Beginning Lecture Course

Week 1

Title: Tangent Vectors and Twisting Planes: An Introduction to Legendrian Knot Theory

Lecturer: Joan Licata, Institute for Advanced Study

Teaching Assistant: Patricia Cahn

Course Description: A contact manifold is an odd-dimensional manifold equipped with some extra geometric structure. When the contact manifold is 3-dimensional, this structure distinguishes a special class of embedded circles, called Legendrian knots. In this course, we'll focus on the Legendrian knot theory associated to Euclidean R3 with the "standard" contact structure. We'll compare the classification of topological and Legendrian knots, paying particular attention to knot projections as a tool for computing invariants. After defining the classical invariants of Legendrian knots combinatorially, we'll interpret them geometrically in order to generalize these definitions to a broad range of contact manifolds.

Pre-requisites: The required background is advanced calculus or elementary analysis (embeddings, parameterized curves, continuity, vector fields). While not required, familiarity with differential forms, manifolds, or topology may enhance the course.

References: TBA

Week 2

Title: From Linear Algebra to the Non-Squeezing Theorem of Symplectic Geometry

Lecturer: Margaret Symington, Mercer University

Teaching Assistant: Joanna Nelson, University of Wisconsin

Course Description: Symplectic geometry is a geometry of even dimensional spaces in which area measurements, rather than length measurements, are the fundamental quantities. This course will introduce symplectic manifolds, starting with symplectic vector spaces and examples in R2n. It will culminate with the foundational non-squeezing theorem that reveals the ever-present tension between the rigid geometric and the soft topological features of symplectic manifolds.

Pre-requisites: Multivariable calculus and linear algebra are both prerequisites for this course. Advanced calculus or an exposure to some topology would be helpful but is not required.

References: Spivak's Calculus on Manifolds, especially Ch 4 and 5 (which is very concise)

Lee's Introduction to Smooth Manifolds, especially Ch 1, 3-5 and 8-9 (which is full of detail).

Week 1

Title: The Flexibility and Rigidity of Lagrangian and legendrian Submanifolds

Lecturer: Lisa Traynor, Bryn Mawr College

Teaching Assistant: Sheila Margherita Sandon

Course Description: Cotangent bundles and 1-jet bundles from classic examples of symplectic and contact manifolds. Working in these standard manifolds, we will examine the flexibility and rigidity of Lagrangian and Legendrian submanifolds. In particular, we will describe how one can define homology groups for Lagrangian and Legendrian submanifolds that can be used to detect rigidity. Our constructions will employ the finite-dimensional technique of generating families and will parallel the constructions of Lagrangian Floer and Legendrian Contact Homology defined through the infinite-dimensional technique of J-holomorphic curves. We will also discuss Lagrangian cobordisms between Legendrian submanifolds; this will involve defining a generating family version of wrapped Floer cohomology.

Pre-requisites: Elementary differential and algebraic topology

References: Introduction to Symplectic Topology, McDuff and Salamon

Mathematical Methods of Classical Mechanics, Arnold

Symplectic Invariants and Hamiltonian Dynamics, Hofer and Zehnder

Morse Theory, Milnor

Morse Homology, Matthias Schwarz

Week 2

Title: Symplectic Techniques: The Space of Holomorphic Curves

Lecturer: Eleny Ionel, Stanford University

Teaching Assistant: Penka Georgieva

Course Description: After discussing more examples and elementary constructions in symplectic geometry, we will turn our attention to the space of J-holomorphic curves, a widely used technique in symplectic and contact geometry. We will explore some of their features and the basic topology, geometry and analysis principles involved in the study of the moduli space of J-holomorphic curves. We will ilustrate some of their applications, for example in the construction of various theories like Gromov-Witten invariants or Floer homologies, and how they were used to prove "classical" results like Gromov's nonsqueezing theorem or the Arnold's conjectures.

Pre-requisites: Elementary differential and algebraic topology; some background in analysis and differential geometry would be useful.

References: TBA