My research papers concern the theory of transformation groups on manifolds and CW complexes.
The mathematical methods which are applied to solve the discussed problems go back to algebraic and geometric topology, group theory and representation theory.
Transformation groups appear in such branches of mathematics as algebra, geometry, topology,
and analysis, and number theory, as well as in theoretical physics; for example, in gauge theory,
quantum mechanics, or Einstein’s theory of general relativity.
In the theory of transformation groups, one studies groups of symmetries of the objects of interest.
In geometry and topology, manifolds play a central role and the discussed groups of symmetries of manifolds are Lie groups. Focusing on Lie groups, also when the manifold in question is not smooth, is justified by the Hilbert-Smith conjecture whose affirmative solution reads as follows.
Every locally compact group acting both continuously and effectively on a topological manifold
is a Lie group. In order to prove that the claim is true, it remains to show that there does not exist such an action in the case where the acting group is the group of p-adic integers for a prime p.
In the theory of transformation groups, a basic problem reads as follows. Let G be a compact Lie group. For a smooth manifold M, describe necessary and sufficient conditions for the existence of
a smooth action of G on a manifold W with specific properties (e.g., with a given homotopy type)
such that Fix(G,W), the set of points in W fixed under the action of G, is diffeomorphic to M.
A more elaborate problem includes in addition a G-vector bundle N over M, and calls for necessary and sufficient conditions imposed on M and N for the existence of a smooth action of G on a specific manifold W such that Fix(G,W) is diffeomorphic to M and the G-equivariant normal bundle of M in W
is stably isomorphic to N.
One may frase the problem other way around. For a given smooth manifold M, which compact Lie groups G act smoothly on specific manifolds W in such a way that Fix(G,W) is diffeomorphic to M?
If M is empty or M is just one point, one obtains in particular the following two questions.
Which compact Lie groups G act smoothly on disks (resp., Euclidean spaces) without fixed points?
Which compact Lie groups G act smoothly on spheres with just one fixed point?
If M consists of exactly two points, any real G-vector bundle over M consists of two real G-modules
. For a compact Lie group G, an example of the more elaborate problem reads as follows.
For a smooth action of G on a sphere with exactly two fixed points, which real G-modules can occur stably as the tagent G-modules at the two fixed points?