A different way to think about something Zoran Sunic did
Hopfian Groups are Complete co-Analytic, Submitted, arχiv)
Frucht's Theorem without Choice (short verson)a versone full of typos I haven't put on arxiv yet
I work on permutation models of ZFA, which are used to show the independence of the axiom of choice. Sometimes, especially without choice, the properties of a structure depend on the ambient set theory in unintuitve ways. We would like this not to happen; for example, we shouldn't care whether real numbers are defind as dadekind cuts or sets of cauchy sequences. By working in ZFA (ZF with atoms), we can often build our structures so all of their elements are atoms. This avoids any ambiguity in how the structure is defined, but also gives us all of our usual tools in ZF set theory to solve them. This is philosophically similar to mathematics done within type theoretic foundations, where the way a structure is encoded as sets isn't part of the type signature, and is inaccessible in proofs.
My dissertation shows that, over ZFA, it is possible to have a group that isn't the automorphism group of a graph. However, this is impossible over ZF, because elements of a group are themselves sets, which, by induction on rank, we can label with graphs. I am working on related problems in borel combinatorics.
I have worked on some problems in group theory and borel combinatorics as well
A different way to think about something Zoran Sunic did
Hopfian Groups are Complete co-Analytic, Submitted, arχiv)
Frucht's Theorem without Choice (short verson)a versone full of typos I haven't put on arxiv yet