2011 Cascade Topology Seminar - Schedue

As we get closer to the conference the schedule will solidify. Below you will find the evolving approximate schedule.

Saturday morning events:

Meet-and-greet 9am SSM A514. Pick up your nametags, schedule, etc.

First morning talk 10am in SSM A110. Speaker: Tom Church

Tea time 11am

Second talk 11:30am in SSM A110. Speaker: Jing Tao

Lunch somewhere on or near campus.

Saturday afternoon events:

First talk 3pm in DSB C118. Speaker: Ian Agol

Second talk 4pm in DSB C118. Speaker: Yi Liu

Third talk 5pm in DSB C118. Speaker: Johanna Mangahas

Saturday evening:

Dinner somewhere in town.

Sunday morning events:

Lounge: 8am meet in SSM A514. Participants who are bringing their luggage to campus can store their luggage in Budney's office (SSM A516) at this time.

First talk 9am in SSM A102. Speaker: Alexandra Pettet

Tea time 10am

Second talk 10:30am in SSM A102. Speaker: Jesse Johnson

Sunday afternoon events:

Lunch, a hike, and dinner in town.


Speaker: Ian Agol

Title: Presentation length and Simon's conjecture

Abstract: We show that any knot group maps onto at most finitely many knot groups. This gives an affirmative answer to a conjecture of J. Simon. We also bound the diameter of a closed hyperbolic 3-manifold linearly in terms of the presentation length of its fundamental group, improving a result of White.

Speaker: Tom Church

Title: Representation theory and homological stability

Abstract: Homological stability is a remarkable phenomenon where for certain sequences X_n of groups or spaces -- for example SL(n,Z), the braid group B_n, or the moduli space M_n of genus n curves -- it turns out that the homology groups H_i(X_n) do not depend on n once n is large enough. But for many natural analogous sequences, from pure braid groups to congruence groups to Torelli groups, homological stability fails horribly. In these cases the rank of H_i(X_n) blows up to infinity, and in the latter two cases almost nothing is known about H_i(X_n); indeed it's possible there is no nice "closed form" for the answers.

While doing some homology computations for the Torelli group, we found what looked like the shadow of an overarching pattern. In order to explain it and to formulate a specific conjecture, we came up with the notion of "representation stability" for a sequence of representations of groups. This makes it possible to meaningfully talk about "the stable homology of the pure braid group" or "the stable homology of the Torelli group" even though the homology never stabilizes. This work is joint with Benson Farb.

In this talk I will explain our broad picture and describe a number of connections to other areas of math, including two major applications. One is a surprisingly strong connection between representation stability for certain configuration spaces and arithmetic statistics for varieties over finite fields, joint with Jordan Ellenberg and Benson Farb. The other is representation stability for the homology of the configuration space of n distinct points on a manifold M.

Speaker: Jesse Johnson

Title: Common stabilizations of Heegaard splittings

Abstract: A Heegaard splitting is a decomposition of a 3-manifold into two simple pieces called handlebodies. It has long been known that any two Heegaard splittings of the same 3-manifold are related by repeating a construction called stabilization. However, the early proofs of this fact gave no suggestion of how many stabilizations might be needed to turn one Heegaard splitting into another. I will describe a new upper bound on the necessary number of stabilizations.

Speaker: Yi Liu

Title: A Jorgensen-Thurston Theorem for Homomorphisms

Abstract: In this talk, we describe the structure of homomorphisms from a finitely generated group to torsion-free Kleinian groups of uniformly bounded covolume. This is an analogy of the Jorgensen-Thurston theorem to homomorphisms.

Speaker: Johanna Mangahas

Title: The geometry of right-angled Artin subgroups of mapping class groups

Abstract: I'll describe joint work with Matt Clay and Chris Leininger. We give sufficient conditions for a finite set of mapping classes to generate a right-angled Artin group quasi-isometrically embedded in the mapping class group. Moreover, under these conditions, the orbit map to Teichmueller space is a quasi-isometric embedding for both of the standard metrics. As a consequence, we produce infinitely many genus h surfaces (h at least 2) in the moduli space of genus g surfaces (g at least 3) for which the universal covers are quasi-isometrically embedded in the Teichmueller space.

Speaker: Alexandra Pettet

Title: Fully irreducible outer automorphisms of the outer automorphism group of a free group.

Abstract: The outer automorphism group Out(F) of a free group F of finite rank shares many properties with linear groups and the mapping class group Mod(S) of a surface, although the techniques for studying Out(F) are often quite different from the latter two. Motivated by analogy, I will present some results about Out(F) previously well-known for the mapping class group, and highlight some of the features in the proofs which distinguish it from Mod(S).

Speaker: Jing Tao

Title: Diameter of the thick part of moduli space

Abstract: We study the shape of the moduli space of a surface of finite type. In particular, we compute the asymptotic behavior of the Teichmuller diameter of the thick part of moduli space. For a surface S of genus g with b boundary components, we show that the diameter grows like logarithm of the Euler characteristic. This is joint with Kasra Rafi.

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