Courses

Course Description: The aim of these talks is to give a quick overview of knot theory. We will review some of the classical knot invariants, Seifert genus, Alexander polynomial, smooth-4-ball genus and the Jones polynomial. In later lectures we will discuss knot homologies, including Khovanov homology and knot Floer homology. We will also discuss topological applications of these invariants, and some open problems.

References for knot theory:


• D. Rolfsen. Knots and Links.

• R. Lickorish. An introduction to knot theory.


References for Knot Floer homology:

• P Ozsváth and Z. Szabó. An overview of knot Floer homology

Reading List:

• Dror Bar-Natan: On Khovanov's categorification of the Jones polynomial.

• Raymond Lickorish: An introduction to knot theory.

• Jacob Rasmussen: Knot polynomials and knot homologies.

• Peter Ozsvath and Zoltan Szabo: An overview of knot Floer homology.

Course Description: Heegaard Floer homology is a tool for studying low-dimensional manifolds, using ideas inspired by symplectic geometry. This 3-lecture semi-mini-course will serve as an introduction to these ideas. A loose lecture plan will be to:

• Describe the structure of the theory.

• Give some motivation for the introduction of the invariants.

• Describe variations of the construction, and its relationship with other invariants.

Time permitting, I will also describe constructions in grid homology, where holomorphic aspects of theory can be described in purely combinatorial terms.

Suggested reading:

Topology background:    

• J. Milnor. Topology from a differentiable point of view, for a rapid and very elegant introduction to differential topology.     

• Milnor. Morse Theory, for further background in topology.     

• R. Bott and L. P. Tu. Differential forms in Algebraic Topology for further reading in topology.

Symplectic Geometry background:     

• M. Audin and M. Damian. Morse theory and Floer homology

Heegaard Floer homology surveys:     

• P Ozsváth and Z. Szabó. An introduction to Heegaard Floer homology (lecture notes from another lecture series similar to this one)
     

• P Ozsváth and Z. Szabó. An overview of knot Floer homology

Course Description: We will discuss ways to construct and view surfaces in 4-dimensional manifolds with emphasis on the case of 2-spheres in the 4-sphere.

Suggested Reading: 

• S. Carter, S. Kamada, & M. Saito, "Surfaces in 4-space"

• D. Gabai, Self-referential discs and the light bulb lemma 

• M. Hughes, S. Kim, M. Miller, "Isotopies of surfaces in 4-manifolds via banded unlink diagrams" 

• R. Litherland, "Deforming twist-spun knots"

• J. Meier & A. Zupan, “Bridge trisections of knotted surfaces in S4” 

• F. Swenton, “On a calculus for 2-knots and surfaces in 4-space” 

• E. Zeeman, "Twisting spun knots"

Other course materials:

Slides from last lecture

Course Description: Moduli spaces of solutions to nonlinear elliptic pdes (anti-self-dual connections, monopoles, pseudo-holomorphic curves, etc.) are a fundamental tool in low-dimensional and symplectic topology.  I will discuss foundational aspects of moduli spaces of pseudo-holomorphic curves.  Topics may include Uhlenbeck/Gromov compactness, gluing, and derived (Kuranishi) structure.  I will also spend some time on purely topological topics such as orbifolds, smooth stacks, log smooth manifolds, derived smooth manifolds, and possibly infinity-categories.

Course Description: This mini-course will be an introduction to hyperbolic knot theory with a focus on the Volume Conjecture. We will discuss when hyperbolic structures exist for knot complements in S^3, how to construct them, and how to compute their volumes. In the second half of the course, we will discuss the colored Jones polynomial and how its asymptotics conjecturally compute the volume of a knot complement. In particular, we will give a proof that the conjecture holds of the figure-eight knot complement.

Suggested reading:

• J. Prucell. Hyperbolic Knot Theory (https://arxiv.org/pdf/2002.12652.pdf)

• R. Lickorish. An introduction to knot theory.

• Thang T. Le, The colored Jones polynomial and the AJ conjecture. (https://letu.math.gatech.edu/Papers/Lectures_Luminy_2014_new.pdf)

Course Description: In this course, we will learn the basics of the Manolescu-Ozsvath-Szabo surgery formulas. We will focus on building algebraic tools, such as understanding various versions of the "exact triangle detection lemma," in both the category of chain complexes and also in the Fukaya category. We will outline some aspects of the proof of their formula, highlighting basic examples. Additionally, we will learn some basics about A_infty modules and some interesting and fun tools, such as homological perturbation theory and also Koszul duality in some basic contexts. Time permitting we hope to discuss the link Floer homology of algebraic links and how it fits into this story.



Background reading: See background reading for Ozsváth’s mini-course on Heegaard Floer homology. This mini-course will assume the material covered in that course.

Course Description: Floer homology is an important construction defined for a pair of n-dimensional Lagrangian submanifolds of a symplectic (2n)-manifold. In the first half of this course, we will discuss Floer homology in the particularly simple case that n=1–that is, we discuss the Floer homology of curves in surfaces. In this case, Floer homology can be defined combinatorially, so we will avoid any need for the machinery from symplectic geometry, but we still get a taste of the rich structures that result. In the second half of the course, we will see how the Floer homology of curves in surfaces arises naturally when working with certain invariants of 3-dimensional manifolds. In particular, the Heegaard Floer homology of a manifold with torus boundary can be interpreted as an immersed curve in the punctured torus, and when two such manifolds are glued, the Heegaard Floer invariant of the resulting closed manifold is obtained by taking Floer homology of curves. We will see that perspective leads to simple geometric proofs of some nice results.

Suggested Reading:

• D. Auroux. A beginners introduction to Fukaya categories, A good introduction to Floer homology and Fukaya categories, assumes basic knowledge of symplectic geometry 

• M. Abouzaid. On the Fukaya categories of higher genus surfaces, Good description of Floer homology for curves in surfaces, described combinatorially (symplectic background is helpful but not essential)

• V. de Silva, J. Robbin, and S. Salamon. Combinatorial Floer homology. Fully combinatorial construction of Floer homology for embedded curves in surfaces, long and detailed but fully accessible without symplectic background.

See also suggested readings for the course on Heegaard Floer homology. For the later part of this mini-course, familiarity with the material covered in Ozsváths mini-course will be assumed.

Course Description: Symplectic geometry is a field in pure mathematics that arose from classical mechanics. The main objects of study are symplectic manifolds, which are even dimensional manifolds with an additional structure given by a certain differential 2-form. These manifolds admit an important class of dynamics, called Hamiltonian flows. Many questions in symplectic geometry concern the relation between such  dynamics and the topology or geometry of the manifold. In this course I will try to present some of the fundamental theorems and ideas in the field, and explain relations between dynamical and geometric aspects.

Prerequisites: Basic analysis on manifolds, differential forms, basic ODE. (Recommended but not necessary:  Riemannian geometry, basic algebraic topology, De Rham cohomology)

Suggested reading:
A. Canas da Silva, Lectures on Symplectic Geometry (https://people.math.ethz.ch/~acannas/Papers/lsg.pdf)