SPEAKERS & ABSTRACTS
Ivan Contreras (Amherst College)
Title: “Symplectic and Poisson Geometry in Classical Field Theory”
Abstract: Lagrangian field theories have been used to describe classical physics, inspired by the Lagrangian formulation of classical mechanics. On the other hand, symplectic and Poisson geometry provide a language for Lagrangian and Hamiltonian mechanics, which is the groundwork of classical field theory. In this mini-course we will introduce some of the basic concepts and constructions in symplectic and Poisson geometry which appear naturally in field theory, such as Darboux coordinates, Moser’s trick, Weinstein’s tubular neighborhood theorem, symplectic reduction, symplectic foliation of Poisson manifolds, among others.
Following Weinstein’s motto: “Everything is Lagrangian”, we will pay close attention to Lagrangian submanifolds and their role in classical field theory. We will discuss specific examples, including classical mechanics, scalar field theory and the Poisson sigma model, a 2-dimensional field theory of interest to both mathematicians and physicists. This mini-course is intended for undergraduate students and therefore no prior exposure to classical field theory or differential geometry is expected.
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Juanita Pinzón Caicedo (University of Notre Dame)
Title: “Knots and Surfaces”
Contents:
- Knots and Seifert surfaces,
- Knot invariants using Seifert surfaces
- Surfaces in 4D and group structure
- inequalities that can be used to compute the three-genus of knots
Abstract: Knot theory is the subarea of topology that studies mathematical knots, that is, embeddings of the circle S^1 into 3-dimensional space. Proving that two knots are distinct (or equivalent) is the main problem knot theorists deal with. Using the help of surfaces and their classification, it is possible to a) define knot invariants that detect differences, and b) define an equivalence relation on the set of knots that turns it into a group. In this mini-course we will discuss methods used to distinguish knots, mostly from the perspective of the surfaces that the knots bound.
Bruce Bartlett (Stellenbosch University)
Title: “A crash course in geometric quantization for undergrads”
Abstract: Kahler geometric quantization is the mathematical procedure of producing a quantum-mechanical Hilbert space from underlying classical-mechanical data, namely a Hermitian line bundle with connection over a compact complex manifold. Moreover, it comes with a canonical way to turn geometric data (points, or Bohr-Sommerfeld loops, in the classical phase space) into vectors in the quantum Hilbert space. In this course, we will explain how all of this works in the setting of simple examples (the plane, the sphere, the torus). We will review Riemann surfaces and then cover line bundles and connections, holomorphic line bundles, their spaces of sections, the Chern connection, coherent states, and coherent loop states.