\titleofpresentation{Geometry of classical groups over rings}

\presenter{Alexander A. Lashkhi}

      Classical Groups can be defined when the scalars only form a
      ring (commutative in most cases). The methods used in their
      study when the scalars form a field can be slightly extended to
      local rings; but for more general rings, they don't apply any
      more, and new ideas were needed. They were brilliantly provided
      by O. T. O'Meara, J. Tits and F. Veldkamp. More recently,
      unexpected connection of Classical Groups with $K$-theory has been
      discovered (A. Hahn, D. James, U. Brehm, A. Bak and others)
      \cite{lashkhi1}--\cite{lashkhi5}. 

One of the main problems in this field of study is to translate the
     specific maps from the geometrical point of view
     (perspectivities, collineations, harmonic maps) into algebraic
     language (representation by the [semi]-linear isomorphisms)
---the
     Fundamental Theorems of Geometric Algebra. For different
     geometries (affine, projective, symplectic, orthogonal, unitary)
     and different rings we will consider this and other similar
     problems.  

A.  Defining the perceptivity (in the classical way) it's essential
                                that the geometry is the lattice with
                                complements. This property is false
                                for rings. Consequently, it's
                                necessary to change the definition of
                                perspectivities. Note that none single
                                ring versions of the theorem about
                                perspective maps is not known. With
                                this goal should be important the
                                definition: submodule $A\subseteq X$
                                is $D$-complement 
                                to the submodule $B\subseteq X$  if
$A\cap B=0$ and $\operatorname{PG}(K,A\oplus B)\cong\operatorname{PG}(K,X)$.
        Taking into consideration this one for the left Ore domains
                                desirable results could be obtained. 

     B. For the projective line harmonic maps and existence of K. von
                                Staudt's theorem was considered by
                                various of authors. Positive answers
                                were given for some specific
                                rings. However for the (commutative)
                                principal ideal domains the theorem of
                                K. von Staudt is invalid. Defining the
                                projective space as the set of all
                                free one-dimensional submodules for
                                some general noncommutative rings in
                                condition for non-bijective harmonic
                                maps is given and the complete analog
                                of von Staudt's theorem is obtained. 

     C.  For free modules over the rings with invariant basis property
                                (IBN-ring) the collineations and
                                lattice isomorphisms of affine spaces
                                and affine geometries was studied and
                                thus the fundamental Theorem of Affine
                                Geometry was proved. 
       
       \begin{ourbiblio}{9}
\bibitem{lashkhi1}
F. Buekenhout. (eds.) Handbook of Incidence geometry. Buildings and Foundations. Elsevier, 
      Amsteram--Lausanne--New York--Oxford--Shanno--Tokyo, 1995. 

\bibitem{lashkhi2}
U. Brehm. Representation of Sum-preserving mappings between submodule lattices. Manuscript.
      Hawaii, 1984, see also J. Math Sci. plenum Publ. CO. New York. 2006  (in printing).

\bibitem{lashkhi3}
J. Dieudonn\'e. La Geometrie des Groups Classiques. Springer, Berlin -
Heidelberg--New York, 1963. 

\bibitem{lashkhi4}
A. J. Hahn, O. Timothy O'Meara. The Classical Groups and
K-Theory. Springer-Verlag, Berlin--Heidelberg--New
York--London--Paris--Tokyo, 1989. 

\bibitem{lashkhi5}
J. Tits. Buildings of Spherical Type and Finite BN-pairs. Lecture
      Notes in Math. v. 386, Springer,  
      Berlin--Heidelberg--New York, 1974.
       \end{ourbiblio}

\presenteraddress{Batumi State University}

\email{lashkhi@spider.gtu.edu.ge}
