Georgia Institute of Technology College of Sciences

The notion of distance for a Heegaard splitting of athree-dimensional manifold $M$, introduced by John Hempel, has provedto be a very powerful tool for understanding the geometry and topologyof $M$. I will describe how distance, and a slight generalizationknown as subsurface projection distance, can be used to explore theconnection between geometry and topology at the center of the moderntheory hyperbolic three-manifolds.In particular, Schalremann-Tomova showed that if a Heegaard splittingfor $M$ has high distance then it will be the only irreducibleHeegaard splitting of $M$ with genus less than a certain bound. I willexplain this result in terms of both a geometric proof and atopological proof. Then, using the notion of subsurface distance, Iwill describe a construction of a manifold with multiple distinctlow-distance Heegaard splittings of the same (small) genus, and amanifold with both a high distance, low-genus Heegaard splitting and adistinct, irreducible high-genus, low-distance Heegaard splitting.