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Scientific Mission

Geometry and Theoretical Physics have a long history of fruitful interaction. Sometimes physics finds the appropriate mathematical context for formulating its results already exists and was developed by mathematicians for internal mathematical reasons. Two examples of this are (i) the formulation of quantum mechanics in terms of operators on a Hilbert space; and (ii) Einstein’s general theory of relativity which is formulated in terms of (a Lorentzian version) of Riemannian geometry. Sometimes physics leads to the discovery of new geometric objects of central importance, for example the Laplacian, the Dirac operator and spin structures, Brownian motion, and the heat kernel. The last 30 years or so have seen a new type of interaction. Beginning with its use in the construction of what is now the Standard Model of particle physics — or even earlier in the context of Quantum Electrodynamics — gauge theory, also known as non-abelian Yang-Mills theory, has been central in physics. At the classical level, gauge theory is, from a mathematical point of view, the theory of connections on principal bundles and their curvatures; in the context of physics, gauge theory is treated quantum mechanically. Here we see the basic context for modern theoretical physics being expressed in terms of classical mathematical structures, yet the passage in physics from the classical to the quantum world for these objects has no parallel in mathematics. Physics gives hints about the nature of the mathematics that will be required in the analogue and also produces conjectures and questions about the purely classical mathematical objects (think for example of Gromov-Witten invariants and quantum cohomology, of mirror symmetry relating algebraic manifolds and symplectic ones, of the description of the Jones polynomial in terms of the Chern-Simons gauge theory). This is a rich area of interaction between geometry and physics and is one focus of the Simons Center. One can list many other topics of a similar nature: AdS/CFT correspondence in physics and its relation to hyperbolic geometry, conformal field theory and its appearance in scaling limits, homotopy theory and algebraic topology with applications to condensed matter physics, general relativity, string theory and questions its raises about moduli space of curves, string compactifications and relations to Calabi-Yau three-folds and G2 manifolds, invariants of low dimensional manifolds and their relationship to gauge theories. This list is not meant to be exhaustive, rather it is presented in order to give a sense of the areas of mathematics and physics that will be the central foci of the Simons Center.

The topics listed above give a sense of the intellectual terrain in which the Center will operate but there is another crucial aspect to the intellectual focus of the Center. The Center believes that much progress can be made in each domain, geometry and theoretical physics, by serious, substantive interchanges between mathematicians and physicists around fundamental questions of common interest. Enhancing these interactions will be always at the forefront as the Center plans and carries out its activities. Thus, the Center focuses on activities of common interest to mathematicians and physicists and invites members of both groups to each of its programs and workshops. The Center also searches out mathematicians and physicists who have exhibiting an willingness and an ability to work across the subject divide that separates their fields to invite to join the Center as permanent members, as members of its oversight bodies, as organizers of its activities, and as participants in the activities. Our program organizing committees, in almost every case, have both mathematicians and physicists as members and one of their charges will be to construct programs and workshops that are accessible to and appeal to both groups. In the case of those topics that are clearly primarily in one of the two areas, Center will look for ways to broaden the topic in such a way that it has appeal to some members of the other group and then actively recruit the members of the minority group to participate in the activity.

–John Morgan, Founding Director of the SCGP 2010