Helmut Hofer
Institute for Advanced Study
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Primary Section: 11, Mathematics Secondary Section: 13, Physics Membership Type:
Member
(elected 2008)
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Research Interests
As a mathematician I have been interested in partial differential equations, dynamical systems and symplectic geometry. A central object in symplectic geometry and Hamiltonian mechanics is the symplectic diffeomorphism group. Symplectic structures have a very persuasive occurrence in the equations of physics, but also arise in the study of the structure of three-dimensional and four-dimensional spaces. About twenty years ago I introduced a new infinite-dimensional geometry, meanwhile called 'Hofer Geometry', which by now is the standard tool for approaching a broad range of questions in Hamiltonian dynamics and symplectic geometry. Jointly with Y. Eliashberg and V. Givental we developed a theory of symplectic invariants called 'Symplectic Field Theory'. It has relations to string theory, low-dimensional topology, and Hamiltonian dynamics. The details of symplectic field theory are still being worked out, and require the development of new tools in nonlinear analysis, which is being done jointly with K. Wysocki and E. Zehnder.
