[geometry-ml:00129]

[Atsushi KATSUDA]

% 幾何学メーリングリストの皆様

% 以下のような磁場その他に関連する幾何の小研究会を開催いたします。
% ご興味のある方はご参加ください。

% 勝田 　篤　（岡山大学理学部）

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\begin{center}
\textbf{\Large 幾何学小研究集会　岡山２００３}
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\begin{flushleft}
日時：2003年9月8日(月)10:30〜9月9日(火)12:00\\
場所：岡山大学理学部幾何ゼミナール室 (A308) \\
（変更の可能性がありますがその場合はこの室の入口に掲示します。）
\\
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\begin{center}
\textbf{Program}
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\begin{tabbing}
\textbf{8 September}
\end{tabbing}

\begin{tabbing}
\=10:30--12:00 \= \ \  Urs Frauenfelder (Hokkaido University　北大)\\
\quad\quad Twisted cotangent bundles and periodic orbits \quad\quad
\end{tabbing}

\begin{quotation}
I will talk of a joint work with Felix Schlenk. We prove
existence of contractible closed orbits of a charge in a nonvanishing
magnetic field on almost all small energy levels. The proof is based on the
construction of some spectral norm. Using a separation result of
L.Polterovich we can show that the spectral norm is uniformly bounded. This
result enables us to estimate some version of Hofer-Zehnder capacity from
above and hence we can derive almost existence of periodic orbits.
\end{quotation}




\begin{tabbing}
\=13:30--15:00 \= \ \  Toshiaki Adachi  足立　俊明 (名工大)\\
\quad\quad  A report on K\"ahler magnetic fields \quad\quad
\end{tabbing}

\begin{quotation}
Let $(M,J)$ be a K\"ahler manifold with complex structure $J$.
We call a closed 2-form ${\bf B}_{\kappa} = \kappa {\bf B}_J$,
which is a constant multiple of the K\"ahler form ${\bf B}_J$, a {\it
K\"ahler magnetic field} on $M$. A smooth curve $\gamma$ parameterized by its arclength is called a 
trajectory for ${\bf B}_{\kappa}$ if it satisfies
 $\nabla_{\dot{\gamma}} \dot{\gamma} = \kappa J \dot{\gamma}$.
I will talk about my study on K\"ahler manfolds from Riemannian geometric
point of view.
\end{quotation}

\begin{tabbing}
\=15:30--17:00 \= \ \  Ruishi Kuwabara 桑原　類史　(徳島大) \\
\quad\quad Eigenvalues associated  with a periodic orbit
of the magnetic flow \quad\quad
\end{tabbing}

\begin{quotation}
The so-called Bohr-Sommerfeld-Maslov quantization condition plays
an important role in the semi-classical analysis for the completely
integrable systems. In this case we have an approximate quantum
energy associated with each invariant torus satisfying BSM condition.
On the other hand, motivated by Voros' heuristic work on the quantization
along a closed orbit for the non-integrable systems, Guillemin and
Weinstein, and Ralston have shown that there exists a sequence of eigenvalues of the Laplace-
Beltrami operator associated to a stable periodic orbit of the
geodesic flow.
  In this talk we consider the case of the magnetic flow, where the system
is not homogeneous. By introducing the lifted system in the frame work of the
principal $U(1)$ bundle, we can show the existence of  a sequence of eigenvalues
associated with a stable periodic orbit satisfying a ``quantization
condition".
\end{quotation}




\begin{tabbing}
\textbf{9 September}
\end{tabbing}

\begin{tabbing}
\=10:00--12:00 \= \ \  Tetsuya Tate 楯　辰哉　(慶応大)\\
\quad\quad Distribution laws for integrable eigenfunctions
 \quad\quad
\end{tabbing}

\begin{quotation}
We determine  the asymptotics  of the joint eigenfunctions
of the torus action on a toric Kahler variety. Such varieties are
models of completely integrable systems in complex geometry.  We
first determine the pointwise asymptotics of the eigenfunctions,
which show that they behave like  Gaussians centered at the
corresponding classical torus. We then
 show that there is a  universal Gaussian scaling limit of
the distribution function near its center. We also determine the
limit distribution for the tails of the eigenfunctions  on large
length scales. These are not universal but depend on the global
geometry of the toric variety and in particular on the details of
the exponential decay of the eigenfunctions away from the
classically allowed set.
\end{quotation}



\smallskip


\noindent 連絡先： 岡山市津島中３−１  岡山大学理学部数学科　勝田　篤　\\
\quad\quad\quad\quad ({\tt tel 086(251)7793 e-mail katsuda ＠ math.okayama-u.ac.jp})\\

\smallskip


\noindent 会場は以下の
\begin{verbatim}
  http://www.okayama-u.ac.jp/Location/tsushima_j.html
\end{verbatim}

\noindent の案内図で理学部の建物の北側部分の３階、ちょうど``理学部"とかいて
ある部分の``部 "の字の北側あたりです。

\noindent また、岡山大学津島キャンパスへのアクセスは
\begin{verbatim}
  http://www.okayama-u.ac.jp/Location/location_j.html
\end{verbatim}

\noindent を参照してください。
\end{document}

----
Atsushi KATSUDA  katsuda ＠ math.okayama-u.ac.jp