**Title of talk: **An SL(2, R) Casson-Lin invariant and applications

**Abstract: **When M is the exterior of a knot K in the 3-sphere, Lin showed that the signature of K can be viewed as a Casson-style signed count of the SU(2) representations of the fundamental group of M where the meridian has trace 0. This was later generalized to the fact that signature function of K on the unit circle counts SU(2) representations as a function of the trace of the meridan. I will define the SL(2,R) analog of these Casson-Lin invariants, and explain how it interacts with the original SU(2) version via a new kind of smooth resolution of the real points of certain SL(2, C)-character varieties in which both kinds of representations live. I will use the new invariant to study leftorderability of Dehn fillings on M using the translation extension locus I introduced with Marc Culler, and also give a new proof of a recent theorem of Gordons on parabolic SL(2,R) representations of two-bridge knot groups. This is joint work with Jake Rasmussen.

Conference Organizers:

- David Gay
- Gordana Matic
- Rachel Roberts
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