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 on the same day as the seminar talk. Titles of the colloquium talks are also given below where they apply.
This seminar is partly funded as one of Dean's Speaker Series 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.
January 28, 4:30–5:30 pm, Dean's Speaker Series in Geometry/TopologyAbstract: We will describe a class of quasi-invariant measures on the circle and on the interval related to the Wiener measure. Some of their unusual properties will be explained. These measures are defined on the group of type Diff1 and are quasi-invariant with respect to the action of the subgroup of type Diff3. Some applications to the representation theory, quantum field theory, and amenability will be mentioned.
Abstract: We will discuss speaker's approach to proving amenability of Thompson's group F based on use of quasi-invariant measure from the previous talk, and properties (A) and (B) that will be defined for subgroups of Diff and verified in the case of F.
Abstract: We will give a preliminary report on sufficient conditions for subgroups to be invariant under quasi-isometries. That is, given a quasi-isometry f: G → G', we will show that if H is a subgroup of G that satisfies certain conditions, then f(H) is a finite Hausdorff distance from a subgroup of G'.
Joint work with Ross Geoghegan.
Abstract: In the first talk I will introduce the notions of self-similar groups, explain their basic properties, give examples, motivate interest in them, and formulate several open questions. One particular application I am going to concentrate on is a relatively recent discovery (mainly by Nekrashevych) of the beautiful connection between self-similar groups and holomorphic dynamics realized by iterated monodromy groups. Here is a rough idea: with every (nice) d-fold self-covering of a (nice) topological space X one can associate the group acting on the tree of the preimages of any point in X by automorphisms. It can be shown that under certain labeling of the tree this group is self-similar. This may provide some information on the self-covering itself. More details will be provided on the talk.
Abstract: L-presented groups are finitely generated groups that admit a presentation involving finitely many relators and their iterations by substitution. The presence of L-presentation is important from different points of view. Such presentations are at the first level of complexity after the finite presentations and quite often provide the simplest way to describe a group that is not finitely presented. Further, such presentations can be used to embed a group into a finitely presented group in a way that preserves many properties of the original group. It is known that many iterated monodromy groups admit L-presentation. In this talk I will concentrate on a particular example of this sort — iterated monodromy group of the map z → z2+i. I will show how to compute the automaton generating this group and give an idea how to obtain an L-presentation.
Abstract: This talk will discuss using uniformly finite homology or “Ponzi schemes” as a tool to study the geometry of Thompson's group F. We will see how we can use this tool with two-way forest diagrams to show that certain subgraphs of the Cayley Graph of F are not amenable. This gives some sense of what a positive measure subset of F would have to look like if F is amenable.
Abstract: A real arrangement of hyperplanes is a collection of finitely many hyperplanes in a real vector space. It is known that the combinatorics of the intersections contains substantial information about the topology of the complement of the hyperplanes in the complexified space. For example, the cohomology of the complement can be expressed in terms of the intersection lattice associated with the arrangement. The face poset of an arrangement defines a simplicial complex (Salvetti's complex) which has the homotopy type of the complement.
In the same spirit, I define the notion of an arrangement of submanifolds and its complexification, and I investigate whether there is any relationship between the combinatorics of the intersections and the topology of the complement. The aim of this talk is to introduce this generalization of hyperplane arrangements and report current results.
Abstract: (Joint with Bruce Hughes.) Let T be a locally finite rooted simplicial tree. The space of ends of T, which we denote X, is a compact ultrametric space. A finite similarity structure S(X) assigns a finite (possibly empty) set S(B1,B2) of surjective similarities j: B1 → B2 to each pair of balls B1, B2 ⊂ X. The union of all sets S(B1,B2) (for varying B1 and B2) is also assumed to satisfy certain groupoid-like properties.
We are interested in the groups determined by finite similarity structures. Given S(X) (as above), we let G(S) denote the group of all self-homeomorphisms of X that are locally determined by S(X). That is, if h ∈ G(S), then, for every x ∈ X, there are balls B1 and B2, and a j ∈ S(B1,B2), such that x ∈ B1 and j|B1=h|B1.
For example, if we let T be the ordered infinite rooted binary tree, X be its space of ends, and, for any balls B1 and B2, S(B1,B2) be the singleton set containing the unique order-preserving similarity from B1 to B2, then G(S) is Thompson's group V. It appears that there are many other examples as well.
I will discuss joint work in progress with Bruce Hughes, in which we argue that a fairly general class of the groups G(S) have type F∞, i.e., have a classifying complex with finitely many cells in each dimension.
Abstract: An arrangement of linear hyperplanes in Cn generates subspaces defined by intersections of hyperplanes. A choice of such subspaces is an arrangement of subspaces. When the subspaces are hyperplanes the arrangement's characteristic polynomial carries information about the topology of M, the complement of the subspaces in Cn; this is a fundamental theorem of Orlik and Solomon.
An arrangement of subspaces of the braid arrangement (the arrangement that consists of all hyperplanes with equations xi=xj) can be encoded by an edge-colored hypergraph. The characteristic polynomial of this type of subspace arrangement is given by a generalized chromatic polynomial of the associated edge-colored hypergraph. However, this polynomial is less informative than in the case of hyperplane arrangements. Stronger topological information about M can be found directly in the hypergraph.
A Massey product is an algebraic simplification in cohomology. I will present a sufficient condition for the existence of non-trivial Massey products in the cohomology of M. The condition is proved by studying a spectral sequence associated to the Lie coalgebras of Sinha and Walter. These coalgebras are constructed from the cohomology of M.
If time permits I will construct a family of subspace arrangements whose intersection lattices have the shape of Pascal's triangle. Even though the intersection lattices are not geometric, the complex complements of the arrangements have the property of rational formality, i.e., their homotopy type is determined by their rational cohomology.
Everything I will talk about is combinatorial. Some of the topics are in combinatorial topology, but I will try my best to not be overly technical.
Abstract: We will give an introduction to the notion of an expander. The families of graphs which have come to be known as expanders have some amazing and often counter-intuitive properties. Study of these objects has bridged many disciplines, including theoretical computer science, geometric group theory, graph theory, and functional geometry. We will present the basic definitions and a few properties of expanders, with an eye towards a recent generalization of an expander, to be covered in a future talk.
October 22, 4:30–5:30 pmAbstract: This is a joint work with A. Borisov and I. Spakulova. We prove that almost all 1-related groups with at least 3 generators are residually finite and coherent. The proof uses geometric group theory, algebraic geometry, and probability theory (Wiener's measure).
Abstract: If X is a PD3-complex which is indecomposable as a connected sum then either it is aspherical or its fundamental group is virtually free, and hence is the fundamental group of a reduced finite graph of finite groups, by work of Turaev and Crisp. We show that if X is an indecomposable PD3-complex and π1(X) = πG, where G is such a graph of groups, then either Z nor Z⊕Z/2Z or X is orientable, the underlying graph is a tree, the vertex groups have cohomological period dividing 4 and all but at most one of the edge groups is Z/2Z. If there are no exceptions then all but at most one of the vertex groups is dihedral of order 2m with m odd. Every such group is realized by some PD3-complex. Otherwise, one edge group may be Z/6Z. We do not know of any such examples.
Abstract: I will mainly talk about (my viewpoint at) a method for proving property T started by Dymara and Januszkiewicz. Their original motivation came from groups acting on dimensional buildings, but the refined idea does not use anything more than angles between subspaces in an finite dimensional Euclidian space. Parts of the talk are based on a work of M. Ershov and A. Jaikin.
Abstract: Finite automata capture the notion of computation with a finite amount of memory and are a fundamental notion in computer science. In this talk, I will discuss encoding automata as representations of a monoid in a monoidal category. A morphism of representations can be interpreted as a proof that two automata accept the same language.
Using this encoding, I will show that many constructions from the theory of automata are in fact instances of general constructions in monoidal categories and explain the relation between automata and representations of a bialgebra. I will explain how this work relates to uses of coalgebras in computer science and indicate how theorems about representations of a bialgebra might shed light on questions of complexity theory.
I will not assume any knowledge of computer science in this talk.
Abstract: Tessera and Ostrovskii have independently introduced a generalized notion of expander in terms of probability measures on metric spaces. In joint work with Jerry Kaminker, we analyze certain classes of these generalized expanders. In this context we study, among other results, the obstruction to being able to uniformly embed a metric space into a Hadamard manifold.
Abstract: Let M be a closed 3-manifold with a given Heegaard splitting. We show that after a single stabilization, some core of the stabilized splitting has arbitrarily high distance with respect to the splitting surface. This generalizes a result of Minsky, Moriah, and Schleimer for knots in S3. We also show that in the complex of curves, handlebody sets are either coarsely distinct or identical. We define the coarse mapping class group of a Heegaard splitting, and show that if (S,V,W) is a Heegaard splitting of genus at least 4, then CMCG(S,V,W)=MCG(S,V,W). This is joint work with Marion Moore.
Friday, November 13, 4:30–5:30 pmAbstract: A conjugation space is a space X with involution, where the cohomology mod 2 of the fixed set is the same as the cohomology of the space after doubling dimensions. The first example is X = complex projective space, with the involution given by complex conjugation. In the talk I will describe the relation between smooth conjugation 4-manifolds and knotted surfaces in mod 2 homology 4-spheres. This is joint work with Jean-Claude Hausmann.
Abstract: In recent joint work with R. Charney, N. Stambaugh and A. Vijayan we prove that a finite index subgroup of the automorphism group of a certain type of graph product of finitely generated abelian groups is again a graph product of finitely generated abelian groups. The necessary condition is the “no SILS” condition discovered in previous joint work of mine with A. Piggott and M. Gutierrez. If the original graph product has all finite vertex groups, then this condition implies that the outer automorphism group is finite. One consequence of the new result is that these automorphism groups are virtually CAT(0).
December 3, 4:30–5:30 pmAbstract: In the mid 1970's, Jaco and Shalen, and independently Johannson, proved that Haken 3-manifolds have a canonical submanifold (possibly empty) called the characteristic submanifold, and they showed that this submanifold has some important properties. Starting in the mid 1980's, there has been a whole series of algebraic results which describe analogous results in group theory. However, none of these results yields as much information as the topological result when applied to the fundamental group of a Haken 3-manifold.
In this talk, I will discuss joint work with Gadde Swarup of the University of Melbourne in which we give a new approach to this area. We obtain new algebraic results which yield the topological result when applied to the fundamental group of a Haken 3-manifold.
The aim of the talk will be to convey the basic ideas of our approach rather than all the details!
Abstract: I will discuss joint work with Hass on a combinatorial approach to harmonic maps. This work is still in progress. Such an approach has been used previously by other authors for computational reasons, but we are developing our approach with an eye to theoretical applications. Some applications in low dimensional topology will be discussed.
Abstract: Let ρ be an action by a finitely generated group G on a locally finite tree T. Viewing points of ∂T (the visual boundary of T) as directions in T, we can use ρ to lift this sense of direction to G and ask, for each end point E ∈ ∂T, whether G is “connected in the direction of E”. If so, we say that ρ is controlled connected over E. The Bieri-Geoghegan invariant Σ1(ρ) ⊆ ∂T consists of those points over which ρ is controlled connected. This is a generalization of the classical Bieri-Neumann-Strebel invariant.
In this talk, I will introduce a class of actions for which Σ1 can be calculated by analyzing certain quotient maps between trees and show that under reasonable hypotheses, Σ1 consists of no more than a single point of ∂T for these actions. I will illustrate the result with some examples.