Abstracts for Topics in Topology

Abstract: There are a lot of questions related to affine algebraic  geometry which have very simple formulations but which, sometimes, are  very difficult to answer. For example, are the surfaces given by  $x^ny=z^2-1$ isomorphic for different $n$? or is a hypersurface given  by $x+x^2y+z^2+t^3=0$ isomorphic to three-dimensional affine space?
In my talk I'll answer these and some other questions with the help of  automorphism groups of these objects.
 

We consider an infinite symmetric group $G$. For some its
subgroups $K$ double cosets $K\setminus G/ K$ admit a natural  structure of
a semigroup (remark for experts: this semigroup acts in  $K$-fixed vectors
of representations of $G$).  Elements of $K\setminus  G/ K$ can be
interpreted as two-dimensional polygonal bordisms, the  multplication is
the product of bordisms. We also
produce a family of constructions in the spirit of 'topological field
theories' (i.e., for each bordism we write a matrix, product of  bordisms