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In the past decades, we have witnessed rapid development in the construction of minimal surfaces with controlled topology by Simon-Smith min-max theory. In this talk, I’ll discuss the existence of a number of genus 2 minimal surfaces in a 3-sphere with a positive-Ricci-curved metric. This is based on the recent work joint with Adrian Chu and Yangyang Li.

The Menasco-Reid conjecture supposes a negative answer to the question: Is there a hyperbolic knot complement which contains a closed embedded totally geodesic surface? Another Kirby problem asks which hyperbolic knot complements admit hidden symmetries? Here a manifold $M$ admits hidden symmetries, if $M$ covers an orbifold $Q$ and  $Q$ is not the quotient of $M$ by symmetries. Historically, there were three knot complements known to have hidden symmetries, and a conjecture Neumann and Reid states these are the only such examples.  After giving some of the relevant background, we will construct examples of knot complements that are counterexamples to both conjectures. Each of these knot complements has the property that it admits a decomposition into geometric prisms.  This is joint work with Jason DeBlois and Arshia Gharazolou and has appeared on the arxiv: arXiv:2507.01263.

TBA