Geometry and Topology Seminar

Thursday Apr 23, 16:00, room B3.03

Tara Brendle (Glasgow)

The symmetric Torelli group

Abstract: We will discuss work in progress with Dan Margalit on the subgroup of the mapping class group consisting of elements which commute with a fixed hyperelliptic involution and which act trivially on homology. We will describe various aspects of this group including connections with braid groups and with spin subgroups of mapping class groups.

Thursday Apr 30, 16:00, room B1.01

Walter Neumann (Columbia)

Quasi-isometry classification of 3-manifold groups

Abstract: Joint work with Jason Behrstock. The talk will describe the current status of the quasi-isometric classification of 3-manifold groups (or, what is equivalent, the bi-Lipschitz classification of universal covers of 3-manifolds). This has involved a large number of researchers and is now nearing completion.

Thursday May 7, 16:00, room B1.01

Dieter Kotschick (München)

Domination by products versus negative curvature in geometry, topology and group theory

Abstract: The existence of a map of non-zero degree defines an interesting transitive relation, called the domination relation, between homotopy types of closed oriented manifolds. In dimension two the relation coincides with the ordering given by the genus. We study this relation in higher dimensions, with special emphasis on the case where the domain is a non-trivial product and the target has a large universal covering in a suitable sense, e.g. the target could be negatively curved. In many such situations we prove that there are no maps of non-zero degree. We shall extract some purely group-theoretic manifestations of negative curvature at the level of fundamental groups. Similar arguments, and related but weaker notions of negative curvature, also allow us to give obstructions to the existence of dominant rational maps from products to certain complex algebraic varieties.

Thursday May 14, 16:00, room B1.01

Mahan Mj (Vivekananda)

Model geometries and the Cannon-Thurston map

Abstract: I shall spend some time discussing the proof of the existence of a Cannon-Thurston map for a specific model geometry (called amalgamation geometry). I'll also indicate how a general surface group has a model geometry that is a slight generalization of amalgamation geometry.

Thursday May 28, 16:00, room B1.01

John Jones (Warick)

The solution of the Kervaire invariant problem

Abstract: The Kervaire invariant first arose in Kervaire's construction, published in 1960, of a 10 dimensional closed (i.e. compact with no boundary) topological manifold with no smooth structure. One of the key ingredients is a Z/2 valued invariant defined for closed smooth framed manifolds of dimension 4n+2; this is the Kervaire invariant. In a classic paper "Groups of Homotopy Spheres: I" published in 1963, Kervaire and Milnor showed the importance of this invariant for the classification of homotopy spheres closed smooth manifolds homotopy equivalent to a sphere.

This all leads to the Kervaire invariant problem: "In which dimensions of the form 4n+2 does there exist a framed manifold with Kervaire invariant one?" By the early 1980's it had been established that the Kervaire invariant can only be non-zero in dimensions of the form 2^m - 2 (that is 2 less than a power of 2) and that it is non-zero in dimensions 2, 6, 14, 30, 62.

There has been no progress on this classical problem for the last 25 or so years until last month when it was announced that Mike Hill, Mike Hopkins, and Doug Ravenel have proved that the Kervaire invariant is zero in dimensions strictly greater than 126. This settles the Kervaire invariant problem completely except in one irritating/interesting case, dimension 126.

In this talk I will explain more about the Kervaire invariant problem and set it in context. I will also make one or two comments on the nature and significance of the Hill-Hopkins-Ravenel proof.

Tuesday Jun 9, 16:00, room MS.05

Sara Maloni (Warwick)

The trace formula for a representation of the fundamental group of a surface in PSL(2, C)

Abstract: The aim of this talk is to present a formula for traces for simple closed curves. Using a particular case of Kra's plumbing construction, I will give a ``recipe'' to calculate a representation for the fundamental group of a surface into PSL(2, C) which depends on the pants decomposition chosen, but which is independent of the labeling chosen for the standard chart. The trace of this representation is a polynomial in the plumbing parameters and the two highest order terms of this polynomial depend only on the Dehn Thurston coordinates. The proof of this result will be done by induction, so I will also present some examples and the calculus of the holonomy of some particular paths. Before presenting my results, I will review the definition of the Dehn-Thurston coordinates for simple closed curves on a surface and of Kra's horocyclic coordinates.

Tuesday Jun 16, 15:00, room MS.05

David Dumas (UI Chicago)

Kaehler structures on ML

Abstract: The space ML of measured geodesic laminations on a hyperbolic surface S has a natural symplectic structure, which was described by Thurston using train track coordinates. On the other hand, for any complex structure X on S, there is a natural identification between ML and the vector space Q(X) of holomorphic quadratic differentials on X. We show that this identification is compatible with both the symplectic structure of ML and the complex structure of Q(X), inducing a (stratified, singular) Kaehler structure on each of them.

Tuesday Jun 16, 16:30, room MS.05

Ian Agol (UC Berkeley)

The virtual fibering conjecture and related questions

Abstract: Thurston asked a bold question of whether finite volume hyperbolic 3-manifolds might always admit a finite-sheeted cover which fibers over the circle. This talk will review some of the progress on this question, and discuss its relation to other questions including residual finiteness and subgroup separability, the behavior of Heegaard genus in finite-sheeted covers, CAT(0) cubings, the RFRS condition, and subgroups of right-angled Artin groups. In particular, hyperbolic Haken 3-manifolds with LERF fundamental group are virtually fibered. Some applications of the techniques will also be mentioned.

Thursday Jun 25, 16:00, room B1.01

TBA (TBA)

TBA

Abstract: TBA