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I am interested in learning Mirror Symmetry, both from the SYZ and Homological point of view. I am taking a reading course in Mirror Symmetry, which will focus on the SYZ side. I know basic Complex geometry, Kahler manifolds, Symplectic manifolds in the geometric side and also reading some material for my course on SYZ conjecture. My major concern is Homological side, about which I have little knowledge.

I am seeking a list of good references for SYZ conjecture, Homological Mirror Symmetry, physics of the theory, modern developments and on its relation to other areas of mathematics and some original papers (preferably in Chronological order).

What are your views about the Claire Voisin's book on Mirror Symmetry. And what is the present status of research in Mirror Symmetry, I mean what type of problems are people working on.

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  • $\begingroup$ Seidel has a book on Fukaya categories. $\endgroup$ – S. Carnahan Sep 27 '10 at 1:45
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    $\begingroup$ I recommend reading Kontsevich's original paper "Homological algebra of mirror symmetry". $\endgroup$ – Kevin H. Lin Sep 27 '10 at 16:37
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Auroux's notes for a course on mirror symmetry at Berkeley.

They look interesting and they cover a lot of material.

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  • $\begingroup$ thanks for the reference, this,I believe, will be immensely helpful. $\endgroup$ – J Verma Sep 27 '10 at 19:52
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Can't really say anything about the physics ... :-)

But for Kontsevich's HMS conjecture, my personal (very biased) all-time-favorite list is:

  • Two great survey papers to start with are: "A beginner's introduction to Fukaya categories" and "A symplectic prolegomenon".
  • Following that, there are Denis's notes from a graduate class he taught two times. Notes can be found in: 18.969 (MIT)/Math 277 (Berkeley).
  • Another great source is Nick Sheridan's lectures at IAS and Jussieu summer school. They are extremely clear and also include some overview of the operadic background.
  • James Pascaleff's class M392C (UT Austin) covers a lot of the basic aspects of Lagrangian Floer (co)homology (which is a neccesary component in defining the Fukaya category) beyond the usual 1-hour lecture. He also connects the actual theory with the standard heurestics ("Floer theory is Morse theory for the action functional!").
  • Sheel's graduate class Math 257B has a really nice summary of the analysis and algebra needed to define a Fukaya category with an eye toward LG-models
  • Seidel's notes from a topics class on equivariant mirror symmetry ("Lectures on Categorical Dynamics and Symplectic Topology") include a wealth of material on almost every subject you can imagine from Derived Picard groups, via Hochschild homology and cohomology, and to the actual setup of a Fukaya category for the surface case ... all in modular, concise (and dense) lectures.
  • Hiro Lee Tanaka taught a graduate class on Fukaya categories called: Fukaya Categories, Sheaves, and Cosheaves. It includes some nice examples of HMS, and also discusses the bigger abstract framework (like TFT's, $(\infty,1)$-categories, CY-categories,...)
  • For SYZ (besides Auroux's class), Siu Cheung Liu taught a couple of classes on mirror symmetry for an SYZ perspective a couple of years ago. The notes are on his website.
  • A really nice book on derived categories is "Fourier-Mukai transforms in Algebraic Geometry". Sheel Gantara's thesis (more generally, the "Preliminaries" section in all the papers of Gantara-Perutz-Sheridan) provide a lot of background and references on $A_\infty$-algebra and noncommutative geometry with a mirror symmetry application in mind.

After that, one should probably start reading research papers (e.g. Paul's early papers like "A long exact sequence for symplectic Floer cohomology" and "Graded Lagrangian submanifolds" are a good source for the nuts and bolts of Floer theory, culminating in the definition of the Fukaya category in the exact setting).

To understand the Fukaya category for a closed, non-monotone, symplectic manfold you will probably need virtual perturbation techniques. But this is a different story altogether ....

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For an understanding of the SYZ conjecture it is necessary to understand the framework of D-branes. I believe that a very good introduction, still unmatched in its comprehensiveness, is the book by Hori et al. on "Mirror Symmetry". This book has been made freely available by the Clay Math Institute and can be downloaded from their website http://www.claymath.org/library/monographs/cmim01.pdf. [ or from http://www.worldcat.org/title/mirror-symmetry/oclc/491393219 ] A more recent reference is the book by Joyce on "Riemannian Holonomy Groups and Calibrated Geometry", which contains a discussion of the SYZ conjecture from a more rigorous point of view.

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The book "Dirichlet Branes and Mirror Symmetry" by Aspinwall et. al. seems to fit quite well with your request. It discusses SYZ, Homological Mirror Symmetry and its physical origin.

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My master's thesis, An Introduction to Homological Mirror Symmetry and the Case of Elliptic Curves, might provide a piece of what you're looking for. It mostly concerns itself with the symplectic side of HMS (cause I have only a very superficial knowledge of algebraic geometry), but it includes a good amount background and some history. I tried to make it a useful document for other beginners to read... though then I let it rot on JSTOR for two years before posting it on arXiv :)
Oh well, it's up there now. I hope it helps!

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Homological Mirror Symmetry : New Developments and Perspectives by A.Kapustin,M.Kreuzer,K.G.Schlesinger (Eds.) may be useful to you.

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