In this case study Jennifer Pi studies the complexity of the "classification programme" for certain objects called C$^*$-algebras. The work was done together with Michal Szachniewicz and Mira Tartarotti [1].
What do I mean by "classification programme"? A simple example is the classification of triangles; for example, we know any two triangles with the same internal angles must be similar (i.e. one triangle must be a scaled-up version of the other). In this classification, the objects are triangles, which we classify up to similarity, and the invariant is the angle measurements. In my area of research, the classification programme deals with C$^*$-algebras as the objects, which we classify up to isomorphism, and the invariant consists of $K$-theory, traces, and the interaction between these two. [It is okay for the purposes of this brief case study that you don't know what $K$-theory and traces are.]
C$^*$-algebras are norm-closed $*$-subalgebras of bounded operators on a Hilbert space. Some straightforward examples of these include matrix algebras in the finite-dimensional case, and $C[0,1]$ (or indeed, continuous functions over any compact space) in the commutative case. There are many infinite-dimensional noncommutative C$^*$-algebras too, for example the Toeplitz algebra which is generated by the one-sided shift operator on $\ell^2(\mathbb{N})$. I won't write too much more about these objects, as there is so much to say that it could (and does) occupy entire careers! The important things to note are that they were originally motivated by mathematical physics, and that classifying them up to isomorphism is a difficult task that mathematicians have been working on since the 1970s.
The classification of an appropriate class of C$^*$-algebras has been completed recently (see the survey [2] for history and context), but a reasonable question to ask is how complex this classification is. Certainly, the invariant of "$K$-theory, traces, and their interaction" sounds much more complex than just the invariant of angles used for triangles!
We study the complexity of classification of C$^*$-algebras using certain games from mathematical logic called Ehrenfeucht-Fraïssé games. Formally, for the case of discrete (classical) logic, these games can be described as follows: Given structures $A$ and $B$ of some discrete, relational language $L$ and $n<\omega$, the game $\text{EF}_n(A,B)$ is played between two players: Player I (the challenger) and Player II (the duplicator). The challenger wants to prove that $A$ and $B$ are different (and that the considered logic can detect this difference), while the duplicator wants to show that they are similar. In round $k$, Player I chooses an element of $A$ or $B$, and Player II responds by choosing an element of the other structure. The played elements $a_k\in A$ and $b_k\in B$ are recorded, and after $n$ rounds, Player II wins if the resulting sequences $a\in A^n$, $b\in B^n$ satisfy the same quantifier-free $L$-formulas.
The key idea is: Players I and II take turns playing elements from $A$ and $B$, and check at the end of the game whether the underlying logic can see any difference between the played elements.
For simplicity, I've described the game above for discrete, purely algebraic structures. There is a variant of this game which works for continuous structures (like C$^*$-algebras), and also incorporates varying game length via the use of a game clock that Player I controls. The game clock starts at some countable ordinal $\alpha$, and at round $k$ of the game, Player I is allowed to set a new "time" on the clock: $\alpha_k < \alpha_{k-1}$. Again the key is that the clock is strictly decreasing, so every play of the game is finite, but this clock allows for more flexibility because Player II does not know a priori how long the game will last.
What we show in our paper is that, essentially, the classification of $C^*$-algebras is not hopelessly complex, in the sense that these mathematical games played on the invariant ($K$-theory and traces) can be transferred to mathematical games played on the $C^*$-algebras. For the purposes of this case study, we write $A \equiv_{\alpha} B$ for two objects $A, B$ if Player II (duplicator) has a winning strategy for every game played between $A$ and $B$ with starting clock $\alpha$.
Theorem
Let $A$ and $B$ be classifiable $C^*$-algebras, and let $KT_u(\cdot)$ denote the invariant involving $K$-theory and traces. Then there exist functions $\theta$ and $\theta'$ on countable ordinals such that \begin{align*} KT_u(A) \equiv_{\theta(\alpha)} KT_u(B) \quad &\implies \quad A \equiv_\alpha B, \text{ and } \\ A \equiv_{\theta'(\alpha)} B \quad &\implies \quad KT_u(A) \equiv_\alpha KT_u(B). \end{align*}
Upshot: We use purely abstract techniques from mathematical logic to probe the complexity of the classification (up to isomorphism) of a large class of C$^*$-algebras.
Jennifer PI is a Postdoctoral Research Associate here in Oxford Mathematics and a member of the Functional Analysis Group.
References
[1] Jennifer Pi, Michał Szachniewicz, and Mira Tartarotti. A Game-Theoretic Unital Classification Theorem for C$^*$-Algebras. 2026. arXiv: 2601.01735 [math.OA]. URL: https://arxiv.org/abs/2601.01735.
[2] Stuart White. Abstract classification theorems for amenable C*-algebras. 2023. arXiv: 2307.03782 [math.OA]. URL: https://arxiv.org/abs/2307.03782.