September 19, 2003

G.Khimshiashvili
"Quaternions in some problems of low-dimensional geometry"

We discuss the topological and geometric structure of certain
differentiable mappings which are constructed using quaternions. For
example, it turns out that, for a unilateral (i.e., variable commutes with
the coefficients) quaternionic polynomials in one variable, the structure of
its zero set can be described in a quite effective way. Related results
about the structure of stable perturbations of a quaternionic branch point
and some conjectures about the structure of stable perturbations of
quaternionic polynomials will be also presented. It will be also shown that
quaternions may be used for constructing analytic discs in loop spaces
and conformal codimension one immersions of two-dimensional closed surfaces.
Another two visual applications of quaternions are related to the so-called
Weisbach's theorem (called sometimes the main theorem of axonometry) and
to the description of configuration spaces of polygons in five-dimensional
Euclidean space.
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http://www.pdmi.ras.ru/~lowdimma