# Lectures on Homological Mirror Symmetry

### Time and place

Mondays, 3.30-5pm, in Simonyi 101.### Notes

11/4: Introduction (notes).

Here's the paper by Witten I referred to in answer to Tom's question: Mirror manifolds and topological field theory.

11/11: Closed-string invariants 1 (notes).

Reference: D. McDuff and D. Salamon, J-holomorphic curves and symplectic topology American Mathematical Society Colloquium Publications, 52, AMS, Providence, RI, 2004.

11/18: Closed-string invariants and operads (notes).

References: On operads: J. P. May, Definitions: operads, algebras and modules, in Operads: Proceedings of Renaissance Conferences, J-L. Loday, J. D. Stasheff, A. A. Voronov, editors, Contemporary Mathematics 202, AMS, Providence, RI, 1997.

On their application to Gromov-Witten theory: M. Kontsevich and Yu. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164, pp. 525-562, 1994.

11/25: Closed-string invariants and operads continued (notes).

References: M. Kontsevich and Yu. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164, pp. 525-562, 1994.

Fulton, W.; Pandharipande, R. Notes on stable maps and quantum cohomology. Algebraic geometry - Santa Cruz 1995, 45–96, Proc. Sympos. Pure Math., 62, Part 2, Amer. Math. Soc., Providence, RI, 1997.

12/2: Gerstenhaber algebras and the B-model (notes).

References: S. Barannikov and M. Kontsevich, Frobenius Manifolds and Formality of Lie Algebras of Polyvector Fields. Internat. Math. Res. Notices 1998, no. 4, 201–215.

12/9: Gerstenhaber algebras and the B-model (notes).

References: S. Barannikov and M. Kontsevich, Frobenius Manifolds and Formality of Lie Algebras of Polyvector Fields. Internat. Math. Res. Notices 1998, no. 4, 201–215.

2/10: Batalin-Vilkovisky algebras and symplectic cohomology (notes).

References: P. Seidel, A biased view of symplectic cohomology. Current Developments in Mathematics 2006:211–253, (2008). 87.

For the relation between BV algebras and homology of framed configuration spaces: E. Getzler. Batalin-Vilkovisky algebras and 2d Topological Field Theories. Commun. Math. Phys, 159:265–285, 1994.

2/17: Open-string invariants (notes).

References: M. Gerstenhaber. The cohomology structure of an associative ring. Annals of Math., 78:267–288, 1963.

For Lagrangian Floer cohomology: D. Auroux, A beginner's introduction to Fukaya categories, 2013, arXiv link

2/24: Lagrangian Floer cohomology (notes).

References: D. Auroux, A beginner's introduction to Fukaya categories, 2013, arXiv link

3/3: The Fukaya category (notes).

References: D. Auroux, A beginner's introduction to Fukaya categories, 2013, arXiv link

3/10: Closed-open maps (notes).

4/14: Coherent sheaves, examples of HMS (notes).

References: D. Huybrechts, Fourier-Mukai Transforms in Algebraic Geometry, Clarendon Press, 2006.

The homotopy theory of dg-categories and derived Morita theory, Inventiones Mathematicae 167, Issue 3, pp. 615-667, 2006.

A. Polishchuk and E. Zaslow, Categorical mirror symmetry: the elliptic curve, Adv. Theor. Math. Phys. 2, pp. 443-470, 1998.

### Links

Notes from Denis Auroux's graduate topics course on mirror symmetry.

Notes from Paul Seidel's Talbot workshop on Fukaya categories.

Notes from Paul Seidel's graduate course on Categorical Dynamics and Symplectic Topology (has concise summaries of many relevant concepts).

### Bibliography

D. Auroux, A beginner's introduction to Fukaya categories arXiv:1301.7056

M. Kontsevich, Homological algebra of mirror symmetry arXiv:alg-geom/9411018

I. Smith, A symplectic prolegomenon, arXiv:1401.0269

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

R. P. Thomas, The geometry of mirror symmetry, Encyclopedia of Mathematical Physics, Elsevier, 2006, pp. 439-448; arXiv:math.AG/0512412