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.
Abstract: We investigate the non-bounded generation of subgroups of mapping class groups through the hierarchy in curve complexes developed by Masur and Minsky (2000). We compare the subsurface projections to nearest point projections in curve complexes and extend Behrstock's inequality to include geodesics in curve complexes of subsurfaces in the Inequality on Triples in Bestivina-Bromberg-Fujiwara (2010). Based on this inequality, we can estimate translation lengths of words in the form when is sufficiently large for any given sequence in a mapping class group. With a growth argument, we further show that any subgroup of a mapping class group is boundedly generated if and only if it is virtually abelian.
Abstract: A dynamical system is partially hyperbolic if it has three invariant directions E^s, E^c and E^u, being E^s uniformly contracting, E^u uniformly expanding while E^c has an intermediate behavior. The study of partially hyperbolic systems has been one of the most active topics in dynamics in the last two decades. The purpose of this talk will be to present the state of the art in the study of the ergodicity of conservative partially hyperbolic diffeomorphisms on three dimensional manifolds. Interestingly, 3-dimensional topology is a crucial ingredient in the study of such systems. In a previous work (joint with Jana and Federico Rodriguez Hertz) we proved the Pugh-Shub conjecture for partially hyperbolic diffeomorphisms with 1-dimensional center, i.e. stably ergodic diffeomorphisms are dense among the partially hyperbolic ones. In subsequent results, we obtained, jointly with the same co-authors, a more accurate description of this abundance of ergodicity in dimension three. We will describe these results, some recent advances and the main open problems and conjectures on the subject.
Abstract: When a group acts on a simply connected non-positively curved space, such as Euclidean space, Hyperbolic space or a tree, the orbit of a point can have limit points at infinity. There are various definitions of what should be called a limit point. Some go back to the 19th century, some modern, and each invented for a good reason. My talk will explore some of these ideas with emphasis on new work I have been doing with Robert Bieri.
Abstract: Homoclinic points describe the asymptotic behavior of group actions on spaces and play an important role in general theory of dynamical systems. In 1999, Doug Lind and Klaus Schmidt established relations between homoclinic points and entropy properties for expansive algebraic actions of Z^d. Their proof depends heavily on the commutative factorial Noetherian ring structure of the integral group ring of Z^d.
In a joint work with Hanfeng Li, we extend their results to expansive algebraic actions of polycyclic-by-finite groups. We use three ingredients to do this: characterizations of expansive algebraic actions, local entropy theory for actions of countable amenable groups on compact groups, and comparison between entropies of dual algebraic actions.
Applying our results to the field of von Neumann algebras, we get a positive answer to a question of Deninger about the Fuglede-Kadison determinant to the case group is amenable. We also prove that for an amenable group, an element in the integral group ring is a non-zero divisor if and only if the entropy of corresponding principal algebraic action is finite.
Abstract: Farrell and Jones constructed codimension one Anosov diffeomorphisms on manifolds that are homeomorphic to tori but have exotic smooth structures. In a joint work with Tom Farrell we use a different method to construct Anosov diffeomorphisms of higher codimension on manifolds that are homeomorphic to infranilmanifolds yet have irreducible exotic smooth structures.
Abstract: acts naturally on several geometric objects. There is a proper isometric action on Outer Space which plays the role of a homogeneous space of the Lie group in our setting. There are also several simplicial complexes with natural actions. Three important examples are: the free factor complex - which was recently shown to be Gromov hyperbolic (Bestvina-Feighn), the separating splitting complex - which has quasi-flats of unbounded rank (Sabalka-Savchuk), and the free splitting complex - which even more recently was shown to be Gromov hyperbolic (Handel-Mosher). It is still not fully understood how these complexes relate to each other and to Outer Space. I will present a proof that the free splitting complex is homeomorphic to the simplicial part of the metric completion of Outer Space. As an application, I will present a new proof of a theorem of Francaviglia-Martino: The isometry group of Outer Space is for and for .
Abstract: We construct a new functional on the space of metrics and show that it achieves its minimum in a conformal class under some non-degeneracy assumptions. As an application we get a new family of conformal metrics on compact Riemann surfaces.
Abstract: The a priori expectation of first year elementary school students who were just introduced to the natural numbers, if they would be ready to verbalize it, must be that soon, perhaps by second grade, they'd master the theory and know all there is to know about those numbers. But they would be wrong, for number theory remains a thriving subject, well-connected to practically anything there is out there in mathematics.
I was a bit more sophisticated when I first heard of knot theory. My first thought was that it was either trivial or intractable, and most definitely, I wasn't going to learn it is interesting. But it is, and I was wrong, for the reader of knot theory is often led to the most interesting and beautiful structures in topology, geometry, quantum field theory, and algebra.
Today I will talk about just one minor example: A straightforward proposal for a group-theoretic invariant of knots fails if one really means groups, but works once generalized to meta-groups (to be defined). We will construct one complicated but elementary meta-group as a meta-bicrossed-product (to be defined), and explain how the resulting invariant is a not-yet-understood generalization of the Alexander polynomial, while at the same time being a specialization of a somewhat-understood "universal finite type invariant of w-knots" and of an elusive "universal finite type invariant of v-knots".
See more details at http://www.math.toronto.edu/~drorbn/Talks/Binghamton-1203/
Abstract: In the 1960's, Richard J. Thompson described three groups F, T, and V, which act by homeomorphisms on the interval, the circle, and the Cantor set, respectively. In this talk, I will describe an analogous group that acts by homeomorphisms on the Basilica Julia set. This group can also be described as a group of piecewise-linear homeomorphisms of the unit circle that preserves the invariant lamination determined by the Basilica. I will sketch a proof that this group is finitely generated and virtually simple, and discuss possible generalizations to other Julia sets. This is joint work with Bradley Forrest.
Abstract: String topology is the study of the free loop space of a manifold LM. The loop product, defined on the homology of LM, is described intuitively as a combination of the intersection product on M and loop concatenation in the based loop space of M. However, since the intersection product is well-defined only on transversally intersecting chains, this description is incomplete. Brown's theory of twisting cochains provides a chain model of a bundle in terms of the chains on the base and chains on the fiber. We extend this theory so that it can be applied to provide a model of the free loop space. We give a precise definition of the loop product defined at the chain level.
Abstract: My talk will involve two concepts which are apparently very different. The character variety of a surface S, consisting of homomorphisms from the fundamental group of S to a Lie group G, arises in many different branches of mathematics. The classical Kauffman bracket is an invariant of knots and links in space, closely related to the Jones polynomial. When G = SL2(C), Turaev showed that the character variety can be quantised by a generalisation of Kauffman brackets to the surface S. I will discuss the classification problem for Kauffman brackets on S, with results, conjectures and interesting examples.
Abstract: A free splitting a finite rank free group F is a minimal action of F on a simplicial tree T with trivial edge stabilizers. The free splitting complex of F is a simplicial complex with one 0-simplex for each free splitting of F having 1 orbit of natural edges (modulo F-equivariant homeomorphism), and more generally one k-simplex for each free splitting with k+1 orbits of natural edges. We prove that the free splitting complex of F, when equipped with its simplicial metric, is a hyperbolic metric space. This is a joint work with M.Handel.
Abstract: We construct and describe smooth forward Ricci flow evolutions of singular initial metrics resulting from rotationally symmetric neckpinches, without performing an intervening surgery. In the restrictive context of rotational symmetry, our construction gives evidence in favor of Perelman's hope for a "canonically defined Ricci flow through singularities." (This is joint work with Sigurd Angenent and Cristina Caputo.)
Abstract: I will describe two classes of finitely presented groups. In the first class, the isomorphism problem is solvable, but the commensurability problem is unsolvable. In the second class, the commensurability problem is solvable, but the isomorphism problem is unsolvable. This is joint work with Goulnara Arzhantseva and Ashot Minasyan.
Abstract: An origami manifold is a manifold equipped with a closed 2-form which is symplectic everywhere except on a hypersurface, where it is a folded form whose kernel defines a circle fibration. In this talk, I will explain how an origami manifold can be unfolded into a collection of symplectic pieces and conversely, how a collection of symplectic pieces can be folded (modulo compatibility conditions), into an origami manifold. Using equivariant versions of these operations, we will see how classic symplectic results of convexity and classification of toric manifolds translate to the origami world. There will be pictures resembling paper origami, but no instructions on how to fold a paper crane. I will attempt to make this talk symplectically-self contained.
Abstract: Let M be a smooth compact Riemannian manifold. An Anosov diffeomorphism is a diffeomorphism from M to itself such that the tangent bundle of M splits into an invariant sum of the stable subbundle and the unstable subbundle. The diffeomorphism contracts the stable subbundle and expands the unstable subbundle exponentially fast. Anosov diffeomorphisms form a C^1 open set in Diff(M) and structural stability asserts that two Anosov diffeomorphisms which are C^1 close are conjugate. The conjugacy is a homeomorphism that typically fails to be C^1. In this talk we will discuss the structure of smooth conjugacy classes of Anosov diffeomorphisms.
Abstract: The homology and singular sets of quotients of spheres by linear actions of tori can be computed using the Tutte polynomial of a matroid.
Abstract: The theorem says this: let m and n be natural numbers with m ‹ n. Suppose you have a group G which admits a presentation with n generators and m relators. Then for any set Y of generators of G, there is a subset of n-m elements of Y that freely generate a free group of rank n-m. It is proved by using ordered groups and embeddings in division rings to reduce it to the following statement about finite dimensional vector spaces: if V is an n dimensional vector space and U is an m-dimensional subspace then any subset Y of of V which spans V modulo U contains a subset of n-m vectors which span a complement to U in V.
Abstract: String topology is basically the study of algebraic structures present in the free loop space of a manifold. The seminal paper of Chas and Sullivan which started this topic was motivated by works of Goldman and Turaev. We shall review the background of Goldman and Turaev's work for surfaces and lead to its more modern interpretations which has significant connections to other topics of current research.
Abstract: We define and study certain geometric loops, called transversal strings, which satisfy some specific boundary conditions. In particular, we consider smooth paths in MxM that start and end on the diagonal and only intersect the diagonal non-tangentially, including the end points. Such strings can be naturally split at the intersection points giving rise to a differential graded coalgebra. We'll analyze where this coalgebra lives and discuss further algebraic structures in this setting. As an application, aided by homological algebra, we can recover the homotopy type of the complement of the diagonal in MxM which is known not to be an invariant of the homotopy type of M.
Abstract: I will discuss a method of coding of geodesics on surfaces of constant negative curvature using boundary maps and "reduction theory". For compact surfaces these maps are generalizations of the Bowen-Series map. For the modular surface they are related to a family of (a,b)-continued fractions. In special cases, when an (a,b)-expansion has a so-called "dual", the coding sequences are obtained by juxtaposition of the boundary expansions of the fixed points, and the set of coding sequences is a countable sofic shift. I will also give a dynamical interpretation of the "reduction theory" which underlines these constructions and its relation to the attractor of a certain associated natural extension map. The talk is based on joint works with Ilie Ugarcovici.
Abstract: The period of a compact Riemann surface induces the period map from the moduli space of compact Riemann surfaces to the Siegel modular variety, and the classical Schottky problem is to characterize the image of the period map, or equivalently to characterize Jacobian varieties among principally polarized abelian varieties. The coarse Schottky problem is to describe the image of the period map from the point of large scale geometry when the Siegel modular variety is considered as a noncompact metric space with respect to the natural metric.
One purpose of this talk is to discuss a solution to the Coarse Schottky problem.
Another purpose is to discuss the analogous problem for the period map from the moduli space of compact tropical curves (or compact metric graphs) and possible application to the outer space of marked metric graphs, which is an analogue of Teichmuller spaces for the outer automoerphism group of free groups.
Abstract: The set G(m) of all isomorphism classes of m-generator groups can be identified with the set of all normal subgroups S of the free group F of rank m. G(m) is endowed with the CHABAUTY-topology, which is based on the sets of all S containing one and avoiding a second given finite subset of F. It is a remarkable fact that whether a group G in G(m) is isolated, or is a condensation point - or, for that matter, any other local property of the point G in G(m) - is independent of the generating set of G and of m; hence those are group theoretic properties. I will present examples and condensation criteria, one of which is best understood in terms of the Geometric Invariant Σ(G) from joint work with Walter Neumann and Ralph Strebel. The presented work is a joint work with Luc Guyot, Yves de Cornulier and Ralph Strebel.
Abstract: We will present a proof of the Strengthened Hanna Neumann Conjecture (SHNC), and some more general results. We will mention trees, forests, flowers, gardens, and leafages. Submultiplicativity is a generalization of the statement of SHNC from graphs to complexes, and from free groups to more general groups. Submultiplicativity holds for complexes under an additional assumption: the deep-fall property. This property is related to the Atiyah Conjecture, a problem from analysis.
Abstract: I will describe some wild geometry that arises in an apparently benign group theoretic setting: I will exhibit a family of groups that are CAT(0), bi-automatic, 1-relator, and free-by-cyclic, and yet have free subgroups of huge (Ackermannian) distortion. I will show how these lead to examples of hyperbolic groups with finite-rank free subgroups of similarly huge distortion. The origin of the extreme behaviour lies in a simple computational game - a realisation of Hercules' battle with the hydra, played out in manipulations of strings of letters.