Course Descriptions

Uhlenbeck Lecture Course: From Area to Action: A Symplectic Story
Lecturer: Eva Miranda, Universitat Politècnica de Catalunya and CRM
Teaching Assistants: Ipsita Datta and Isaac Ramos, ETH Zürich

Abstract: From Artemis’ journey to the Moon to the ancient problem of planetary motion, many of the most emblematic stories of physics are, at their core, symplectic stories. Symplectic geometry provides the language in which Hamiltonian systems describe motion through positions, momenta, area, and action. In this course, we will follow the search for understanding trajectories: first through completely integrable systems, where symmetries organize motion into tori and provide action-angle coordinates, and then beyond integrability, where periodic orbits emerge as the guiding traces through which Poincaré taught us to understand dynamics. This path culminates in the Weinstein and Arnold conjectures and in Floer homology, revealing symplectic geometry as a vibrant meeting point of topology, dynamics, and physics.

Prerequisites: Familiarity with smooth manifolds, differential forms, vector fields, and basic notions from differential geometry at the level of a master’s or first year graduate course. Previous exposure to Hamiltonian mechanics or dynamical systems is recommended but not required.


Terng Lecture Course: The Best Kind of Symmetry: From Archimedes' Hat-Box to Toric Manifolds 
Lecturer: Ana Cannas, ETH Zurich
Teaching Assistants: Pazit Haim-Kislev and Luya Wang, IAS

Abstract: Archimedes discovered — without calculus and without coordinates — that the area of a sphere between two parallel planes depends only on the distance between the planes, not on where they slice the sphere. This is his "hat-box theorem": band for band, the sphere has exactly the same surface area as the cylinder that snugly contains it.

In the 1980s, this fact was recognized as a small shadow of a much larger phenomenon. Take a 2n-dimensional manifold equipped with an area-like structure (called a symplectic form), and enough rotational symmetry (called a torus action; think of n independent circles' worth of spinning at once). In the best scenario, most of the manifold's geometry collapses onto a simple n-dimensional convex polytope. The sphere, spinning around its north–south axis, is the first case, collapsing onto its north-south height segment. Those "best" manifolds are called symplectic toric manifolds. They sit at the crossroads of symplectic geometry, algebraic geometry, and mathematical physics, and are a favorite testing ground for ideas in all three — precise enough to compute with, rich enough to be interesting.

In this course, we will construct symplectic toric manifolds essentially from scratch — starting from a polytope and a complex vector space with a linear action — and get a first look at the remarkable dictionary between their geometry and simple combinatorial data. Guided exercises will complement the exposition. No prior exposure to symplectic geometry is assumed.

Prerequisites: Differential forms and vector fields on smooth manifolds, complex numbers and complex vector spaces, and basic point-set topology (proper maps, quotient topology). Comfort with exterior derivative, Lie derivative, and Cartan's magic formula; elementary knowledge of Lie group actions or other group actions; and some exposure to bundles beyond tangent and cotangent bundles would be helpful.