West Coast Pre-Workshop

A pre-workshop on the West Coast was held on May 5 at Stanford University. Notes, courtesy of Cédric De Groote are available: just click the speaker’s name.

1) [Maxim Jeffs (Berkeley) – Floer theory and symplectic topology]  An introduction to pseudo-holomorphic curves, as well as grading and the signs in Floer theory. This will be important for our workshop. A sample application could be versions of the PSS isomorphism (e.g. HF(L,L) = H(L) for a compact exact Lagrangian). Define symplectic cohomology.
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2) [Cédric De Groote (Stanford) – Introduction to Fukaya categoriesWe will define the Fukaya category of a symplectic manifold, talk about the wrapping, and introduce various algebraic notions that revolve around it: A_\infty structures, twisted complexes and the generation criterion.
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3) [Laurent Côté (Stanford) – Overview of the progress on the nearby Lagrangian conjectureA general overview of the nearby Lagrangian conjecture, what’s been done so far, with all the names, and maybe a run down of the special cases. Maybe a very brief sketch of Fukaya-Seidel-Smith here, and possibly a nod to the micro-local sheaf and Lefschetz fibration parts, and sketch the category theoretic part.
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4) [Ipsita Datta (Stanford) – Spectral sequences An introduction to spectral sequences: how they work, the Serre spectral sequence, and examples of applications such as the one coming from the filtration of a complex. (https://math.berkeley.edu/~hutching/teach/215b-2011/ss.pdf) Include the example of the homology of the loop space of spheres and/or the Hopf fibration over complex projective spaces. (https://en.wikipedia.org/wiki/Serre_spectral_sequence#Example_computations)
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5) [No talk, but notes by Catherine Cannizzo (Berkeley) – Local Systems] Define local systems both in terms of locally constant sheaves and in terms of representations of the fundamental groupoid of a space; explain the correspondence between both (this is the important bit). Explain what is homology with coefficients in a local system: homology of a cover of the space with an action by deck transformations (chapter 5 of http://www.indiana.edu/~jfdavis/teaching/m623/book.pdf). State Poincaré duality for non-orientable manifolds, i.e. with coefficients in the orientation local system. State how homology pertaining to the locally constant sheaf is the same as the topological description of homology of the space with coefficients in the corresponding local system. If there remains time, explain how one can incorporate local systems into Lagrangian Floer homology (remark 2.11 in https://arxiv.org/pdf/1301.7056.pdf) (or section 3 of Fukaya-Seidel-Smith’s “The symplectic geometry of cotangent bundles from a categorical viewpoint”).

 

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