Affine manifolds and Mirror Symmetry
Workshop
April 25 - 27, 2008
University of
Michigan
Ann Arbor, Michigan
Resources:
Andrey's notes on Affine Geometry
Airport shuttle number:
734-394-1665
(reserve 24h in advance)
Organizers:
Renzo
Cavalieri
Hannah Markwig
For website problems, contact webmaster
This year the goal of the workshop is to try and get a grasp of the circle of ideas contained in recent work of Mark Gross and Bernd Siebert:
- Affine manifolds, log structures, and mirror symmetry (survey)
- The SYZ conjectures: from torus fibrations to degenerations (survey)
- From real affine geometry to complex geometry
The Gross-Siebert mirror symmetry program, inspired by the Strominger-Yau-Zaslow conjecture, builds on the idea that the fundamental objects controllingmirror symmetry are integral affine manifolds: real manifolds with transition maps being integral affine transformations. In toy situations,
given such a manifold B without singularities, one obtains easily from it two manifolds: one symplectic and one complex. These form a mirror pair.
However, this leads to few interesting examples, and B must be allowed to have singularities where the affine structure is not defined in order to get interesting examples of mirror symmetry, such as for hypersurfaces in toric varieties.
The Gross-Siebert program then is an approach for dealing with this singular case, and at its heart lies a reconstruction theorem, stating that given some additional combinatorial structure and assumptions on B, it is possible to construct a family of degenerating Calabi-Yau varieties over the spectrum of a complete local ring. The goal of this seminar is to understand the motivation for this result and some of the ideas behind the proof, which appears in ``From Real Affine Geometry to Complex Geometry.'' This paper gives a very explicit description of the family using tropical data on B, hinting at a tropical description of the B-model side (complex structure side) of mirror symmetry.
Mark Gross has kindly agreed to participate and help out find a path among the vast amount of material involved. We intend this workshop as an occasion to bring toghether people from various areas of algebraic geometry (and mathematics in general). For this reason, the assumed background will be minimal. In the spirit of a "learning-seminar", most of the talks will be given by non-experts.
The first talks will be accessible to graduate students in algebraic geometry and related fields.
| SPEAKER | FROM | TENTATIVE TALK TITLE |
| Arend Bayer | U. Utah | The SYZ conjectures and homological mirror symmetry |
| Dan Budreau | UCSD | Batyrev's contruction |
| Renzo Cavalieri | U. Michigan | Overview |
| Mark Gross | UCSD | Everything is illuminated |
| Hannah Markwig | U. Michigan | Tropical curves |
| Andrey Novoseltsev | U. Washington | Affine geometry and the Legendre transform |
| Nick Proudfoot | U.Oregon | Toric degenerations and the dual intersection complex |
| Helge Ruddat | U. Freiburg | The various Hodge numbers |
| David Speyer | CLAY | Log geometry in the G-S program |
SCHEDULE:
all talks - and refreshments - will be in EH 4088
|
Day |
Time |
Speaker or Event |
|
April 25 th |
|
|
|
Apr 26 |
|
|
Apr 27
|
|
|
Some funding may be available for outside participants, through the department's NSF RTG grant. Contact Hannah Markwig or Renzo Cavalieri for information.
Department
of Mathematics | 2074
East Hall | 530
Church Street
Ann Arbor, MI 48109-1043
Phone:
734.764-0335 | Fax:
734.763-0937
Site errors should be directed to math-webmaster@umich.edu