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Connections in Geometry and Physics: 2011

was held on May 13-15, 2011, at the Fields Institute, Toronto, Canada.

Organizing Committee:   Marco Gualtieri (Toronto), Spiro Karigiannis (Waterloo), Ruxandra Moraru (Waterloo), Rob Myers (Perimeter), McKenzie Wang (McMaster).

Principal Speakers:
(60 Minute Talks)
  • Michael Anderson (Stony Brook)
  • Octav Cornea (Montréal)
  • Robin Graham (Washington)
  • François Lalonde (Montréal)
  • John Lott (UC Berkeley)
  • Robert McCann (Toronto)
  • Tom Mrowka (M.I.T.)
  • André Neves (Imperial)
  • Natasa Sesum (Rutgers)
  • Pedro Vieira (Perimeter)
  • Katrin Wehrheim (M.I.T.)
  • Xi Yin (Harvard)

Local Area Postdocs:
(30 Minute Talks)
  • Ken Chan (Waterloo)
  • Dezhong Chen (Toronto)
  • Roman Golovko (Montréal)
  • Hongnian Huang (CRM-Montréal)
  • Ben Mares (McMaster)
  • João Penedones (Perimeter)
  • Sinem Onaran (Waterloo)

Audio and Slides of talks: Fields Institute Lecture Audio and Slides (GAP 2011).

Article about the conference from the newsletter of the Fields Institute: GAP 2011 Article.

Final list of all participants with affiliations: GAP 2011 Participant List.

GROUP PHOTO for the conference: GAP 2011 Group Photo.

Final schedule (with abstracts) for the conference: GAP 2011 Schedule.

Poster for "Connections in Geometry and Physics 2011".

Overview of this year's topics:

1. Advances in Floer theory. Floer theory began in the mid-1980's with the foundational work of Andreas Floer, who combined ideas from Morse theory, partial differential equations, and index theory to develop certain kinds of homology theories which detect "half-infinite" dimensional cycles in infinite-dimensional spaces. These may be thought of as infinite-dimensional versions of Morse theory. Floer theory has found its greatest application in the field of symplectic geometry, where Floer originally used it to prove the Arnol'd conjecture concerning the counting of intersections between Lagrangian submanifolds of symplectic manifolds. However, Floer theory has been extended in many different directions, and has found applications in the study of 3-manifolds and knot invariants, as well as the study of contact manifolds, Seiberg-Witten theory of 4-manifolds, and the study of mirror symmetry, where it plays a role in symplectic geometry analogous to the Ext groups of algebraic geometry. The foundational work of Floer was tied to further foundational work by Gromov and Witten, and this ushered in a decade of spectacular results in the above-mentioned fields.

In the past 15 years, a much more detailed understanding of the depth of the subject has been achieved, through the work of many analysts and geometers, especially the Fukaya, Oh, Ono, and Ohto group, as well as Seidel and his collaborators, Hofer, McDuff and Salamon, Kronheimer and Mrowka, and many others. This has led to all the major results in homological mirror symmetry in the past 10 years, culminating with the work of Seidel proving Kontsevich's mirror symmetry conjecture in several important cases, including that of the quartic surface.

Even more recently, in a landmark 2007 paper, Wehrheim and Woodward established the beginnings of a functorial description of Lagrangian Floer theory, placing many analytical results into a clean categorical framework, and realizing the dream of Weinstein dating from the 1970's of having a true symplectic category, where Lagrangian correspondences function as morphisms and the composition law is well-defined. In subsequent papers of Wehrheim-Woodward and Ma'u- Wehrheim-Woodward, the theory has been extended to cover a larger class of symplectic manifolds and the Floer theory has been further "categorized" in the sense of including Floer cochains in an A categorical structure, rather than only the Floer homology groups.

The foundational basis of Floer theory has also been undergoing several simultaneous shifts. To cope with the complexities of compactifying moduli spaces and wrestling with the singularities which ensue, proposals for the enhancement of the notion of moduli space have been made, for example, by Joyce (in the form of Kuranishi spaces) and Hofer (polyfolds). Most intriguing perhaps is the recent work of Mrowka and Lipyanskiy, which places Floer theory on a different analytical foundation - that of polarized Hilbert manifolds. By taking seriously the notion, suggested by Atiyah, that Floer theory describes half-infinite homology cycles, they have shown that many of the analytic difficulties may be sidestepped.

2. Geometric flows. The study of geometric flows (also called geometric evolution equations) was initiated by Richard Hamilton in the early 1980's when he first considered the Ricci flow. This is the evolution of a Riemannian metric on a manifold in the direction of its Ricci curvature tensor. Such a flow tends to "flatten" the metric, and is qualitatively very similar to the diffusion of heat via the classical heat equation. The intended application of the Ricci flow was to prove the Geometrization conjecture of Thurston: that all 3-manifolds can be decomposed into pieces which admit certain geometric structures. In fact, this conjecture can be viewed as a generalization of the classical uniformization theorem, which states that every surface admits a metric of constant scalar curvature, and which can be proved using the Ricci flow. Through the work of Hamilton and Perelman, we now have a complete proof of the geometrization conjecture using Ricci flow, and this includes the famous Poincaré conjecture as a special case.

The idea of evolving certain tensors on manifolds in natural, geometrically defined directions has now become a very active and extremely fruitful avenue of research. One notable example is the mean curvature flow of immersed submanifolds, which is a tool used to construct minimal surfaces or more generally constant mean curvature submanifolds. In fact a variant of this, the inverse mean curvature flow, was used by Huisken and Ilmanen to provide the first proof of the Riemannian Penrose Inequality in general relativity, which relates the minimum mass of a single black hole to the area of its outermost minimal surfface. Another very successful geometric flow is the Kähler-Ricci flow, which has led to new proofs of the existence of Kähler-Einstein metrics under certain conditions. This is work by Cao, Chen-Tian, and others. It is hoped that the Kähler-Ricci flow could lead to a kind of "geometrization" of Kähler manifolds in the future.

Today the subject of geometric flows is an industry in itself. Whenever a natural functional exists on a certain geometric moduli space, one can try to flow in the direction of the gradient of this functional. Examples of these include the Willmore flow and the Yamabe flow. In general, geometric flows (when properly interpreted, modulo the natural equivalence) tend to be parabolic partial differential equations, and are studied using various tools from geometric analysis, such as the maximum principle (as generalized to tensors by Hamilton) and the Harnack inequalities of Li-Yau-Hamilton type.

3. AdS/CFT correspondence. The AdS/CFT correspondence refers to a conjecture made by J. Maldacena around 1997 that type IIB superstring theory on the background AdS5 X S5 is dual to the conformally invariant N=4 supersymmetric Yang-Mills theory on the conformal boundary of anti-de Sitter space AdS5. By now a whole series of conjectures have emerged whereby string theories/M-theory on different backgrounds are associated to various conformally invariant quantum field theories. These conjectures are having profound effects in both theoretical physics and differential geometry. Research activity is intense and highly interdisciplinary in nature. For example, the research of almost all current members of the Perimeter Institute touch upon this correspondence.

Mathematically speaking, the geometry of AdS5 that is relevant for AdS/CFT is, after a Wick rotation, the theory of conformally compact Einstein manifolds. Main mathematical contributors here include C. Fefferman, R. Graham, J. Lee, M. Anderson, O. Biquard, A. Chang-J. Qing-P. Yang, R. Mazzeo-F. Pacard. On the other hand, the essence of the geometry of S5 is Sasakian-Einstein geometry in low dimensions. This type of Einstein geometry is necessitated by the supersymmetry of the physical theory. A general theory of Sasaki-Einstein spaces has been developed by Boyer-Galicki, who also constructed numerous families of quasi-regular examples. Irregular Sasakian Einstein spaces have recently been studied by Gauntlett, Martelli, Sparks, Waldram, and Yau.

Another ingredient critical to the AdS/CFT correspondence is the theory of branes and the associated flux fields. Mathematically, branes are calibrated submanifolds in (possibly singular) Einstein spaces with special holonomy. The mathematical theory was originally developed by Harvey-Lawson, R. Bryant, R. McLean, and more recently by D. Joyce, N. Hitchin, Schoen-Wolfson. Furthermore, high-dimensional black hole solutions (with associated black branes) have recently become important in certain AdS/CFT dualities.

By bringing together physicists working on AdS/CFT with differential geometers, who tend to work on the above subtopics separately (witness the disjoint list of names above), it is hoped that a more comprehensive understanding will emerge of how the different geometries are interwoven by requirements of the physics. The interaction may lead to new mathematical questions about the inter-relationships between these different geometries.

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