Generalized periods and mirror symmetry in dimensions n>3

The predictions of the Mirror Symmetry are extended in dimensions n>3 and are proven for projective complete intersections Calabi-Yau varieties. Precisely, we prove that the total collection of rational Gromov-Witten invariants of such variety can be expressed in terms of certain invariants of a new generalization of variation of Hodge structures attached to the dual variety. To formulate the general principles of Mirror Symmetry in arbitrary dimension it is necessary to introduce the ``extended moduli space of complex structures'' M. An analog M\to H*(X,C)[n] of the classical period map is described and is shown to be a local isomorphism. The invariants of the generalized variations of Hodge structures are introduced. It is proven that their generating function satisfies the system of WDVV-equations exactly as in the case of Gromov-Witten invariants. The basic technical tool utilized is the Deformation theory.Comment: 51 pages, LaTe

Frobenius Manifolds and Formality of Lie Algebras of Polyvector Fields

We construct a generalization of the variations of Hodge structures on Calabi-Yau manifolds. It gives a Mirror partner for the theory of genus=0 Gromov-Witten invariantsComment: 12 pages, AMS-TeX; typos and a sign corrected, appendix added. Submitted to IMR

Solving the noncommutative Batalin-Vilkovisky equation

I show that a summation over ribbon graphs with legs gives the construction of the solutions to the noncommutative Batalin-Vilkovisky equation, including the equivariant version. This generalizes the known construction of A-infinity algebra via summation over ribbon trees. These solutions give naturally the supersymmetric matrix action functionals, which are the gl(N)-equivariantly closed differential forms on the matrix spaces, which were introduced in one of my previous papers "Noncommmutative Batalin-Vilkovisky geometry and Matrix integrals" (arXiv:0912.5484, electronic CNRS preprint hal-00102085(28/09/2006)).Comment: 17 pages, electronic CNRS preprint hal-00464794 (17/03/2010

Invariants of Morse complexes, persistent homology and applications.

International audienceThe algorithm for calculation of "canonical form" = "persistence barcodes/diagrams" invariants from S.Barannikov "The Framed Morse complex and its invariants" Adv. in Sov. Math., vol 21, AMS transl, (1994), is described