Rank three instantons, representations and sutures

Yang--Mills gauge theory with gauge group SU(2) has played a significant role in the study of the topology of 3- and 4-manifolds. However, there is not much known about applications of gauge theory with other gauge groups in the study of low dimensional manifolds. In this talk, I will discuss a program to give a gauge theoretical proof of the Smith conjecture using SU(n) Yang--Mills gauge theory. I will also explain how to carry out this program in the case n=3 and obtain n=3 version of the Smith conjecture as a consequence of the following result: the knot group of any knot in any integer homology sphere admits a non-abelian representation into SU(3) such that meridians are mapped to matrices whose eigenvalues are the three distinct third roots of unity. This talk is based on a joint work with Nobuo Iida and Chris Scaduto.