The Hebrew University of Jerusalem
The Hebrew University of Jerusalem
Welcome to the Einstein Institute of Mathematics
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The Einstein Institute of Mathematics continues the
Landau lecture series,
a series of three lectures by a distinguished mathematician on the subject of
his choice.It is usually suggested that one lecture be accessible to the university scientific community at large and the additional two lectures may be more specialized and of interest only to professional mathematicians and specialists.
Past lectures (1991-2010):
Edmund Landau (1877-1938) |
2011: Prof. Helmut Hofer (School of Mathematics, Institute for Advanced Study, Princeton) June 2nd, 5th and 6th
- Lecture 1 From Celestial Mechanics to a Geometry Based on Area
Thursday, June 2nd Department Colloquium- The mathematical problems arising from modern celestial mechanics, which originated with Isaac Newton's Principia in 1687, have led to many mathematical theories. Poincaré (1854-1912) discovered that a system of several celestial bodies moving under Newton's gravitational law shows chaotic dynamics. Earlier, Euler (1707-83) and Lagrange (1736-1813) found instances of stable motion. For example a spacecraft in the gravitational fields of the sun, earth, and the moon provides an interesting system which can experience stable as well as chaotic motion.
These seminal observations have led to the theory of dynamical systems and to the field of symplectic geometry, which is a geometry based on area rather than distance. It is somewhat surprising that these fields have developed separately, since both, in their modern form, have their origin with Poincaré, who had an highly integrated view point. Given the highly developed states of both fields, and the background of some promising results, the time seems ripe to bring them together around the core of Hamiltonian mechanics in a field which perhaps should be called "Symplectic Dynamics". The talk will conclude by giving some ideas what this field would be about.
- Lecture 2 & 3 Holomorphic Curves in Symplectic Geometry and a Generalization of Fredholm Theory I & II
Sunday-Monday, June 5th & 6th Seminar Talks- A meanwhile standard idea for producing geometric invariants (f.e. Donaldson Theory, Gromov-Witten Theory, Symplectic Field Theory) consists of counting solutions of nonlinear elliptic systems associated to the geometric data. Although the basic idea is easy, the implementation can be very difficult and involved, due to a usually large number of technical issues, which in more classical approaches to such type of problems are more than "painful".
In symplectic geometry the partial differential equation in question is the nonlinear Cauchy-Riemann equation. If there weren't these inherent compactness and transversality problems, the solution sets of the nonlinear Cauchy-Riemann operator would be nice manifolds or orbifolds, and the invariants could be achieved by integration of suitable differential forms over them. As it turns out, the arising difficulties can be overcome by a drastic generalization of nonlinear Fredholm theory and new methods for implementing it in concrete problems.
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Last updated: April 10th, 2011