| Time | Speaker | Title | Resources | |
|---|---|---|---|---|
| 09:30 to 11:00 | Michael G. Sullivan (University of Massachusetts, USA) |
Introduction to legendrian contact homology using pseudo-holomoprhic disks and Chekanov combinatorics 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: Related topics/applications that I may cover if time presents itself: arXiv: 1502.04939 Augmentations are Sheaves. Lenhard Ng, Dan Rutherford, Vivek 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. |
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| 11:00 to 11:30 | -- | Tea/Coffee | ||
| 11:30 to 13:00 | Mahuya Datta (ISIK, India) |
Isocontact and isosymplectic immersions and embeddings 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. |
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| 13:00 to 14:30 | -- | Lunch | ||
| 14:30 to 15:30 | Somnath Basu (IISERK, India) |
A quick review of infinity algebras 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. |
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| 15:30 to 16:00 | -- | Tea/Coffee | ||
| 16:00 to 17:30 | Mohammed Abouzaid (Columbia University, USA) |
Floer homotopy theory 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 $A_\infty$. |
| Time | Speaker | Title | Resources | |
|---|---|---|---|---|
| 09:30 to 11:00 | Yasha Savelyev (University of Colima, Mexico) | Fukaya category of a Hamiltonian fibration (Lecture 2) | ||
| 11:00 to 11:30 | -- | Tea/Coffee | ||
| 11:30 to 13:00 | Ezra Getzler (Northwestern University, USA) | Gromov–Witten Invariants and the Virasoro Conjecture. III | ||
| 13:00 to 14:30 | -- | Lunch | ||
| 14:30 to 15:30 | Sushmita Venugopalan (IMSc, India) | Lagrangian Floer theory (Lecture 2) | ||
| 15:30 to 16:00 | -- | Tea/Coffee | ||
| 16:00 to 17:30 | Janko Latschev (University of Hamburg, Germany) | Symplectic homology, algebraic operations on it and their applications (Lecture 2) | ||
| 18:15 to 19:15 | Chris Wendl (Humboldt University, Berlin) |
Transversality and super-rigidity in Gromov-Witten Theory 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: |