Topology is the study of space up to continuous or differentiable deformations — roughly speaking, you can stretch and bend space, but you may not cut space apart or glue parts together. For example, the surface of the Earth is not perfectly round, but it is topologically the same as a sphere, because it can be deformed into one. One interesting problem is understanding the possible topological shapes of the universe we live in. Of course, the question of what the actual shape is belongs to the general field of physics, and is not likely to be answered any time soon.
The common research of the members of the group is low-dimensional topology — the study of spaces of dimension up to 4. Dimensions 3 and 4 are particularly important because these are the dimensions of space and spacetime that we live in. Such spaces are impossible to draw, but they can be described mathematically through knot and link theory.
Vladimir Chernov studies ordinary and virtual links. Another part of his research is about the relationship between causal structures on spacetimes and the theory of Legendrian knots coming from a branch of differential geometry called contact geometry. He is also interested in Lorentz geometry and smooth structures on spacetimes.
Ina Petkova studies Heegaard Floer homology and its applications to low-dimensional topology. The original definition involves heavy analysis, whereas recent computational methods rely on algebra and combinatorics. The variant of Heegaard Floer homology for links categorifies the famous Alexander polynomial. Ina is also interested in the connections between the variant for tangles (pieces of a knot) and categorified representation theory.