"The literature is rather chaotic and quite technical, so I won't assume or expect that people have read any of it. Just so that people have some preliminary frame of reference here is a list of papers that I may (will) touch upon:"

• "Mirror Symmetry" by Kentaro Hori and Cumrun Vafa
• "Mirror symmetry for weighted projective planes and their noncommutative deformations" by Denis Auroux, Ludmil Katzarkov, and Dmitri Orlov
• Lectures on dg-categories", Sedano winter school on K-theory , Sedano, January 2007
• "Cartier isomorphism and Hodge Theory in the non-commutative case" by Dmitry Kaledin
• "Hodge theoretic aspects of mirror symmetry" by Ludmil Katzarkov, Maxim Kontsevich, and Tony Pantev
• "Examples of non-commutative Hodge structures" by Claus Hertling and Claude Sabbah