Abstracts for Extra talk

The title may sound too much of an old issue. However  recently in the case of positive characteristic cases we found new phenomena and interesting techniques much more than the case of characteristic zero. For this rare precious opportunity, I do not want  to spend much time on  the history of the topic nor about possible applications. I will focus my effort to explain all the essence of the proofs in positive characteristics and arithmetic cases. If my talk becomes too technical to any one among the audience, please stop me and question me.

Recently, many efforts have been made to integrate a Courant algebroid, namely to find a global object associated to a Courant algebroid (For example, a global object corresponds to a Lie algebra is a Lie group). One of the reasons is probably that the standard Courant algebroid serves as the generalized tangent bundle of a generalized complex manifold of Hichin and Gualtieri. Thus the integration will help to understand the global symmetry of such manifolds.

 When a stringed instrument is played, the sound we hear consists of a fundamental tone and infinitely many harmonic tones.  Mathematically, what we ``hear'' is the spectrum of the Laplace operator.  In one dimension, the only geometric quantity to be heard is the length of the string.  However, in higher dimensions a natural question is:  What can we hear?  In the 1960s, a popular variation of this question in two dimensions was posed by M.