Geometry Summer School

Lisbon, July 13-17, 2009


Program

There will be five minicourses aimed at graduate students:

Toric Kähler-Sasaki geometry

by Miguel Abreu (Instituto Superior Técnico)

Abstract:
In the same way that a contact manifold determines and is determined by a symplectic cone, a Sasaki manifold determines and is determined by a Kähler cone. Kähler-Sasaki geometry is the geometry of such a pair.
In this course, after an introduction, I will present an action-angle coordinates approach to toric Kähler geometry and how it was recently generalized, by Burns-Guillemin-Lerman and Martelli-Sparks-Yau, to toric Kähler-Sasaki geometry. As an application, I will describe how this approach can be used to relate a recent new family of Sasaki-Einstein metrics constructed by Gauntlett-Martelli-Sparks-Waldram in 2004, to an old family of extremal Kähler metrics constructed by Calabi in 1982.

Curves on algebraic varieties

by János Kollár (Princeton University)

Abstract:
The aim of these lectures is to give an introduction to the basic theory of families of curves on smooth algebraic varieties. We will emphasize the parallels between the algebraic and symplectic theories, with special attention to varieties with many rational curves on them.
As an introductory text we recommend:
Araujo, Carolina and Kollár, János: Rational curves on varieties. In: Higher dimensional varieties and rational points (Budapest, 2001), 13-68, Bolyai Soc. Math. Stud., 12, Springer, Berlin, 2003.
Those with solid algebraic geometry background may find the following useful:
Kollár, János: Rational curves on algebraic varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 32. Springer-Verlag, Berlin, 1996.

Integrals on moduli spaces of bundles and Grassmannian TQFTs

by Alina Marian (University of Illinois, Chicago)

Abstract:
Some of the most interesting intersection numbers on the moduli space of rank r bundles on a Riemann surface satisfy straightforward degeneration rules, and can therefore be integrated in a 2d topological quantum field theory. This TQFT is based on representations of the unitary group U(r) and is closely related to the quantum cohomology of a suitable Grassmannian. I will describe the theory and discuss this circle of ideas, which originated in a classic paper of Witten.

Symplectomorphims of the plane and curve counts

by Rahul Pandharipande (Princeton University)

Abstract:
I will talk about 1-parameter families of symplectomorphisms of the complex plane and their relationship to Euler characteristics of quivers varieties and enumerations of rational curves. The course will cover recent results of Reineke, Gross-Siebert, Kontsevich-Soibelman, etc.

Symplectic categories

by Alan Weinstein (University of California, Berkeley)

Abstract:
I will speak in these lectures about work (much of it still in progress) with Henrique Bursztyn, Alberto Cattaneo, Benoit Dherin, Shamgar Gurevich, and Ronny Hadani, as well as work of others.
Quantization problems suggest that the category of symplectic manifolds and symplectomorphisms should be augmented by the inclusion of more general morphisms, namely canonical relations, i.e. lagrangian submanifolds of products. It is well known that these relations compose well when a transversality condition is satisfied, but the failure of this condition to hold in general means that they do not comprise the morphisms of a category.
I will discuss several existing and potential remedies to the transversality problem. Some of these involve restriction to classes of lagrangian submanifolds for which the transversality property automatically holds. Others involve allowing lagrangian "objects" more general than submanifolds.
I will also mention another meaning of the term "symplectic category", namely a category in which the morphism spaces Hom(X,Y), rather than the individual objects X and Y, are symplectic manifolds, and the composition operation Hom(X,Y) x Hom(Y,Z) --->  Hom(X,Z) is a morphism in one of the categories of the preceding paragraph. These can produce associative algebras upon quantization.

Schedule

Monday Tuesday Wednesday Thursday Friday
10-11 am Kollár Weinstein Marian Abreu Pandharipande
11:30-12:30 Weinstein Abreu Kollár Pandharipande Marian
2:30-3:30 pm Marian Pandharipande FREE Kollár Abreu

There will be coffee breaks daily 11-11:30am and 3:30-4pm (except Wednesday afternoon).

Monday at 9:30am there will be a reception with coffee and snacks in the hallway by the lecture room.

School social activities will be posted here in early July.

Venue

All talks take place in room PA1, located on floor -1 (a.k.a. 01) of the IST mathematics building, Pavilhão de Matemática.

The IST Alameda campus is a large heart-shaped block bounded by four streets (see map under Local Information):
  • South - Av. Rovisco Pais (continues as Av. Duque d'Ávila)
  • West - Rua Alves Redol
  • North - Av. António José de Almeida
  • East - Av. Manuel da Maia
There is one campus gate at each of those streets, the west and eastern ones being at the top of stairs.

Pavilhão de Matemática is the greyish building at the Southwest corner of campus and can be entered only from campus.