Math 277 - Section 3 - Topics in Differential Geometry - Fall 2009
D. Auroux -
Tue. & Thu., 9:30-11am, Room 3 Evans
Course outline
This course will focus on various aspects of mirror symmetry. It is
aimed at students who already have some basic knowledge in symplectic
and/or complex geometry (Math 242 helpful but not required).
The geometric concepts needed to formulate various mathematical
versions of mirror symmetry will be introduced along
the way, in variable levels of detail and rigor. The main topics will be
as follows:
1. Hodge structures, quantum cohomology, and mirror symmetry
Calabi-Yau manifolds; deformations of complex structures,
periods; pseudoholomorphic curves, Gromov-Witten invariants,
quantum cohomology; large complex structure limits;
mirror symmetry at the level of periods and quantum cohomology.
2. A brief overview of homological mirror symmetry
Coherent sheaves, derived categories; Lagrangian Floer homology and
Fukaya categories (in a limited setting); homological mirror symmetry
conjecture; example: the elliptic curve.
3. Lagrangian fibrations and the SYZ conjecture
Special Lagrangian submanifolds and their deformations;
Lagrangian fibrations, affine geometry, and tropical geometry;
SYZ conjecture: motivation, statement, examples (torus, K3);
challenges: instanton corrections, ...
4. Beyond the Calabi-Yau case: Landau-Ginzburg models and
mirror symmetry for Fanos (if time permits)
Matrix factorizations; admissible Lagrangians; examples
(CP1, CP2);
the superpotential as a Floer
theoretic obstruction; the case of toric varieties.
Bibliography
This very incomplete list tries to provide some of the more accessible references
on the material. There are many other excellent references, but those often
require a higher level of dedication.
Books:
- M. Gross, D. Huybrechts, D. Joyce, Calabi-Yau manifolds and related
geometries, Lectures from the Summer School held in Nordfjordeid, June
2001, Universitext, Springer, 2003.
- D. A. Cox, S. Katz, Mirror symmetry and algebraic geometry,
Mathematical Surveys and Monographs 68, AMS, 1999.
- D. McDuff, D. Salamon, J-holomorphic curves and symplectic
topology, AMS Colloquium Publ. 52, AMS, 2004.
Papers:
- R. P. Thomas, The geometry of mirror symmetry,
Encyclopedia of Mathematical Physics, Elsevier, 2006,
pp. 439-448;
arXiv:math.AG/0512412
- A. Polishchuk, E. Zaslow, Categorical mirror symmetry: the elliptic
curve, Adv. Theor. Math. Phys. 2 (1998), 443-470;
arXiv:math.AG/9801119
- R. P. Thomas, Derived categories for the working mathematician,
Winter school on mirror symmetry (Cambridge MA, 1999), AMS/IP Stud. Adv. Math.
23, AMS, 2001, pp. 363-377;
arXiv:math.AG/0001045
- D. Auroux, Mirror symmetry and T-duality in the complement of an
anticanonical divisor,
J. Gökova Geom. Topol. 1 (2007), 51-91;
arXiv:math.SG/0706.3207