Fall Redbud Topology Conference
The talks are SCEN 408 on the University of Arkansas campus. Abstracts are below.
Church, Stanford University
- Title: A survey of representation stability
- Abstract: I will give a gentle survey of the theory of representation
stability, viewed through the lens of its applications in topology and
elsewhere. These applications include: homological stability for
configuration spaces of manifolds; understanding the stable (and
unstable) homology of arithmetic lattices; Charney's theorem and Hecke
eigenclasses in stable mod-p cohomology; uniform generators for
congruence subgroups and "congruence" subgroups; and distributional
stability for random squarefree polynomials over finite fields.
- Matt Day,
University of Arkansas
- Title: On the second homology group of the Torelli subgroup of Aut(Fn)
- Abstract: I will discuss joint work with Andrew Putman in which
we find an infinite generating set for
H2(IAn; ℤ), the second
homology of the Torelli subgroup (a.k.a. IAn) of the automorphism group of the
free group. This generating set comes from a new infinite
presentation for IAn. There is a natural action of
GL(n, ℤ) on
H2(IAn; ℤ) and our generating set is the orbit of finitely many
"basic generators" under this action. Our generators "look the
same" for all n starting with n = 6, proving a surjective
representation stability conjecture for H2(IAn; ℤ).
Durham, University of Michigan
- Title: Convex cocompactness and stability in mapping class groups
- Abstract: Originally defined by Farb−Mosher to study hyperbolic extensions of surface subgroups, convex cocompact subgroups of mapping class groups have deep ties to the geometry of Teichmuller space and the curve complex. I will present a strong notion of quasiconvexity in any finitely generated group, called stability, which coincides with convex cocompactness in mapping class groups. I will also discuss some work in progress and future directions. This is joint work with Sam Taylor.
- Robert Tang, University of Oklahoma
- Title: Shadows of Teichmueller discs in the curve complex
- Abstract: Given a flat structure on a surface, we can deform the metric using elements of SL(2,ℝ) to obtain different metrics. The SL(2,ℝ)−orbit under this action is called a Teichmueller disc. The set of systoles arising on some flat metric on a Teichmueller disc gives a quasiconvex subset in the curve graph associated with the surface. We describe nearest point projections to the systole set using a generalised notion of Masur−Minsky's balance time. We will also describe other sets of curves associated to a given Teichmueller disc. This is joint work with Richard Webb.
- Pengcheng Xu, Oklahoma State University
- Title: P−moves between pants block decompositions of
- Abstract: A pants block decomposition of a compact hyperbolic 3−manifold is a decomposition of the 3−manifold which cuts the manifold into fundamental pieces called pants blocks. This is similar to a triangulation, which cuts the 3−manifold into tetrahedra. In this talk we will discuss how to relate two pants block decompositions of a manifold with a sequences of P−moves, which are similar to Pachner moves between triangulations.
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