SCIENTIFIC ACTIVITIES - Fields Institute - University of Toronto

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GAP 2 Conn CLASSICALLY, IT WAS IMPOSSIBLE to distinguish between theoretical physics and pure mathematics. For example, consider Newton, Lagrange, and Hamilton. Were they physicists or mathematicians? In later years, however, as research in both disciplines became more specialized, the two sides began to drift apart. Since the 1950s and 1960s there has been a gradual reconciliation, resulting in spectacular success both in physics and in mathematics, particularly in geometry and topology. More recently, the unexpected phenomenon of “mirror symmetry” of Calabi-Yau manifolds was predicted by string theorists, and in many cases verified rigorously by mathematicians. Many aspects of the current research in differential and algebraic geometry are being fueled by predictions from theoretical physics, opening new and exciting rigourous mathematical roads that may otherwise have been left undiscovered. In order to take advantage of this symbiotic relationship between geometry and physics, on May 7–9, 2010, the Perimeter Institute hosted the second Connections in Geometry and Physics (GAP) conference, with major financial support from the Fields Institute and the Faculty of Mathematics of the University of Waterloo. The organizers of this year’s conference were Jaume Gomis (Perimeter), Marco Gualtieri (Toronto), Spiro Karigiannis (Waterloo), Ruxandra Moraru (Waterloo), Rob Myers (Perimeter), and McKenzie Wang (McMaster.) As was the case during the 2009 conference, this meeting was an opportunity to increase the interaction, both locally and globally, between theoretical physicists and mathematicians, particularly those working in differential and algebraic geometry. The list of principal speakers included a healthy mixture from all three themes, 10 FIELDSNOTES | FIELDS INSTITUTE Research in Mathematical Sciences both mathematicians and physicists, and also consisted of both local area researchers and international experts. In particular, we had a significantly increased international participation this time. Every year, the organizers of GAP choose three main themes for the conference, emphasizing areas where physics and geometry have enjoyed much recent mutual benefit. Here is a short overview of this year’s three themes: 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 (Alexakis, Anderson, Herzlich), 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 higherdimensional black hole geometry (Gibbons- Hartnoll-Page-Pope, Myers-Perry). The last two topics are related to the AdS/CFT (anti-de-Sitter space/conformal field theory) correspondence in conformal field theory. Many more exciting research directions come into focus if one ventures beyond the Lorentzian setting to the Riemannian setting: for example, metrics with special holonomy, Sasakian-Einstein geometry, and quasi-Einstein metrics. Mathematical Relativity is an area in which historically there has been substantial Canadian representation. Besides current contributors such as Niky Kamran (McGill), Richard Mann (Waterloo), Robert Myers (Perimeter), Don Page (Alberta), one should note the classical works of W. Israel and J. L. Synge. 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 lowdimensional topology. The successes of gauge theory in uncovering deep structure in these fields are too numerous to list. They include the understanding of flat
010 ections in Geometry and Physics 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 Kronheimer, Mrowka, and Taubes using monopoles; and finally the recent revitalizing of the geometric Langlands program by Gukov, Kapustin, and Witten, using topological supersymmetric 4-dimensional 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 spectacular developments using twistor theory to calculate amplitudes in supersymmetric Yang-Mills theory. MIRROR SYMMETRY Discovered by physicists as a duality between string theories with spacetimes associated to 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 Workshop participants 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 data that were previously thought to be inaccessible. The first mathematical explanation of mirror symmetry was proposed by Strominger, Yau, and Zaslow, who outlined a way of establishing the duality using special Lagrangian fibrations of Calabi-Yau manifolds, and involving both the Legendre and Fourier transforms. This has led to a very successful program, 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-Yau manifolds 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. We were very fortunate to attract many excellent world-renowned researchers to GAP 2010, including Shing-Tung Yau (Fields Medal 1982, Wolf Prize 2010); David Morrison (Clay Mathematics Institute Senior Scholar 2005); and Nikita Nekrasov (Hermann Weyl Prize 2004). We were also pleased once again to have heavy participation by local area graduate students in both mathematics and physics. At least one third of the registered participants were students. GAP 2010 featured six short talks by local area postdoctoral fellows, intending to showcase these promising young researchers to help them succeed in the next stage of their professional careers. Spiro Karigiannis (Waterloo) FIELDS INSTITUTE Research in Mathematical Sciences | FIELDSNOTES 11
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Page 3 and 4: This summer, 20 undergraduate stude
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