Monday, 25 December 2017
Discuss the $A_\infty$ operad in spaces. Explain how the based loop space is an $A_\infty$ algebra. Derive from this the corresponding statement for the space of based discs.
I will briefly survey the basic notions in contact geometry. I will focus on Legendrian submanifolds of contact manifolds and neighborhood theorems that will be used in other talks.
In this lecture I shall explain the notion of h-principle for differential relations and then discuss Gromov's theorem for open, invariant relations on open manifolds.
Tuesday, 26 December 2017
Introduce the basics of Lagrangian Floer theory (holomorphic discs, compactification, orientation). Describe curved $A_\infty$ algebras, and explain how Floer theory equips a twisted homology group on the loop space with a deformed product.
Will derive h-principle of Symplectic and Contact forms on open manifolds from Gromov's theorem. . Also, discuss Eliashberg's theorem on classification of overtwisted contact structures on 3-manifolds.
I will cover the basics of Legendrian and transverse knot theory. This will set up the background for Michael Sullivan's talk on knot contact homology.
Give a brief overview of spectra, focusing on the Thom spectrum. Explain how the Thom spectrum on an H-space acquires a multiplicative structure if its classifying map is also multiplicative.
Wednesday, 27 December 2017
In these lectures I will give several different, but equivalent, approaches to defining the Legendrian contact homology of a Legendrian submanifold inside of a contact manifold. I begin with the original Chekanov-Eliashberg combinatorial definition for 1-dimensional Legendrian knots in standard contact $R^3$. Then I sketch the general definition which uses pseudo-holomorphic curves, and which itself is a part of the more general symplectic field theory introduced by Eliashberg-Hofer-Givental. In the case of certain 2-dimensional Legendrian surfaces, I discuss how this version recovers the combinatorial/topological knot contact homology invariants of smooth 1-dimensional knots in $R^3$ introduced by Ng. The final approach applies to a general 2-dimensional Legendrian surfaces, whereby I discuss its cellular contact homology. This last version is combinatorial with an algebraic topology flavor. Along the way and depending on time, applications, and connections of Legendrian contact homology to other areas will be discussed. These can include generating families, augmentation varieties, and sheaves.
ArXiv references:
math/9709233 Differential algebras of Legendrian links. Yuri Chekanov [offers the original definition of Legendrian contact homology for Legendrian knots.] arXiv:1210.4803 A topological introduction to knot contact homology. Lenhard Ng [provides an (brief) overview of the holomorphic curve definition of Legendrain contact homology, as well as the alternative topological formulations of knot contact homology.] arXiv:1608.02984 Cellular Legendrian contact homology for surfaces, part I. Dan Rutherford, Michael G Sullivan
[defines the combinatorial reformulation for Legendrian surfaces.]
Related topics/applications that I may cover if time presents itself:
arXiv:1703.04656 Generating families and augmentations for Legendrian surfaces. Dan Rutherford, Michael G Sullivan
Dan Rutherford, Michael G Sullivan
[connects generating families and augmentations for Legendrian surfaces.]
arXiv: 1502.04939 Augmentations are Sheaves. Lenhard Ng, Dan Rutherford, Vivek
Shende, Steven Sivek, Eric Zaslow
[connects augmentations and sheaves for Legendrian knots.]
The following 3 papers can be skipped but offer more details behind "A topological introduction to knot contact homology" for the adventuresome reader. For more details on the holomorphic version of Legendrian contact homology, see Part 1 of math/0210124 Legendrian Submanifolds in $R^{2n+1}$ and Contact Homology. Tobias Ekholm, John Etnyre, Michael G. Sullivan.
For more details on the topological/combinatorial approach to knot contact homology, as well as the augmentation variety see math/0407071 Framed knot contact homology. Lenhard Ng. For a proof of why the topological/combinatorial version is equivalent to the holomorphic version, see arXiv:1109.1542 Knot contact homology. Tobias Ekholm, John Etnyre, Lenhard Ng, Michael Sullivan.
In this lecture I will discuss an extension of Gromov's theorem to prove that isocontact and isosymplectic immersions satisfy the h-principle. Also consider the embedding problems briefly.
We will give a short introduction to $A_\infty$ spaces leading to $A_\infty$ algebras and the underlying combinatorial polytopes that given such operations. The homotopy Lie version will also be discussed. Time permitting, we will talk about $A_\infty$ operads.
Introduce the index bundle of the dbar operator as a stable bundle on the space of based discs. Formulate the notion of "curved $A_\infty$ operad." Explain how Floer theory makes the Thom spectrum of the index bundle into a curved $A_\infty$.
Thursday, 28 December 2017
Topological strings are simplified string theories which are often solvable. Yet they encapsulate many of the novel features of string theory such as mirror symmetry, strong-weak dualities and gauge-string dualities. I will try to give a flavour of what topological strings are in the first lecture and how they give rise to more and more surprising results in enumerative geometry as we climb up the duality ladder. In the second lecture I shall try to explain why open-closed string dualities might be the most mysterious of such dualities and perhaps yield the most profound mathematical insights - with wide ramifications.
We will give a brief introduction to moduli space of stable maps and use it to define Gromov-Witten invariants. If time permits we will discuss examples from enumerative geometry.
Rack invariant of topological knots (up to isotopy) in 3-sphere is known to be a complete invariant up to mirror image. Hence, it turned out to be useful theoretically. In recent times computer assisted theorem provers have been making progress towards applications of racks in practice to distinguish knots topologically.
In this talk, I will describe the challenges in developing rack theoretic invariants for Legendrian knots. I will mention some recent results obtained in a joint work with Dr. Prathamesh along the same direction. At the end, I will pose a few relevant questions which may be worth exploring in the near future.
Friday, 29 December 2017
We will give a brief introduction to moduli space of stable maps and use it to define Gromov-Witten invariants. If time permits we will discuss examples from enumerative geometry.
This will be a brief introduction to the moduli space of smooth algebraic curves and their Deligne-Mumford compactification. We shall also see connections with the moduli of stable maps and Gromov-Witten theory.
Saturday, 30 December 2017
As prerequisites, virtual fundamental cycles and equivariant localization will be important.
Monday, 01 January 2018
Floer defined Lagrangian Floer cohomology groups in order to tackle the Arnold conjecture regarding the number of intersection points of Lagrangians. The initial definition required restrictive hypothesis in order to rule out disk bubbling and sphere bubbling. When disk bubbling is allowed the differential no longer squares to zero, however it turns out that the resulting structure is that of an A-infinity algebra. On the other hand sphere bubbling makes it hard to achieve 'transversality' of moduli spaces. In this introductory talk, I describe how the Fukaya algebra of a Lagrangian, and the Fukaya category of a collection of Lagrangians is defined using counts of curves. I will state a theorem of Fukaya and collaborators about Lagrangians with non-trivial Floer homology in symplectic toric manifolds.
We present a construction of a kind of Fukaya "category" of a Hamiltonian fibration. This leads to some new Floer theoretic invariants not just in symplectic topology but also for classical smooth manifolds. And which may indeed be considered a kind of "quantization" of the theory of Pontryagin classes, leading to some mysterious problems. There are other interesting connections with modern algebraic topology that we shall touch on, like Lurie's theory of infinity categories.
Tuesday, 02 January 2018
This lecture series will start out as a general introduction to the Gromov-Witten invariants of symplectic manifolds and proceed to explain some recent progress on fundamental transversality questions that arise in this context. The basic technical problem is well known: while Gromov-Witten theory attempts to define intersection numbers between moduli spaces of holomorphic curves and cycles in their target spaces, the moduli spaces are only smooth objects of the "correct" dimension if multiply covered curves can be eliminated. The standard solution (going back to work of Ruan and Tian in the mid 1990's) is to introduce inhomogeneous perturbations of the Cauchy-Riemann equation so that all symmetry is broken and transversality is achieved, which works well enough to define the invariants on semipositive symplectic manifolds (in particular for all symplectic manifolds of dimension at most 6). But there are situations in which breaking the symmetry destroys interesting information: one sees this for example in the definition of Taubes's Gromov invariant for symplectic 4-manifolds (which includes a count of multiple covers), and in the geometric motivation for the Gopakumar-Vafa formula on Calabi-Yau 3-folds as a relation between counts of embedded curves and their "multiple cover contributions". The overarching goal of these lectures will be to provide an answer to the following question: "When transversality fails for a multiple cover, what is true instead?" The answer is sometimes that transversality holds in spite of symmetry, or that there is a well-defined obstruction bundle whose Euler class determines everything we want to know, thus producing a complete "localization" of the Gromov-Witten invariants. To understand this, we will talk a bit about the analysis of Cauchy-Riemann type operators in general and then, for the case of operators defined by holomorphic curves with symmetries, introduce a splitting of such operators determined by the irreducible representations of the symmetry group. The main technical result is that this splitting determines a stratification of the moduli space of multiply covered curves, which can then be used to prove transversality (or alternatively, "super-rigidity") results using dimension-counting arguments. If time permits, we will also discuss the outlook for developing a general bifurcation theory of multiply covered holomorphic curves under generic 1-parameter deformations.
First lecture details
Moduli spaces of closed J-holomorphic curves, main idea of the Gromov-Witten invariants, index formula, Gromov compactness, definition of the invariants for semipositive symplectic manifolds, notions of transversality and super-rigidity.
References:
The primary reference for this material is the recent paper https://arxiv.org/abs/1609.09867. For background material on closed J-holomorphic curves and Cauchy-Riemann operators, my lecture notes (Link) or the standard book by McDuff and Salamon should suffice.
Wednesday, 03 January 2018
Consequences of super-rigidity (multiple cover contributions and obstruction bundle calculations), quick review of the Sard-Smale theorem and transversality proof for simple holomorphic curves, properties of Cauchy-Riemann type operators, stratification and proofs of transversality and super-rigidity for degree 2 covers.
Generalized symmetry groups of branched covers, twisted Cauchy-Riemann operators and their indices, general stratification theorem and how it implies a general transversality and super-rigidity results.
Thursday, 04 January 2018
Quadratic unique continuation lemma, Cauchy-Riemann operators with invertible anti-linear part, proof of the stratification theorem, remarks on wall-crossing and bifurcation theory.