**Leonidas
Alaoglu Memorial Lecture**
in Mathematics
Tuesday, February 9, 2010
**4:15
p.m. 151 Sloan**
**Leonid Polterovich**
Professor of Mathematics
Tel Aviv University
University of Chicago
MSRI
**Symplectic
rigidity and quantum-classical correspondence**
**Abstract: **Consider
the algebra of quantum observables, that is of Hermitian operators
acting on a Hilbert space. A quasi-state on this algebra is a
real-valued function linear on every commutative subspace and taking
non-negative values on non-negative operators. In 1957 Andrew Gleason
proved the following remarkable theorem which had a significant impact
on mathematical foundations of quantum mechanics: every quasi-state
must be linear on the whole Hilbert space provided the dimension is at
least three. The quantum-classical correspondence takes Hermitian
operators to functions on a symplectic manifold (classical
observables), and commutators to Poisson brackets. It turns out that on
certain symplectic manifolds, such as projective spaces, one has the
following "anti-Gleason" phenomenon: the algebras of functions on these
manifolds admit non-linear quasi-states. Interestingly enough, in
dimension four and higher all known non-linear quasi-states come from
Floer theory, the cornerstone of ``hard" symplectic topology. In the
lecture, which is based on a series of joint works with Michael Entov,
I shall discuss origins and applications of symplectic quasi-states as
well as the link between quasi-states and group theory.
Dr. Leonid Polterovich was born in Moscow in 1963. Dr. Polterovich received his MSc in mechanics at Moscow State University in 1984. His research interests include symplectic and contact topology and dynamical systems. While he was working in industry until emigration to Israel, he continued research within Sinai's and Arnold's seminars. Dr. Polterovich received his PhD in 1990 at Tel Aviv University under the supervision of Yakov Sinai and Vitali Milman. Since then he has been at Tel Aviv University where he currently holds the Gordon Chair in Symplectic Topology and Hamiltonian dynamics. In the Fall of 2009 Dr. Polterovich joined the Department of Mathematics at University of Chicago. During the Winter quarter of 2010 he will be visiting MSRI as a Simons Professor. Dr. Polterovich has been awarded The European Congress of Mathematicians Award for Young Mathematicians (1996), Erdos prize of the Israel Mathematical Union (1998) and the Michael Bruno Memorial Award (2003). In 1998 he delivered an invited address to the International Congress of Mathematicians, Berlin.
*The
Leonidas Alaoglu Memorial Lecture was established by friends and family
of the late Leonidas Alaoglu in recognition of his great talents,
distinguished contributions to mathematics, and long friendship with
Caltech. The Institute is privileged to honor his memory with a lecture
each year by an outstanding mathematician.* |