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Program

status March 21, 2001.

Sunday
March
25th

Expected arrival at Karlskrona. Accommodation at Hotel Conrad.

Monday
March
26th

08.30-09.30 Breakfast
09.30 Local transportation from Downtown Bus Stop not Far from Hotel Conrad
to Blekinge Institute of Technology (Bus Stop Annebo).

10.00-11.00 M. Brack, Introduction to Periodic Orbit Theory.

I give a brief introduction to the periodic orbit theory initiated by M. Gutzwiller
thirty years ago in the form of the semiclassical trace formula. This approach
allows one to approximate the quantum-mechanical density of states of a Hamiltonian
system in terms of a sum over the periodic orbits of the corresponding classical system,
whereby only properties derived from the classical equations of motion are required.
I present applications of the trace formula to an exactly solvable integrable system,
to experimentally observable quantum shell effects in realistic physical systems
(including chaotic dynamics), and point out an interesting connection to the density
of the non-trivial zeros of the Riemann zeta function.

11.30-12.30 M. de Gosson, Maslov Indices on the Universal Covering Group of Sp(n);
Relation with the Metaplectic Representation.

We present older work of ours from a new viewpoint, and try to relate
our constructions to the Brack-Gutwiller theory of periodic orbits
for Hamiltonian systems.

13.00 No-Host Lunch

14.15-15.00 K. Habermann, A Fourier Transform for Symplectic Spinor Fields.

A Fourier transform for symplectic spinor fields will be presented.
Using this Fourier transform, I will derive some consequences
for symplectic Dirac operators. Furthermore, applications in
mathematical physics will be given.
This talk is based on a joint work with Andreas Klein, Berlin.

15.15-16.00 M. Brack, Maslov Indices in Hamiltonian Systems with Mixed Dynamics.

I discuss the role of Maslov indices in the semiclassical trace formula
and present some physicists' recipes for their calculation. As a prime
example of a non-integrable system with mixed dynamics, I investigate the
H'enon-Heiles potential and the Maslov indices of some of its periodic
orbits appearing in connection with a cascade of orbit bifurcations which
forms the transition to chaos in a way reminiscent of the "Feigenbaum
scenario". As a byproduct, I present a realisation of the periodic Lam'e
functions whose zeros can be directly related to the Maslov indices.

16.15-17.00 H. Frisk, WKB for Multicomponent Wave Fields.

The work in WKB for multicomponent wave fields during the last decade
will be reviewed. Concepts as Berry's phase, Poisson curvature and
level-crossing will be discussed. Numerical experiments are presented
and philosophical remarks are given.

17.15-18.00 W.J. Schempp, Symplectic Spinors. I.

Keppler's phoronomy is basically a physics of phases and frequencies
whereas Newton's dynamics basically forms a physics of accellerations
and gravitational forces. Both fields have their own specific epistemic
flavours and therefore should not be confused. The purpose of the present
lecture is to outline the spectral aspects of the Kepplerian phoronomy
which are based on the orbit method. It uses the lowest weight sl(2,R)
module decomposition of the standard complex Hilbert space L²(R)
associated with the metaplectic representation ω in order to
understand the third Kepplerian law of planetary motion as a Bohr-
Sommerfeld quantization rule for symplectic spinors. This rule is
deduced from the tracial character formula of the real Heisenberg nilpotent
Lie group G. In a forthcoming paper, the same spectral principles of
the Kepplerian phoronomy will be applied to deduce the isotropic
Schwarzschild metric of relativistic astrophysics.

19.00 No-Host Supper

Tuesday
March
27th

08.30-09.30 Breakfast

09.30 Local transportation from Downtown Bus Stop not Far from Hotel Conrad
to Blekinge Institute of Technology (Bus Stop Annebo).

10.00-11.00 B. Hiley, Spinors in Physics.

We discuss the notion of symplectic spinors from the point of view
of quantum mechanical "shadow phase spaces".

11.30-12.30 C. Farsi, Orbifold Eta Invariants.

In this talk we will present an index theorem for orbifolds with boundary
and introduce orbifold eta invariants.

13.00 No-Host Lunch

14.30-15.30 M. Lesch, On the Maslov Index.

The Maslov Index for paths of Lagrangian subspaces is a significant
homotopy invariant in symplectic analysis. There exists a rich literature
and various (≥10) definitions. In my talk I will add another definition
(not completely new but with its own twist) to this list. My presentation
will be completely self--contained. I will work in the setting of
symplectic functional analysis; this setting is suitable also for
applications like the study of boundary value problems for Dirac--type
operators. The talk will also touch such applications.
My talk will be based on joint work with Paul Kirk, Bloomington.

16.00-17.00 K. Furutani, Spectral Flow and Maslov Index.

I would like to explain the role of the product structure of elliptic operators
in a splitting formula of the spectral flow and a related reduction theorem
of the Maslov index of Cauchy data spaces.

17.15-18.00 W.J. Schempp, Symplectic Spinors. II.

19.00 Visit of the Karlskrona Leonardo da Vinci Museum

19.30 Conference Dinner

Wednesday
March
28th

07.30-08.30 Breakfast

08.45 Local transportation from Downtown Bus Stop not Far from Hotel Conrad
to Blekinge Institute of Technology (Bus Stop Annebo).

09.15-10.00 J. E. Björk, Microdifferential Opertators.

We present a survey of the work of Sato, Kawai and Kashiwara.

10.15-11.00 G. Tuynman, R^{2n} is a Universal Model for Symplectic Reduction.

Reduction of symplectic manifolds with respect to a (hamiltonian) group action
is an important procedure. More general, and also important especially
for mechanical systems with constraints, is the symplectic reduction
by a coisotropic submanifold. I will show that all symplectic manifolds
can be obtained by symplectic reduction from the standard model R^{2n}
for a suitable choice of the coisotropic submnifold. This universality holds
even when the symplectic manifold comes with a hamiltonian action of a
compact group, in which case the equivaraint reduction starts from R^{2n}
with an orthogonal action of the compact group in question. These results
should be seen as the symplectic counter part of the Whitney embedding theorem
(any manifold can be seen as a submanifold of some R^{n})
and the extension by Mostow and Palais to manifolds with a compact group action.

11.30-12.15 J. Toft, Young Type Inequalities for a Family of Distribution Spaces,
Defined on Symplectic Vector Spaces.

We consider an increasing family s_p, p ε [1,∞], of Banach algebras
under the twisted convolution. Moreover, s_p is the set of all symbols
such that the corresponding Weyl operators, in pseudo-differential calculus,
are Schatten-von Neumann operators of order p. We prove that the s_p-spaces
satisfy some Young type conditions for dilated multiplications and
convolutions.

12.30 No-Host Lunch

14.00-15.00 B. Ørsted, The Maslov Index and Jordan Algebras.

In this talk we give a new definition and derivation of the properties of the classical
Maslov index, using the geometry of the Hermitian symmetric space for the symplectic
group. At the same time we generalize the Maslov index, replacing the symplectic group
by the Koecher-Tits group for a Euclidian Jordan algebra and the space of Lagragian
subspaces by the Shilov boundary of the corresponding Hermitian symmetric tube-type
domain. This is joint work with J.-L. Clerc.

Departure

March 2001, Bernhelm Booss-Bavnbek

Sunday March 25th
		Expected arrival at Karlskrona. Accommodation at Hotel Conrad.
Monday March 26th
08.30-09.30		Breakfast
09.30		Local transportation from Downtown Bus Stop not Far from Hotel Conrad to Blekinge Institute of Technology (Bus Stop Annebo).
10.00-11.00		M. Brack, Introduction to Periodic Orbit Theory.
		I give a brief introduction to the periodic orbit theory initiated by M. Gutzwiller thirty years ago in the form of the semiclassical trace formula. This approach allows one to approximate the quantum-mechanical density of states of a Hamiltonian system in terms of a sum over the periodic orbits of the corresponding classical system, whereby only properties derived from the classical equations of motion are required. I present applications of the trace formula to an exactly solvable integrable system, to experimentally observable quantum shell effects in realistic physical systems (including chaotic dynamics), and point out an interesting connection to the density of the non-trivial zeros of the Riemann zeta function.
11.30-12.30		M. de Gosson, Maslov Indices on the Universal Covering Group of Sp(n); Relation with the Metaplectic Representation.
		We present older work of ours from a new viewpoint, and try to relate our constructions to the Brack-Gutwiller theory of periodic orbits for Hamiltonian systems.
13.00		No-Host Lunch
14.15-15.00		K. Habermann, A Fourier Transform for Symplectic Spinor Fields.
		A Fourier transform for symplectic spinor fields will be presented. Using this Fourier transform, I will derive some consequences for symplectic Dirac operators. Furthermore, applications in mathematical physics will be given. This talk is based on a joint work with Andreas Klein, Berlin.
15.15-16.00		M. Brack, Maslov Indices in Hamiltonian Systems with Mixed Dynamics.
		I discuss the role of Maslov indices in the semiclassical trace formula and present some physicists' recipes for their calculation. As a prime example of a non-integrable system with mixed dynamics, I investigate the H'enon-Heiles potential and the Maslov indices of some of its periodic orbits appearing in connection with a cascade of orbit bifurcations which forms the transition to chaos in a way reminiscent of the "Feigenbaum scenario". As a byproduct, I present a realisation of the periodic Lam'e functions whose zeros can be directly related to the Maslov indices.
16.15-17.00		H. Frisk, WKB for Multicomponent Wave Fields.
		The work in WKB for multicomponent wave fields during the last decade will be reviewed. Concepts as Berry's phase, Poisson curvature and level-crossing will be discussed. Numerical experiments are presented and philosophical remarks are given.
17.15-18.00		W.J. Schempp, Symplectic Spinors. I.
		Keppler's phoronomy is basically a physics of phases and frequencies whereas Newton's dynamics basically forms a physics of accellerations and gravitational forces. Both fields have their own specific epistemic flavours and therefore should not be confused. The purpose of the present lecture is to outline the spectral aspects of the Kepplerian phoronomy which are based on the orbit method. It uses the lowest weight sl(2,R) module decomposition of the standard complex Hilbert space L²(R) associated with the metaplectic representation ω in order to understand the third Kepplerian law of planetary motion as a Bohr- Sommerfeld quantization rule for symplectic spinors. This rule is deduced from the tracial character formula of the real Heisenberg nilpotent Lie group G. In a forthcoming paper, the same spectral principles of the Kepplerian phoronomy will be applied to deduce the isotropic Schwarzschild metric of relativistic astrophysics.
19.00		No-Host Supper
Tuesday March 27th
08.30-09.30		Breakfast
09.30		Local transportation from Downtown Bus Stop not Far from Hotel Conrad to Blekinge Institute of Technology (Bus Stop Annebo).
10.00-11.00		B. Hiley, Spinors in Physics.
		We discuss the notion of symplectic spinors from the point of view of quantum mechanical "shadow phase spaces".
11.30-12.30		C. Farsi, Orbifold Eta Invariants.
		In this talk we will present an index theorem for orbifolds with boundary and introduce orbifold eta invariants.
13.00		No-Host Lunch
14.30-15.30		M. Lesch, On the Maslov Index.
		The Maslov Index for paths of Lagrangian subspaces is a significant homotopy invariant in symplectic analysis. There exists a rich literature and various (≥10) definitions. In my talk I will add another definition (not completely new but with its own twist) to this list. My presentation will be completely self--contained. I will work in the setting of symplectic functional analysis; this setting is suitable also for applications like the study of boundary value problems for Dirac--type operators. The talk will also touch such applications. My talk will be based on joint work with Paul Kirk, Bloomington.
16.00-17.00		K. Furutani, Spectral Flow and Maslov Index.
		I would like to explain the role of the product structure of elliptic operators in a splitting formula of the spectral flow and a related reduction theorem of the Maslov index of Cauchy data spaces.
17.15-18.00		W.J. Schempp, Symplectic Spinors. II.

19.00		Visit of the Karlskrona Leonardo da Vinci Museum
19.30		Conference Dinner
Wednesday March 28th
07.30-08.30		Breakfast
08.45		Local transportation from Downtown Bus Stop not Far from Hotel Conrad to Blekinge Institute of Technology (Bus Stop Annebo).
09.15-10.00		J. E. Björk, Microdifferential Opertators.
		We present a survey of the work of Sato, Kawai and Kashiwara.
10.15-11.00		G. Tuynman, R^{2n} is a Universal Model for Symplectic Reduction.
		Reduction of symplectic manifolds with respect to a (hamiltonian) group action is an important procedure. More general, and also important especially for mechanical systems with constraints, is the symplectic reduction by a coisotropic submanifold. I will show that all symplectic manifolds can be obtained by symplectic reduction from the standard model R^{2n} for a suitable choice of the coisotropic submnifold. This universality holds even when the symplectic manifold comes with a hamiltonian action of a compact group, in which case the equivaraint reduction starts from R^{2n} with an orthogonal action of the compact group in question. These results should be seen as the symplectic counter part of the Whitney embedding theorem (any manifold can be seen as a submanifold of some R^{n}) and the extension by Mostow and Palais to manifolds with a compact group action.
11.30-12.15		J. Toft, Young Type Inequalities for a Family of Distribution Spaces, Defined on Symplectic Vector Spaces.
		We consider an increasing family s_p, p ε [1,∞], of Banach algebras under the twisted convolution. Moreover, s_p is the set of all symbols such that the corresponding Weyl operators, in pseudo-differential calculus, are Schatten-von Neumann operators of order p. We prove that the s_p-spaces satisfy some Young type conditions for dilated multiplications and convolutions.
12.30		No-Host Lunch
14.00-15.00		B. Ørsted, The Maslov Index and Jordan Algebras.
		In this talk we give a new definition and derivation of the properties of the classical Maslov index, using the geometry of the Hermitian symmetric space for the symplectic group. At the same time we generalize the Maslov index, replacing the symplectic group by the Koecher-Tits group for a Euclidian Jordan algebra and the space of Lagragian subspaces by the Shilov boundary of the corresponding Hermitian symmetric tube-type domain. This is joint work with J.-L. Clerc.
		Departure