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Connections in Geometry and Physics: 2010

was held on May 7-9, 2010; Perimeter Institute for Theoretical Physics, in Waterloo, Ontario, Canada.

Principal Speakers:
(60 Minute Talks)
  • Spyros Alexakis (Toronto)
  • Vincent Bouchard (Alberta)
  • Jim Bryan (British Columbia)
  • Freddy Cachazo (Perimeter)
  • Charles Doran (Alberta)
  • Joel Kamnitzer (Toronto)
  • Melissa Liu (Columbia)
  • David Morrison (Santa Barbara)
  • Nikita Nekrasov (I. H. E. S.)
  • Andrew Neitzke (Texas at Austin)
  • Harvey Reall (Cambridge)
  • Mu-Tao Wang (Columbia)
  • Shing-Tung Yau (Harvard)

Local Area Postdocs:
(30 Minute Talks)
  • Magdalena Czubak (Toronto)
  • Emre Coskun (Western Ontario)
  • Norman Do (McGill)
  • Abhijnan Rej (Fields Institute)
  • Natalia Saulina (Perimeter)
  • David Skinner (Perimeter)

VIDEOS of most of the talks from this conference can be found here: Perimeter Institute Recorded Seminar Archive (GAP 2010).

Article about the conference from the newsletter of the Fields Institute: GAP 2010 Article.

Final list of all participants with affiliations: GAP 2010 Participant List.

GROUP PHOTO for the conference: GAP 2010 Group Photo.

Final schedule (with abstracts) for the conference: GAP 2010 Schedule.

Poster for "Connections in Geometry and Physics 2010".

Overview of this year's topics:

1. Mathematical Relativity. Although General Relativity was formulated over 90 years ago, its impact on mathematical research has only increased steadily with time. This is due largely to the tremendous progress made during the last four decades in our understanding of non-linear partial differential equations and global differential geometry. The Hawking and Penrose singularity theorems result from formulating causal relationships in topological terms and exploiting the relationship between curvature and focal points. The proof of the positive mass theorem by Schoen and Yau uses 3-manifold theory and the theory of minimal surfaces, while the proof by Witten relies on the existence of asymptotically constant harmonic spinors. More recent highlights include the proof by Christodoulou and Klainerman of the global stability of Minkowski space, the proof of the Riemannian Penrose inequality by Huisken-Illmanen and Bray, and the on-going analysis of black-hole dynamics and stability by Finster-Kamran-Smoller-Yau. Here the mathematical tools came from geometric flows (mean curvature and inverse mean curvature flows), Fourier analysis, and the theory of hyperbolic equations. Intense efforts are currently devoted to such topics as the unique continuation of the Einstein equation (Anderson, Herzlich, Alexakis), stability of the Kerr solution (Alexakis-Ionescu-Klainerman), the Einstein constraint equation (Butscher, Chrusciel, Corvino, Isenberg, Pacard, Pollack, Schoen), concepts of quasi-local mass (M. Liu, M.T. Wang, S. T. Yau), Poincaré-Einstein manifolds (Anderson, Biquard, Chrusciel, Chang-Gursky-Qing-Yang, Graham, Fefferman, J. Lee, Mazzeo), and higher-dimensional blackhole geometry (Gibbons-Hartnoll-Page-Pope, Myers-Perry). The last two topics are related to the AdS/CFT correspondence in conformal field theory. Many more exciting research directions come into focus if one ventures beyond the Lorentzian setting to the Riemannian setting, e.g., metrics with special holonomy, Sasakian-Einstein geometry, and quasi-Einstein metrics.

Mathematical General Relativity is an area in which historically there has been substantial Canadian representation. Besides current contributors such as N. Kamran (McGill), R. Mann (Waterloo), R. Myers (PI), D. Page (Alberta), one should note the classical works of J. L. Synge and W. Israel.

2. Gauge theory. As the fundamental basis of the theory of elementary particles, gauge theory in the guise of Yang-Mills theory is at the center of theoretical physics. Since the end of the 1970s, however, it has been a central topic in mathematics as well, especially influencing differential and algebraic geometry as well as low-dimensional topology. The successes of gauge theory in uncovering deep structure in these fields are too numerous to list. They include the understanding of flat connections on surfaces by Atiyah and Bott; the breakthroughs of Donaldson concerning smooth 4-dimensional manifolds using instantons; various flavours of knot invariants starting with Witten's understanding of the Jones polynomial; recent progress in understanding 3-manifolds by Taubes, Kronheimer, and Mrowka using monopoles; and finally the recent revitalizing of the geometric Langlands programme by Gukov, Kapustin, and Witten, using topological supersymmetric 4-d Yang-Mills theory. The general pattern in all of these developments (and many others in gauge theory) is that gauge theory provides us with natural moduli spaces which then yield invariants which we may associate to the original objects of study -- the resulting invariants are, in many cases, extremely deep.

Recently, there has also been a surge of progress in our understanding of gauge theory itself; in particular, the work of Kontsevich and Soibelman, as well as Nakajima, on stability conditions for gauge theories, as well as the work of Costello on the mathematical understanding of Renormalization in 4-dimensional Yang-Mills theory. Finally there are the spectacular developments using twistor theory to calculate amplitudes in supersymmetric Yang-Mills theory.

3. Mirror Symmetry. Discovered by physicists as a duality between string theories with spacetimes associated with different Calabi-Yau manifolds, mirror symmetry has evolved into a rich field within mathematics which involves algebraic geometry, symplectic geometry, and homological algebra. Mirror symmetry is essentially a series of surprising relationships between the complex and symplectic geometry of different Calabi-Yau manifolds, surprising because they seem to be quite indirect and lack an obvious geometric explanation. The relationships are even more remarkable because they enable the calculation of deep and difficult combinatorial and enumerative facts which were previously thought to be inaccessible.

The first mathematical explanation for mirror symmetry was proposed by Strominger, Yau, and Zaslow, who outlined a way of establishing the relationships using special Lagrangian fibrations of Calabi-Yau manifolds, together with T-duality. This has led to a very successful programme, starting with the results of Batyrev-Borisov for Calabi-Yau hypersurfaces in Fano toric varieties, and culminating with the work of Gross and Siebert which involves the use of affine geometry, tropical geometry and the degeneration of Calabi-Yaus to establish a construction of Mirror manifolds with the required properties.

Another approach was suggested by Kontsevich, and is known as homological mirror symmetry. He proposed that a large part of the mirror symmetry relations could be explained as an equivalence of categories between derived categories of coherent sheaves (for a complex manifold) and Fukaya categories (for symplectic manifolds). The conjectured equivalence of categories was then established in many cases by Fukaya and Seidel, and has also led to the use of tropical geometry in the study of Floer theory in symplectic geometry. The homological mirror symmetry approach is notable for its introduction of powerful algebraic techniques in symplectic geometry, which have been used to great effect in many other fields, including categorification and differential topology.

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