial invariants, or more generally, morphisms between representations of classic or quantum groups. The recent discovery of a rotation-invariant sl4 web basis uncovered a new connection between webs and alternating sign matrices. In this mini-course, we will explore both webs and alternating sign matrices, with a goal of understanding their relationship.
Lecture 1 (PDF)
Lecture 2 (PDF)
Lecture 3 (PDF)
Problems 1 (PDF)
Problems 2 (PDF)