Princeton Summer School in Low-dimensional Topology and Symplectic Geometry, June 11–29, 2018

A three-week intensive program (June 1129, 2018) at Princeton University for 25 advanced undergraduates and first year graduate students consisting of courses in low dimensional topology and symplectic geometry. Topics will include low-dimensional topology, gauge theory, and pseudo-holomorphic curves. There will be 6 courses, consisting of five lectures apiece. Daily lectures will be complemented by review and problem sessions.
There will also be two mini-conferences associated with the program, in weeks 2 and 3. 
  • Free Room and board for accepted attendees (on-campus) will be provided from Sunday, June 10 (check-in) through Saturday, June 30th (check-out).
  • Travel reimbursement will be provided up to $500. (Only original receipts will be accepted.)

 

Courses:

First Week Courses Schedule:
 • 4-dimensional knot theory (David Gabai) 

The theory of smooth 2-spheres in the 4-sphere is a beautiful and venerable subject.  In this mini-course we will introduce elements of the classical theory with emphases on examples and contstructions.

 • Seiberg-Witten theory on four-manifolds (Francesco Lin)

The goal of the minicourse is to define the Seiberg-Witten invariants of four-manifolds and discuss their topological applications. In the first part of the course will cover some background topics in differential geometry (Hodge theory, Chern-Weil theory, spin geometry); in the second part we will define the invariants and discuss several computations.

 • Bordered algebras and a bigraded knot invariant (Zoltán Szabó)

The aim of this mini-course is to present some recent advances in knot Floer homology. The main focus will be the use of algebraic techniques and in particular a version of bordered Floer homology. In the course we will discuss certain  algebraic methods (differential graded algebras, D-structures and curved bimodules, box tensor product) and use them to construct a Floer homology invariant for knots in the three-dimensional sphere.

 

Second and Third Week Course Schedule:
An introduction to monopole Floer homology (Francesco Lin)

We will start by motivating monopole Floer homology as a tool to compute the effect of some natural topological operations on the Seiberg-Witten invariants. Then, after discussing its construction in detail, we will focus on applications to three-dimensional topology and triangulations of topological high-dimensional manifolds.

Knot Floer homology and bordered invariants (Peter Ozsváth)

The course will start with a quick overview of Heegaard Floer homology, and its associated knot invariant. After describing this construction, we describe computational techniques, focusing on one inspired by "bordered Floer homology", proving that the invariant constructed in Szabo's course in fact agrees with a version of knot Floer homology.

Geometry and algebra of pseudo-holomorphic curves (John Pardon)

Pseudo-holomorphic curves were introduced into symplectic geometry by Gromov in 1985, and they quickly emerged as a (perhaps the) central tool in the field.  This course will start with a discussion of the moduli properties of pseudo-holomorphic curves (e.g. Gromov compactness, gluing, transversality) and will continue with some of their spectacular applications from Gromov's first paper on the subject (e.g.  Gromov non-squeezing, homotopy type of symplectomorphism groups of CP^2 and S^2xS^2).  If time permits, we will conclude with a brief preview of recent applications of pseudo-holomorphic curves.

Mini-Conferences:

Low-dimensional topology and its interactions with symplectic geometry
The first Summer 2018 Geometry/Topology RTG mini-conference
June 20-22, 2018

Symplectic geometry and its interactions with low-dimensional topology
Summer 2018 Geometry/Topology RTG mini-conference #2
June 28-29, 2018

The first of these mini conferences will emphasize a little more the low-dimensional topological component, and the second will focus more on the symplectic side.

The mini-conferences will focus on recent developments in knot theory, three- and four-dimensional topology, and interactions with gauge theory, symplectic geometry, and Floer homology; as well as fundamental flexibility and rigidity problems in symplectic geometry, and pseudo-holomorphic curve methods.

These mini conferences are supported by an NSF RTG grant.

For more information, visit the Geometry/Topology RTG mini conference website.

Eligibility:

• Open to currently enrolled undergraduate juniors and seniors and first-year graduate students.
• Must be a U.S. citizen or permanent resident.
• Applicants will be considered without regard to race, color, religion, sex, sexual orientation, age, ethnicity, or disability.

Application Requirements:

Complete and submit the online application that will include the following:
• One letter of recommendation from a faculty member well-acquainted with your academic work.
• An unofficial school transcript.
• Personal statement on why you want to attend the program.

Undergraduate Application

Graduate Application

 

Applications for 2018 CLOSED.

The program is sponsored by the National Science Foundation RTG (Research Training Group) in Geometry and Topology at Princeton.
For more information, please contact Jill LeClair, Program Coordinator, at leclair@princeton.edu