Plan of Talks

1. Symplectic and Contact definitions [Vinicius Gripp] notes
   
   • Symplectic, contact, Liouville and Weinstein manifolds, cylindrical ends, cobordisms.
   • Eliashberg, Mishachev: Introduction to the h-principle, Chapters 9 - 12
   • Geiges: Contact Geometry, in: Handbook of Differential Geometry vol. 2 (F.J.E. Dillen and L.C.A. Verstraelen, eds.), North-Holland, Amsterdam (2006), 315-382.
2. Neighborhood theorems [Yang Huang] notes
   
   • Eliashberg, Mishachev; Introduction to the h-principle, Chapters 9 - 12
   • Geiges: Contact Geometry, in: Handbook of Differential Geometry vol. 2 (F.J.E. Dillen and L.C.A. Verstraelen, eds.), North-Holland, Amsterdam (2006), 315-382.
3. Moser's trick and h-principles [Luis Diogo] notes from first talk, notes from second talk
   
   • Eliashberg, Mishachev; Introduction to the h-principle, Chapters 9 - 12
4. Holomorphic curves definitions [Andre Carneiro]
   
   • Definitions of moduli spaces of (pseudo-)holomorphic curves with various features (genus, marked points, boundary, ends on the target) in manifolds with cylindrical ends, especially those that appear in the definitions of symplectic and contact homology
   • Discussion of Fredholm theory, with perhaps the simplest example of an index formula (leaving most of the results to be discussed in Talk 5)
   • Eliashberg, Givental, Hofer, Introduction to Symplectic Field Theory, GAFA 2000 link
   • Salamon, Lectures on Floer homology, in Symplectic Geometry and Topology (Park City, Utah, 1997), IAS/Park City Math. Ser. 7, Amer. Math. Soc., Providence, 1999, 143 - 229 link

Day 2: Tuesday 8/31

1. Gromov compactness in compact manifolds [Sushmita Venugopalan] notes
   
   • Compactness for J-holomorphic curves without boundary in compact symplectic manifolds.
   • Chapter 4 of D. McDuff and D. Salamon: J -holomorphic curves and symplectic topology, volume 52 of American Mathematical Society Col loquium Publications. American Mathematical Society, Providence, RI, 2004.
2. Gromov compactness in manifolds with cylindrical ends [Tim Nguyen] notes
   
   • Focus on Gromov compactness for J-holomorphic curves without boundary in manifolds in cylindrical ends - leave the case of curves with boundary for the next talk.
   • Bourgeois, Eliashberg, Hofer, Wysocki, Zehnder, Compactness results in Symplectic Field Theory, Geometry and Topology 7 (2003) link
3. Boundary degenerations [David Farris] notes
   
   • Gromov compactness for J-holomorphic curves with Lagrangian boundary conditions
   • String-type degenerations that appear in the boundary of moduli spaces of such curves
   • Cieliebak, Ekholm, Latchev, Compactness for holomorphic curves with switching Lagrangian boundary conditions, arxiv link
   • Ekholm, Rational Symplectic Field Theory over Z₂ for exact Lagrangian cobordisms, JEMS (2009) link
4. Dimension formulae [Vera Vertesi] notes
   
   • Maslov and Conley-Zehnder indices, relationship with Fredholm index via spectral flow.
   • General dimension formula for curves with interior and boundary punctures and Lagrangian intersection points.
   • D. McDuff and D. Salamon: J -holomorphic curves and symplectic topology, volume 52 of American Mathematical Society Col loquium Publications. American Mathematical Society, Providence, RI, 2004.
   • Eliashberg, Givental, Hofer, Introduction to Symplectic Field Theory, GAFA 2000 link
   • Cieliebak, Ekholm, Latchev, Compactness for holomorphic curves with switching Lagrangian boundary conditions, arxiv link

Day 3: Wednesday 9/1

1. Floer homology basics [Ramon Horvath] notes
   
   • Basics of Hamiltonian Floer Homology, with a view to defining Symplectic Homology tomorrow.
   • Hutchings, Lecture notes on Morse homology (with an eye towards Floer theory and pseudoholomorphic curves), 2002 link
   • Salamon, Lectures on Floer homology, in Symplectic Geometry and Topology (Park City, Utah, 1997), IAS/Park City Math. Ser. 7, Amer. Math. Soc., Providence, 1999, 143 - 229 link
2. Contact homology [Russell Advek] notes
   
   • Bourgeois, Introduction to Contact Homology. (lecture notes) link
3. Symplectic Field Theory [David Duncan, Tobias Ekholm] notes on Legendrian Contact Homology, notes on invariance
   
   • Prof. Ekholm will talk about the proof of invariance of contact homology.
   • Eliashberg, Givental, Hofer, Introduction to Symplectic Field Theory, GAFA 2000 link
4. Legendrian knots and other examples [Albin Ostman] notes
   
   • Chekanov, Differential algebra of Legendrian links, Invent. Math. 150 (2002) link
   • Other basic examples, e.g., the contact homology of spheres.
5. Orientations [Georgios Rizell] notes
   
   • Ekholm, Etnyre, Sullivan, Orientations in Legendrian contact homology and exact Lagrangian immersions, Internat. J. Math. 16 (2005) link

Day 4: Thursday 9/2

1. Symplectic homology basics [David Jackson-Hanen] notes
   
   • Definition of Symplectic Homology as a limit of Hamiltonian Floer Homology groups.
   • Seidel, A biased view of symplectic homology, Current Developments in Mathematics Volume 2006 (2008) link
   • Bourgeois, Oancea, An exact sequence for contact- and symplectic homology. Inventiones 2009 link
2. Morse-Bott degenerations [Nick Sheridan] notes
   
   • Basic (finite-dimensional) Morse-Bott techniques
   • Morse-Bott descriptions of holomorphic curves with several level curves connected by Morse flows
   • Morse-Bott definition of Symplectic Homology for time independent Hamiltonians
3. Further structure in symplectic homology, product and BV-operator [Sheel Ganatra] notes
   
   • Cieliebak, Latschev, The role of string topology in symplectic field theory link
4. Morse-Bott description of symplectic homology [Keon Choi] notes
   
   • Using the Morse-Bott description of Symplectic Homology (from previous talk) to relate it with Contact Homology
   • Bourgeois, Oancea, An exact sequence for contact- and symplectic homology. Inventiones 2009 link

Day 5: Friday 9/3

1. Legendrian surgery, symplectic and contact homology [James Pascaleff, Maksim Maydanskiy] notes from Max, notes from James
   
   • Bourgeois, Ekholm, Eliashberg, Effect of Legendrian surgery, arxiv link
2. Legendrian surgery and the product in symplectic homology [Tobias Ekholm] notes