Amalgamating groups via linear programming

A compact group $A$ is called an amalgamation basis if, for every way of embedding $A$ into compact groups $B$ and $C$, there exist a compact group $D$ and embeddings $B\to D$ and $C\to D$ that agree on the image of $A$. Bergman in a 1987 paper studied the question of which groups can be amalgamation bases. A fundamental question that is still open is whether the circle group $S^1$ is an amalgamation basis in the category of compact Lie groups. Further reduction shows that it suffices to take $B$ and $C$ to be the special unitary groups. In our work, we focus on the case when $B$ and $C$ are the special unitary group in dimension three. We reformulate the amalgamation question into an algebraic question of constructing specific Schur-positive symmetric polynomials and use integer linear programming to compute the amalgamation. We conjecture that $S^1$ is an amalgamation basis based on our data. This is joint work with Michael Joswig, Mario Kummer, and Andreas Thom.