Topological polymers have architecture that is topologically nontrivial, i.e. cannot be contracted to a point by a smooth transformation (no cutting, but crumpling and stretching is allowed). The most prominent example is a ring (circular) polymer, while ordinary (well studied) linear chain is topologically trivial.

The nontrivial topology impacts many fundamental properties, ranging from the average chain size of isolated topological polymers to the viscoelastic response of their solutions. The latter is particularly interesting in the dense case where many chains share the same volume and have to adapt their conformation and motion accordingly. This fulfillment of the mutual-avoidance and the "topological constraints" create new forms of entanglements that I study.

Here are some of my papers on the topic:

• Soap bubbles on ring polymers

• Supercoiled circular DNAs

• Are rings trees?

Active polymers are ones that are out of equilibrium on the scale of their constituent monomers. This can be achieved by being composed of particles that perform mechanical work on their environment on the expense of some source of free energy, or if the chain is subject to nonequilibrium fluctuations, e.g. by the presence of some elements that exert forces locally on the monomers. A good example is a DNA fiber in living cells that is subject to action of various molecular motors pulling on it.

The energy input on the smallest scale can generate new phenomena, such as conformations or dynamics that are inaccessible in equilibrium and that in turn create novel material properties.

Here are some of my papers on the topic:

• Activity-driven phase separation in polymers

• Active topological glass

• A connection between the two above

Typically polymer conformations (and the corresponding dynamics) are self-similar. Therefore, self-similar objects (fractals) serve well as their mathematical odels of the conformations. These can be characterised by various fractal dimensions, which are the (universal) exponents in the power-law dependence of two quantities of interest, such as 3D size versus polymer length.

Have you ever seen an object that is one-dimensional, three-dimensional and 2.85-dimensional at the same time? What if I tell you there are billions of them in your body?

Here you are:

• Fractal chromosomes

• Space-filling curves with space-filling surfaces (make one yourself)

• How many such curves exist?