Instantons are “finite energy” solutions to a geometric PDE for a connection on a vector bundle. These have their origin in Physics but have also been extensively studied by Mathematicians.

The first instanton on the Euclidean Schwarzschild manifold was found by Charap and Duff in 1977, only 2 years later than the famous BPST instantons on $\mathbb{R}^4$ were discovered. While soon after, in 1978, the ADHM construction gave a complete description of the moduli spaces of instantons on $\mathbb{R}^4$, the case of the Euclidean Schwarzschild manifold resisted many efforts for the past 40 years.

I shall explain, how using a duality between vortices and spherically symmetric instantons, Akos Nagy and I, recently gave a complete description of a connected component of the moduli space of unit energy instantons on the Euclidean Schwarzschild manifold. If time permits I will also explain how to use our techniques to:

1. Find new examples of instantons with non-integer energy;
2. Completely classify spherically symmetric instantons.
3. Give a counterexample to a conjectured on a possible non-Abelian extension of Birkhoff's Theorem.

(joint work with Akos Nagy).