Speaker: Sudipta Ghosh
Date: January 29, 2026
Affiliation: University of Notre Dame

Abstract: The Poincar\'e Conjecture, proved by Perelman, states that every closed, connected three--manifold other than $S^{3}$ has nontrivial fundamental group. A related question, Problem~3.105(A) in Kirby's list, asks whether any three--manifold other than $S^{3}$ admits a nontrivial homomorphism from $\pi_{1}(Y)$ to $\mathrm{SU}(2)$; this problem remains open. Using instanton Floer homology, Zentner showed that for integer homology spheres, the analogous statement holds when $\mathrm{SU}(2)$ is replaced by $\mathrm{SL}(2,\mathbb{C})$. In this talk, I will discuss recent progress on the existence of irreducible $\mathrm{SL}(2,\mathbb{C})$ and $\mathrm{SU}(2)$ representations for rational homology spheres. Some of these results are joint with Steven Sivek and Raphael Zentner, others with Mike Miller Eismeier, and others with Zhenkun Li and Juanita Pinz\'on-Caicedo.