Geometry and Topology Seminar
![[Hopf fibration]](hopf.gif)
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Department of Mathematical Sciences
Geometry and Topology Seminar |
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Unless stated otherwise, the seminar takes place Thursdays at 2:50-3:50 pm in LN-2205 followed by refreshments served from 4:00-4:25 pm in the Anderson Memorial Reading Room.
Directions to the department. LN-2205 is on the same floor as the department offices. Stop at the department offices at LN-2200 and ask directions.
Some seminar speakers will also give a colloquium talk at 4:30 in SW-323 on the same day as the seminar talk. Titles of the colloquium talks are also given below where they apply.
This seminar is funded as one of Dean's Workshops in Harpur College (College of Arts and Sciences) at Binghamton University.
To receive announcements of seminar talks by email, please join the seminar's mailing list.
To subscribe to an on-line calendar with the seminar schedule, please choose a format: iCal or xml.
Abstract: Auter space, a cousin of Marc Culler and Karen Vogtmann's Outer space, was introduced by Allen Hatcher and Vogtmann in 1998. In this introduction, Hatcher and Vogtmann defined subspaces of Auter space called the degree complexes, which act like skeleta for Auter space. In this talk, I will present applications of the degree complexes including a presentation for Aut(Fn) and discuss minimality properties of these complexes. In the spirit of minimizing prerequisites, I will present definitions for both Auter space and the degree complexes.
Abstract: In this talk we will give a complete description of the abstract commensurator of Thompson's group F, and we will prove that it embeds in the quasi-isometry group of F, hence providing a large number of examples of quasi-isometries of F. As a corollary of this result, we classify all finite-index subgroups of F, proving that they are all virtually F, and identifying exactly those which are isomorphic to F itself. (Joint work with Sean Cleary and Claas Röver.)
Abstract: Volume entropy of a Riemannian manifold is the exponential growth rate of the volumes of balls. Entropy rigidity for rank-1 Riemannian manifolds is known: a theorem of Besson-Courtois-Gallot says that the locally symmetric metrics attain minimal volume entropy among all Riemannian metrics. In this talk, we are interested in entropy rigidity for buildings, especially hyperbolic ones. We will give several characterizations of the volume entropy, analogous to the ones for trees, that will help us to find some lower bound on volume entropy. This is a joint work with François Ledrappier.
Abstract: We compare several properties of lattices in semisimple Lie groups and lattices in automorphism groups of polyhedral complexes, such as Davis complexes and right-angled buildings. Questions considered include existence of lattices, their covolumes and (in joint work with A. Barnhill) their commensurators.
Abstract: The Borel Conjecture predicts for two closed aspherical manifolds M and N that they are homeomorphic if and only if their fundamental groups agree and that in this case every homotopy equivalence is homotopic to a homeomorphism. This may be viewed as the topological version of Mostow rigidity. It is more or less closely related to the Farrell-Jones Conjecture about the algebraic K- and L-theory of group rings and the Baum-Connes Conjecture about the topological K-theory of reduced group C*-algebras. We present the recent work together with Bartels, where we prove the Farrell-Jones Conjecture and hence the Borel Conjecture in dimension greater or equal to 5 for a large class of groups which includes word-hyperbolic groups and CAT(0)-groups and is closed under directed colimits, taking subgroups and direct products. This implies that these conjectures are true for certain interesting groups (Tarsky monsters, groups with expanders, limit groups) and those exotic aspherical manifolds constructed by Mike Davis.
Abstract: We calculate the Sigma invariants (aka Bieri-Neumann-Strebel-Renz invariants) of Thompson's group F. This is joint work with Bieri and Kochloukova.
Abstract: We study when a pair of elements in F are the images of the standard generators of F under a self monornorphism. We also characterize all of the finite index subgroups of F that are isomorphic to F.
Abstract: In this talk I will describe the construction of a natural filtration of the classifying space of a group defined by means of the lower central series of free groups. We will discuss some of their main properties, some examples and some (potential) applications.
Abstract: The Kan-Thurston theorem says (roughly) that every connected space has the same homology as a K(G,1) for some group G. Brita Nucinkis and I proved an analogous theorem with the ‘classifying space for proper bundles’ in place of a K(G,1): this space can have any (nice, connected) homotopy type. I will explain these results and a new strengthening of both results which uses CAT(0) cube complexes.
I show that every (nice connected) space has the same homology as a locally CAT(0) cube complex and that every such space has the homotopy type of the quotient of a locally CAT(0) cube complex by an isometric involution.
Abstract:
The classical 3-gap theorem asserts that for each natural number n and
real number x, there are at most three distinct distances between
consecutive elements in the subset of [0,1) consisting of the
reductions modulo 1 of the first n multiples of x. Regarding this as
a statement about rotations of the circle, we find results in a
similar spirit pertaining to isometries of compact Riemannian
manifolds and the distribution of points along their geodesics.
Abstract: The mod 2 homology of the spaces in the connective Omega-spectrum of topological modular forms (tmf) is a Hopf ring. Every mod 2 Hopf ring is equivalent to a Dieudonné ring. We calculate the Dieudonné ring for the non-negatively graded spaces in the spectrum tmf, and show that it splits into two parts that occasionally overlap. One part comes from the homology of the spectrum tmf, and the other part comes from the homology of QSk, the k-th spaces in the Omega-spectrum of the sphere.
Abstract: This talk will give a brief introduction of spectral problems of Laplacian operators on compact manifolds. We'll discuss the Weyl's formula for counting eigenvalues and eigenfunction estimates.
Abstract: Through highly non-constructive methods, works by Bestvina, Culler, Feighn, Morgan, Paulin, Rips, Shalen, and Thurston show that if a finitely presented group does not split over a small subgroup, then the space of its discrete and faithful actions on Hn, modulo conjugation, is compact for all dimensions. We make this result effective for Coxeter groups. We find that either the group splits over a small subgroup or there is a constant C and a point in Hn that is moved no more than C by any generator.
Abstract: A Pi-algebra is an element of the category of universal algebras where the graded homotopy groups of a pointed connected space most naturally lie. An obstruction theory for determining when a commutative diagram of Pi-algebras arises by applying homotopy groups to a strictly commutative diagram of spaces (often called a realization) will be presented. Some techniques for approaching computations of obstructions will also be discussed. This is all part of a broader program, joint with David Blanc and James Turner, which aims to understand higher homotopy operations as obstructions to finding strictly commutative replacements for homotopy commutative diagrams.
Abstract: Dwyer, Weiss, and Williams recently gave a construction of smooth parametrized Reidemester torsion, which is a secondary invariant of bundles of smooth manifolds. A very appealing aspect of their work is that it uses only machinery of homotopy theory, which makes it intuitively simple. The technical side of the construction of torsion is however rather involved. The talk will explain how this construction can be simplified using some results of Waldhausen, and how this new approach can be used to compare the torsion of Dwyer-Weiss-Williams to smooth torsion invariants developed by Bismut-Lott and Igusa-Klein.
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