第１部： "How Poincare became my hero" （ポアンカレ記念特別講演）
Abstract：
I recently discovered littleknown texts of Poincare which include
fundamental results on PDEs together with prophetic insights into
their future impact on various branches of modern mathematics.
第２部： "Can you hear the degree of a map from the circle into itself? An intriguing story which is not yet finished"
Abstract：
A few years ago  following a suggestion by I. M. Gelfand  I
discovered an intriguing connection between the topological degree
of a map from the circle into itself and its Fourier coefficients.
This relation is easily justified when the map is smooth. However,
the situation turns out to be much more delicate if one assumes
only continuity, or even Holder continuity. I will present recent
developments and open problems. The initial motivation for this
direction of research came from the analysis of the GinzburgLandau model.
第３部： "Sobolev maps with values into the circle"
Abstract：
Sobolev functions with values into R are very well understood
and play an immense role in many branches of Mathematics.
By contrast, the theory of Sobolev maps with values into the unit
circle is still under construction. Such maps occur e.g. in the
asymptotic analysis of the GinzburgLandau model. The reason one
is interested in Sobolev maps, rather than smooth maps is to allow
singularities such as x/x in 2D or line singularities 3D which
appear in physical problems. Our focus in these lectures is not
the GinzburgLandau equation per se, but rather the intrinsic study
of the function space W^{1,p} of maps from a smooth domain in R^N
taking their values into the unit circle. Such classes of maps have
an amazingly rich structure. Geometrical and Topological effects
are already noticeable in this simple framework, since S^1 has
nontrivial topology. Moreover the fact that the target space is
the circle (as opposed to higherdimensional manifolds) offers the
option to introduce a lifting. We'll see that "optimal liftings"
are in onetoone correspondence with minimal connections (resp.
minimal surfaces) spanned by the topological singularities of u.
I will also discuss the question of uniqueness of lifting . A key
ingredient in some of the proofs is a formula (due to myself,
Bourgain and Mironescu) which provides an original way of
approximating Sobolev norms (or the total variation) by nonlocal
functionals. Nonconvex versions of these functionals raise very
challenging questions recently tackled together with H.M. Nguyen.
Comparable functionals also occur in Image Processing and suggest
exciting interactions with this field.
