Spring 2026

Abstract: A simplified broken Lefschetz fibration (SBLF) is a type of map from a smooth, compact 4-manifold to a sphere or disk. Unlike Lefschetz fibrations, which have more restricted singular sets, SBLFs are admitted by all closed, smooth 4-manifolds. One implication of this is that we can study the topology of a 4-manifold by examining the data coming from an SBLF it admits. This data comes in the form of an ordered tuple of curves on a closed surface. In this talk, we will focus on SBLFs admitted by closed nonorientable 4-manifolds. We’ll describe how to construct Kirby diagrams corresponding to these SBLFs, as well as their orientation double covers. Using this information, we will characterize all closed nonorientable 4-manifolds admitting genus two SBLFs. This is joint work with İnanç Baykur.

Abstract: Quotients are a powerful tool used for constructing exotic embeddings in groups that act on negatively curved metric spaces.  In 1980s Gromov and Ol’shanskii proposed models for random quotients of free groups to study generic properties of groups such as the hyperbolicity.  I will introduce new models for random quotients of groups with more flexible actions on hyperbolic spaces.  I will describe geometric tools used to establish when these more general forms of hyperbolicity are preserved in random quotients. Our techniques provide new examples of groups that are residually finite and QI rigid and prove the existence of common quotients.  This talk will be based on articles joint with Abbott, Berlyne, Mangioni, and Rasmussen as well as Einstein, Krishna MS, Montee, Steenbock. 

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