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 $g_1^{n_1}\cdots g_k^{n_k}$ when $\sum|n_k|$ is sufficiently large for any given sequence $(g_i)_1^k$ 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.
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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.