Transformations and Singularities - Transformations and Singularities

Project: Research project

Project Details

Description

Transformations of surfaces or hypersurfaces allow to construct new (families of) surfaces or hypersurfaces of the
same kind. For example, an isothermic surface admits a variety of transformations into new isothermic surfaces.
This concept of transformations is rather far-reaching: for example, the classical (local) Weierstrass representation
of minimal surfaces can be interpreted as a special case of the Goursat transformation for isothermic surfaces.
Thus transformations may not just serve to construct new examples or classes of examples of surfaces but may, in
some cases, serve to construct all surfaces of a certain class from simple initial data.
Singularities are, most generally, the bad points of a theory, that is, points at which the methods of the theory
fail. This already constitutes a strong motivation for their study, since it requires to extend the employed methods,
to reconsider the objects studied and hence motivates new viewpoints on the theory. Moreover, studying global
properties of (hyper-)surfaces, singularities are in many cases inevitable: famous examples are the Caratheodory
conjecture, that every (convex) sphere in Euclidean space has at least two umbilics (singularities of its curvature
line net), or Hilbert's theorem on the non-existence of smooth complete pseudospherical surfaces in Euclidean
space.
The principal aim of this project is to study the interplay between transformations and singularities. More
precisely:
-- we aim to understand the singularities in the transformation theories of isothermic and Guichard surfaces and of
conformally flat hypersurfaces, how their transformations behave (or fail to behave) at certain points; and
-- we aim to study how those transformations create (or annihilate) singularities of the transformed (hyper-
)surfaces, and what the nature of the occurring singularities is.
A good understanding of these two aspects of the interplay between transformations and singularities will
ultimately lead to global transformation theories and to natural (global) definitions or characterizations of the
studied (hyper-)surface classes.
The bilateral collaboration between groups in Japan and in Austria will be key to the success of this research
project. In particular, expertise in transformations of surfaces and the integrable systems approach to the relevant
classes of (hyper-)surfaces (Austria) and in singularities of surfaces and global surface theory (Japan) will be
essential for the project; and exchange of diverse viewpoints on discrete differential geometry in both countries will
greatly enrich the related aspects of the project. However, our strategy is to not only bring together experienced
experts in the areas relevant to the problems addressed, but to emphasize support of young researchers and PhD
students in order to create sustainable and long term research collaborations between research groups in Japan and
in Austria.
StatusFinished
Effective start/end date1/07/1431/12/18