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Connections in Geometry and Physics: 2009

This conference was held on May 8-10, 2009

at the Perimeter Institute for Theoretical Physics

Principal Speakers:
(60 Minute Talks)
  • Vestislav Apostolov (U.Q.A.M.)
  • Denis Auroux (M.I.T.)
  • Albert Chau (U.B.C.)
  • Andrew Dancer (Oxford)
  • Veronique Godin (Calgary)
  • Pengfei Guan (McGill)
  • Sergei Gukov (Santa Barbara)
  • John Harnad (Concordia)
  • Colin Ingalls (New Brunswick)
  • Niky Kamran (McGill)
  • James Sparks (Oxford)
  • Ben Weinkove (San Diego)
Local Area Postdocs:
(30 Minute Talks)
  • Chris Brav (Toronto)
  • Andrea Gambioli (U.Q.A.M.)
  • Shengda Hu (Waterloo)
  • Jeffrey Morton (Western Ontario)
  • Benjamin Young (McGill)
  • Fabian Ziltener (Toronto)

VIDEOS of most of the talks from this conference can be found here: Perimeter Institute Recorded Seminar Archive (GAP 2009).

Article about the conference from the newsletter of the Fields Institute: GAP 2009 Article.

Final schedule (with abstracts) for the conference: GAP 2009 Schedule.

Final list of all participants with affiliations: GAP 2009 Participant List.

GROUP PHOTO for the conference: GAP 2009 Group Photo.

Poster for "Connections in Geometry and Physics 2009"

Overview of this year's topics:

1. Elliptic and parabolic equations in geometry. Nonlinear second order elliptic partial differential equations arise naturally in geometry. Some examples of such equations are the Einstein equation, the complex Monge-Ampère equation, and the minimal surface equation. These are special cases of more general equations which arise by prescribing special conditions on the curvature of a Riemannian metric, on the second fundamental form of a Riemannian immersion, or on the curvature of a connection on some vector bundle. In certain cases, one can also consider natural first-order elliptic systems, such as for metrics with special holonomy, or for calibrated submanifolds. It is always of interest to find explicit solutions to such equations, but this is only usually possible on non-compact manifolds, in situations where there is a high degree of symmetry present. On compact manifolds, these geometric elliptic nonlinear equations are much more difficult to solve. Usually the best we can hope for are non-constructive existence results involving very hard analysis, such as in Yau's solution of the Calabi conjecture, or Joyce's construction of compact manifolds with G2 or Spin(7) holonomy.

Nonlinear parabolic equations also arise naturally in geometry, in the context of evolution equations. Some examples are the Ricci flow, mean curvature flow, and the Yamabe flow. Recently the techniques of geometric flows have been exploited with spectacular success by Perelman, Hamilton, and others to prove the Geometrization conjecture in three-dimensional geometry. The subject of geometric flows continues to receive much attention.

Natural geometric PDE's also arise in physics. For example, manifolds with special holonomy exhibit the phenomenon of mirror symmetry, which relates "dual" physical theories. Understanding mirror symmetry involves studying moduli spaces of calibrated submanifolds, as well as moduli spaces of special connections. Another topic related to physics is the AdS/CFT correspondence. Conformally compact manifolds and Sasakian-Einstein geometry are interwoven in this theory.

2. Geometry and topology of moduli spaces. An important problem in geometry is the classification of objects such as curves, varieties, metrics, or vector bundles. Moduli spaces are spaces that parametrise sets of objects of a fixed type (or isomorphism classes of such objects), and therefore arise naturally in classification problems. Some of the key examples are moduli spaces of algebraic curves of a fixed genus, Hilbert schemes of points on a variety, and moduli spaces of connections on a vector bundle. These spaces often have a geometric structure (they are usually manifolds, schemes, or algebraic stacks), and a natural question is to determine what type of additional structures they admit (such as metrics or Poisson structures). In fact, moduli spaces provide many interesting new examples of varieties and stacks.

Moduli spaces have significant applications in both mathematics and physics. For example, in Donaldson and Sieberg-Witten theories, topological invariants of four-manifolds are constructed via moduli spaces of connections, and these invariants have led, over the past twenty years, to great progress in the classification of four-manifolds. Whereas in Gromov-Witten theory, moduli spaces of pointed curves are applied to the study of the topology of symplectic manifolds, as well as to classical counting problems in algebraic geometry such as: How many rational plane curves of degree d pass through n points in general position? It should be noted that the study of these particular moduli spaces has been motivated in part by questions arising in quantum field theory and string theory. In another direction, the Langlands Program connects number theory to the function theory of a special type of moduli spaces, called Shimura varieties, which are generalisations of moduli spaces of elliptic curves.

3. Structures in symplectic geometry. As an old subject with a long history, symplectic geometry encompasses a wide spectrum of mathematical tools, from differential geometry and topology, representation theory, the physics of quantum field theory, and global non-linear analysis in the form of Floer theory. The history of symplectic geometry is in the mathematical description of physical systems and phase spaces in classical mechanics. The geometrical structures encountered there in the description of dynamical systems have been generalized in several different ways. In many cases, the structures are characterized by having a relatively simple local description, at first glance. The challenge is in understaning how the global structure and complexity emerges.

One direction has led to considering structures such as Poisson structures or Dirac structures, which are closely related to symplectic structures, and often arise when considering mechanical systems with various constraints. These are, like symplectic structures, related to semi-classical approximations of quantum field theories, and are involved in the programme of quantization, such as in the work of Kontsevich in finding deformation quantizations of Poisson structures.

Another direction involves studying symplectic manifolds admitting symmetries, via the action of a Lie group. Often our understanding of physical systems may be vastly improved if we may take advantage of a symmetry of the system; the geometric ramifications of this symmetry can be quite subtle, and have involved the introduction of equivariant geometry, the notion of momentum maps, and the development of symplectic groupoids. The focus is in understanding the quotient, or symplectic reduced space, in terms of the equivariant geometry of the original space. Prime examples of such efforts include the surprising convexity results of Atiyah and Guillemin-Sternberg, as well as the results of Kirwan on the descent of cohomology to the quotient.

Finally, a direction which has emerged in the last two decades involves the seemingly innocent question of intersection numbers of Lagrangian submanifolds. This has led to the development, primarily by Floer, of an infinite-dimensional Morse theory on the path space of a symplectic manifold with cohomology given by the Floer homology. This powerful analytic approach to the geometry of Lagrangian submanifolds has finally ushered in the possibility of a completely algebraic representation of a symplectic manifold as a A-category, as described in the work of Fukaya and Seidel. More recently we have seen the realization, by Wehrheim-Woodward, of a symplectic 2-category as envisioned originally by Weinstein more than 30 years ago.

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