​Abstracts and notes:
​
Talk 1: J-holomorphic curves, Index theory, and Elliptic regularity. 
Speaker: Jung Joo Suh
Abstract: The space of J-holomorphic curves can be seen as the zero set of the \bar{\partial}-operator. To understand this zero set, we will study the linearization D of the said operator in a Banach space setting. We will discuss why we need to extend our domain to a Sobolev space, how elliptic regularity ensures our solutions are smooth, and why the linearized operator D must be Fredholm. We will finish with the computation of the Fredholm index of D for the smooth genus=0 case. Time permitting, we will also compute the Fredholm index of D for the nodal case.
Notes for the Talk 1
 
Talk 2: Transversality for the moduli space of simple curves.
Speaker: Harahm Park
Abstract: A simple J-holomorphic curve is one which does not factor through a branched holomorphic cover of Riemann surfaces. We show for a generic choice of compatible almost complex structure on a symplectic manifold that the moduli space of genus 0 simple J-holomorphic curves in a given homology class is a transversally cut-out, smooth manifold.
Notes for the Talk 2
​
Talk 3: Compactness and stable curves.
Speaker: Advika Rajapakse
Abstract: Gromov's compactness theorem is a crucial theorem in symplectic geometry regarding sequences of J-holomorphic curves with bounded area. We will review hyperbolic surfaces and properties of pseudo-holomorphic curves before outlining the proof of Gromov's compactness theorem.
Notes for the Talk 3
​
Talk 4:  Gluing.
Speaker: Roman Krutowski
Abstract:  Last week, we saw that a sequence of regular J-holomorphic curves converges to a nodal J-holomorphic curve. In this talk, we will show that a stable regular nodal J-holomorphic curve with marked points can be obtained as such a limit, which requires "gluing" this curve at the nodes. Specifically, we show that the moduli space of regular genus 0 stable marked J-holomorphic curves is a topological manifold.
Notes for the Talk 4
​
Talk 5: Orbifolds as groupoids.
Speaker:  David Popović
Abstract: We will introduce orbifolds from two different perspectives - first classically as topological objects and then in a more modern way in the language of groupoids.
Notes for the Talk 5
 
Talk 6: Global Kuranishi charts and virtual fundamental class.
Speaker: Victoria Quijano
Abstract: A virtual fundamental class is a choice of homology class for a topological space which can be viewed as a generalization of the notion of a fundamental class for a closed, connected, orientable manifold. We discuss the notion of a virtual fundamental class on a compact Hausdorff space given an implicit atlas. We then define global Kuranishi charts (GKCs) and the definition of a virtual fundamental class for a Hausdorff space with a given GKC before discussing the proof of invariance of this definition under equivalence of GKCs.
Notes for the Talk 6

Talk 7: Moduli of curves in projective space and construction of global Kuranishi chart for Gromov-Witten theory.
Speaker: Yan Tao
Abstract: A fundamental object of study in Gromov-Witten theory is the moduli space of J-holomorphic maps representing a certain homology class. We sketch the construction of a global Kuranishi chart for the moduli space of genus 0 J-holomorphic maps representing a homology class via framed curves. We then (time permitting) discuss how this construction extends to the full moduli space and some applications of the global Kuranishi chart.
Notes for the Talk 7

Talk 8: Hörmander peak sections.
Speaker: Benjamin Slater​
Abstract: We will discuss Hörmander’s theorem, an existence theorem which plays an essential role in the use of the Gromov trick (as we will see next week). We will see how to interpret this theorem as a statement about a certain cohomology theory (Dolbeault cohomology), and we will sketch the proof, which uses tools from functional analysis. Time permitting, we will explain how Hörmander’s theorem is used, in the context of Abouzaid-McLean-Smith, to construct pairs of sequences of holomorphic sections of certain vector bundles which vanish at a set of marked points and whose inner products converge to Dirac distributions.
Notes for the Talk 8

Talk 9: Transversality and gluing for global Kuranishi charts.
Speaker: Roman Krutowski​
Abstract: A virtual fundamental class is a choice of homology class for a topological space which can be viewed as a generalization of the notion of a fundamental class for a closed, connected, orientable manifold. We discuss the notion of a virtual fundamental class on a compact Hausdorff space given an implicit atlas. We then define global Kuranishi charts (GKCs) and the definition of a virtual fundamental class for a Hausdorff space with a given GKC before discussing the proof of invariance of this definition under equivalence of GKCs.
Notes for the Talk 9