Homological Mirror Symmetry and Tropical Geometry

Ricardo Castano-Bernard, Fabrizio Catanese, Maxim Kontsevich, Tony Pantev, Yan Soibelman, Ilia Zharkov (Herausgeber)

Buch | Softcover

XI, 436 Seiten

2014 | 2014
Springer International Publishing (Verlag)
978-3-319-06513-7 (ISBN)

Artikel merken

The relationship between Tropical Geometry and Mirror Symmetry goes back to the work of Kontsevich and Y. Soibelman (2000), who applied methods of non-archimedean geometry (in particular, tropical curves) to Homological Mirror Symmetry. In combination with the subsequent work of Mikhalkin on the "tropical" approach to Gromov-Witten theory and the work of Gross and Siebert, Tropical Geometry has now become a powerful tool. Homological Mirror Symmetry is the area of mathematics concentrated around several categorical equivalences connecting symplectic and holomorphic (or algebraic) geometry. The central ideas first appeared in the work of Maxim Kontsevich (1993). Roughly speaking, the subject can be approached in two ways: either one uses Lagrangian torus fibrations of Calabi-Yau manifolds (the so-called Strominger-Yau-Zaslow picture, further developed by Kontsevich and Soibelman) or one uses Lefschetz fibrations of symplectic manifolds (suggested by Kontsevich and further developed by Seidel). Tropical Geometry studies piecewise-linear objects which appear as "degenerations" of the corresponding algebro-geometric objects.

Oren Ben-Bassat and Elizabeth Gasparim: Moduli Stacks of Bundles on Local Surfaces.- David Favero, Fabian Haiden and Ludmil Katzarkov: An orbit construction of phantoms, Orlov spectra and Knörrer Periodicity.- Stéphane Guillermou and Pierre Schapira: Microlocal theory of sheaves and Tamarkin's non displaceability theorem.- Sergei Gukov and Piotr Sulkowski: A-polynomial, B-model and Quantization.- M. Kapranov, O. Schiffmann, E. Vasserot: Spherical Hall Algebra of Spec(Z).- Maxim Kontsevich and Yan Soibelman: Wall-crossing structures in Donaldson-Thomas invariants, integrable systems and mirror Symmetry.- Grigory Mikhalkin and Ilia Zharkov: Tropical eigen wave and intermediate Jacobians.- Andrew Neitzke: Notes on a new construction of hyperkahler metrics.- Helge Ruddat: Mirror duality of Landau-Ginzburg models via Discrete Legendre Transforms.- Nicolo Sibilla: Mirror Symmetry in dimension one and Fourier-Mukai transforms.- Alexander Soibelman: The very good property for moduli of parabolic bundles and the additive Deligne-Simpson problem.

Erscheint lt. Verlag	16.10.2014
Reihe/Serie	Lecture Notes of the Unione Matematica Italiana
Zusatzinfo	XI, 436 p. 43 illus., 18 illus. in color.
Verlagsort	Cham
Sprache	englisch
Maße	155 x 235 mm
Gewicht	688 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
Schlagworte	14J33,53D37,14T05,14N35,14D24 • Donaldon-Thomas invariants • Fukaya category • Homological Mirror Symmetry • tropical geometry • Wall-crossing formulas
ISBN-10	3-319-06513-0 / 3319065130
ISBN-13	978-3-319-06513-7 / 9783319065137
Zustand	Neuware