Algebraic Geometry / Commutative Algebra Seminar

Fall 2009. Mirror symmetry and tropical geometry
Mirror symmetry is a mysterious correspondence between pairs X, X' of Calabi--Yau manifolds.
(A Calabi--Yau manifold is a compact complex manifold with vanishing first Chern class.)
The complex geometry of X is related to the symplectic geometry of X', and vice versa.
More precisely, Mirror symmetry relates degenerating families of Calabi--Yau manifolds.
Tropical geometry is the study of piecewise linear objects which arise as limits of complex or Lagrangian submanifolds in this context.
Recent work by Kontsevich--Soibelman, Gross--Siebert, and others aims to explain the mirror correspondence using tropical geometry.

Some effort will be made to make talks accessible to graduate students.
Everybody is welcome to contribute talks.

Sep 11. Organizational meeting, 4PM, LGRT 1634.

Sep 18. David Cox, Amherst College. *Overview of classical mirror symmetry*. 2:50PM, LGRT 1634.
Lecture notes.

Sep 25. Peter Dalakov. *Special Lagrangian submanifolds and their local moduli*. 2:50PM, LGRT 1634.
Abstract, Lecture notes.

Oct 2. Peter Dalakov. *Special Lagrangian submanifolds and their local moduli, II*. 2:50PM, LGRT 1634.
Abstract, Lecture notes.

Oct 9. Peter Dalakov. *Special Lagrangian submanifolds and their local moduli, III*. 2:50PM, LGRT 1634.
Abstract, Lecture notes.

Oct 16. Paul Hacking. *The Strominger-Yau-Zaslow conjecture*. 2:50PM, LGRT 1634.
Abstract, Lecture notes.

Oct 23. Paul Hacking. *The Strominger-Yau-Zaslow conjecture, II*. 2:50PM, LGRT 1634.
Abstract, Lecture notes.

Oct 30. No seminar due to AGNES.

Nov 6. Jenia Tevelev. *Reconstruction problem in mirror symmetry*. 2:50PM, LGRT 1634.
Abstract, Lecture notes.

Nov 13. Jenia Tevelev. *Reconstruction problem in mirror symmetry, II*. 2:50PM, LGRT 1634. Lecture notes.

Nov 20. Paul Hacking. *How to glue a K3 surface from flat pieces*. 2:50PM, LGRT 1634.
Abstract, Lecture notes.

Nov 27. No seminar due to Thanksgiving.

Dec 4. Alexei Oblomokov. *The tropical vertex*. 2:50PM, LGRT 1634.
Abstract, Lecture notes.

Dec 11. Paul Hacking. *How to glue a K3 surface from flat pieces, II*. 2:50PM, LGRT 1634.
Abstract, Lecture notes.

Some references and links:

M. Abouzaid, *Morse homology, tropical geometry, and homological mirror symmetry for toric varieties*, arxiv.

D. Auroux, Lecture notes for graduate course on mirror symmetry at MIT in Spring 2009, here.

D. Auroux, L. Katzarkov, and D. Orlov, *Mirror symmetry for del Pezzo surfaces: Vanishing cycles and coherent sheaves*, Invent. Math. 166 (2006), no. 3, 537-582, arxiv.

D. Cox and S. Katz, *Mirror symmetry and algebraic geometry*, Math. Surveys Monogr. 68, AMS 1999.

M.Gross, *The SYZ conjecture: From torus fibrations to degenerations*, in *Algebraic geometry - Seattle 2005*, Proc. Sympos. Pure Math. 80, Part 1, p. 149-192, AMS 2009,
arxiv.

M.Gross, R. Pandharipande, and B. Siebert, *The tropical vertex*, preprint (2009), arxiv.

M. Gross and B. Siebert, *Mirror symmetry via logarithmic degeneration data I*, J. Differential Geom. 72 (2006), no. 2, 169-338, journal, arxiv.

M. Gross and B. Siebert, *An invitation to toric degenerations*, preprint (2008), arxiv.

M. Gross and P. Wilson, *Large complex structure limits of K3 surfaces*, J. Differential Geom. 55 (2000), no. 3, 475-546, journal, arxiv

N. Hitchin, *The moduli space of special Lagrangian submanifolds*, Annali Scuola Sup. Norm. Pisa Sci. Fis. Mat. 25 (1997), 503-515, arxiv.

K. Hori, S.Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa, R. Vakil, and E. Zaslow, *Mirror symmetry*, Clay Math. Monogr. 1, AMS 2003.

M. Kontsevich, *Homological algebra of mirror symmetry*, in *Proceedings of the International Congress of Mathematicians*, Zurich 1994, Vol. 1, 120-139, Birkhauser 1995, arxiv.

M. Kontsevich, Y. Soibelman, *Affine structures and non-archimedean analytic spaces*, in *The unity of mathematics*, p. 321-385, Progr. Math. 244, Birkhauser 2006, arxiv.

G. Mikhalkin, *Enumerative tropical algebraic geometry in R2*, J. Amer. Math. Soc. 18 (2005), no. 2, 313-377, journal, arxiv.

G. Mikhalkin, *Tropical geometry and its applications*, in *Proceedings of the International Congress of Mathematicians*, Madrid 2006, Vol. II, 827-852, Eur. Math. Soc. 2006, arxiv.

G. Mikhalkin and I. Zharkov, *Tropical curves, their Jacobians and theta functions*, in *Curves and abelian varieties*, 203-230, Contemp. Math. 465, AMS 2008, arxiv.

T. Nishinou and B. Siebert, *Toric degenerations of toric varieties and tropical curves*, Duke Math. J. 135 (2006), no. 1, 1--51, journal, arxiv.

A. Polishchuk and E. Zaslow, *Categorical mirror symmetry: The elliptic curve*, Adv. Theor. Math. Phys. 2 (1998), no. 2, 443-470, arxiv.

A. Strominger, S-T. Yau and E. Zaslow, *Mirror symmetry is T-duality*, Nucl. Phys. B 479 (1996), no. 1-2, 243-259, arxiv.

S-T. Yau and E. Zaslow, *BPS states, string duality, and nodal curves on K3*, Nucl. Phys. B 471 (1996), no. 3, 503-512, arxiv.

Tropical geometry and mirror symmetry conference at Kansas state Univ., including series of lectures by M. Gross, 12/13-17/08, here.

Tropical geometry program at MSRI, 8/17/09-12/18/09, here.

Links to the seminar from previous years.

Deformation theory,
Fall 2008

Green's Conjectures,
Spring 2008

Bridgeland Stability,
Fall 2007

Minimal Model Program,
Spring 2007

Commutative
Algebra and Polyhedra Seminar, Spring 2006

Geometry and
Algebra of Polyhedra Seminar, Fall 2005

Commutative Algebra
Seminar, 2004-2005

The page is maintained by Paul Hacking