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| Date | Speaker | Organization | Title | Abstract |
| 1 April | Danny Calegari | Caltech | Planar groups and circular groups. | |
| 8 April | Daryl Cooper | UCSB | A Combination theorem. | Suppose that $A$ and $B$ are two quasi-Fuchsian subgroups of a cocompact Kleinian group and which have intersection $C.$ Then there are subgroups A' and B' of finite index in A and B so that the subgroup generated by A' and B' is the amalgamated free product A' *_C B'. This is joint with Mark Baker. |
| 15 April | Akos Dobay | Universite de Lausanne | Physical and Statistical properties of Random knots | In this talk, I shall introduce you to the concept of random knots. We shall see what kind of mesure we can define to analyse their physical and statistical properties and how we can relate those properties to the world of polymer science. The talk is designed for graduate and undergraduate as well. |
| 22 April | Daniel Groves | Oxford | Obstructions to word-hyperbolicity and mapping tori of free group endomorphisms. | It is known that word-hyperbolic groups admit a finite K(\pi,1) and have no `Baumslag-Solitar' subgroups, but it is not known whether the converse is true. We will discuss this question for mapping tori of injective free group endomorphisms, and prove that the absence of Baumslag-Solitar subgroups is equivalent to being word-hyperbolic for a large class of endomorphisms. We will also discuss when the non-hyperbolic mapping tori admit a quadratic isoperimetric inequality. |
| 29 April | Tao Li | Oklahoma State at Stillwater | An algorithm to recognize small Seifert fiber spaces | We give an algorithm to find vertical essential tori in small Seifert fiber spaces with infinite fundamental groups. This implies that there are algorithms to decide whether a 3-manifold is a Seifert fiber space. |
| 6 May: 3-4 | John Ennis | UCSB | Control over Semilocal Topology from a Ricci Curvature Bound. | The Gromov-Hausdorff metric has many applications in differential geometry. One topic of interest is the topology of limit spaces of manifolds with a uniform curvature bound. For example, if $X=\lim M_i^n$ where $M_i$ are Riemannian manifolds with $\mathrm{Diam}(M_i) \leq D,$ $\mathrm{Ric}(M_i) \geq H(n-1)$ and $\mathrm{Vol}(M_i) \geq V,$ it is unknown whether the universal cover of $X$ is simply connected. In this talk I will discuss how results proven by J.W. Cannon and G.R. Conner clarify the semilocal topology of such limit spaces. |
| 6 May: 4-5 | Matt White | Cal Poly SLO | Maximum Injectivity Radius and Heegaard Genus | In joint work with Daryl Cooper and David Bachman, I will discuss some new bounds for injectivity radius in hyperbolic 3-manifolds. We show if $M$ is a closed, connected, orientable, hyperbolic 3-manifold with Heegaard genus $g$ then $g \ge \frac{1}{2}\cosh (r) - \frac{1}{4}$ where $r$ denotes the radius of any isometrically embedded ball in $M$. Assuming an unpublished result of Pitts and Rubinstein, we can improve this to $g \ge \frac{1}{2}\cosh (r) + \frac{1}{2}.$ |
| 13 May | Marty Scharlemann | UCSB | 3-manifolds with planar presentations | We consider 3-manifolds having a submersion to R
in which each generic point inverse is a planar surface. The standard
height function on a submanifold of S3 is a motivating example.
To (M, h) we associate a connectivity graph Γ. For M
a subset of S3, Γ is a tree if and only if there is a Fox
reimbedding of M which carries horizontal circles to a complete
collection of complementary meridian circles. On the other hand, if
M is a handlebody and the connectivity graph of the complement of M is a tree, then there is a level-preserving reimbedding of M so that its complement is a
connected sum of handlebodies.
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| 20 May | Mohamed Ait Nouh | UCSB | Obtaining graph knots by twisting unknots | This is a joint work with D. Matignon (University of Provence, France) and K. Motegi (Nihon University, Japan) which will appear at Topology and its applications: Let $K$ be a knot in the $3$-sphere $S^3$, and $D$ a disk in $S^3$ meeting $K$ transversely more than once in the interior. For non-triviality we assume that $\vert D \cap K \vert \ge 2$ over all isotopy of $K$ in $S^3 - \partial D$. Let $K_n$($\subset S^3$) be a knot obtained from $K$ by $n$ twisting along the disk $D$. If $K$ is a trivial knot in $S^3$, then $K_n$ is called a twisted knot. We fisrt prove that if $K_n$ is a graph knot, then $ n = \pm 1$ or $K$ and $D$ form a special pair which we call an ``exceptional pair''. As a corollay, if $(K,D)$ is not an exceptional pair, then by twisting unknot $K$ more than once along the disk $D$, we always obtain a knot with positive Gromov volume. We will also show that there are infinity many graph knots each of which is obtained from a trivial knot by twisting, but its companion knot cannot be obtained insuch a manner. The proofs uses heavily Jaco-Shallen-Johansen decomposition and Dehn surgery. |
| 27 May: 2-3 | Kevin Whyte | UIC | Quasi-actions versus Isometric actions : Generalizations of a theorem of Tukia | A basic problem in the study of the large scale geometry of a space X is to decide whether a group of coarse symmetries can be realized as isometric symmetries on X (or a space which closely resembles X). This question turns out to be closely related to several classical problems in dynamics. We will discuss some answers, both positive and negative, and give some applications to Gromov's program of classifying finitely generated groups up to quasi-isometry. |
| 27 May: 3-4 | Raymond Lickorish | Cambridge | Splittings of S^4 | |
| 27 May: 4-5 | David Epstein | Warwick | The Space of Trees. | |
| 3 June | ||||
| 6 June 10am | Jason Manning | UCSB | The Geometry of Pseudocharacters. | Thesis defense. |
| 6 June 1pm | Ian Agol | UIC | Conjectures on minimal surfaces and Ricci flow. | We state various conjectures on the behavior of minimal surfaces and Ricci flow, explaining heuristic motivations for the conjectures and proofs in some very special cases. We also prove a version of Kneser-Haken finiteness for stable minimal surfaces (joint with Hass), i.e. given a closed riemannian 3-manifold, there is a number n such that if one has n disjoint stable minimal surfaces, then two must be parallel. We use this to show finiteness of components of stable minimal surfaces in bumpy metrics, and conjecture finiteness in the case of analytic metrics which are not foliated by compact minimal surfaces. |
| 6 June 2pm | Maggie Tomova | UCSB |
| Date | Speaker | Organization | Title | Abstract |
| Friday, 10 January SH 6635 |
John Meier | Lafayette College | The braided Thompson group | Thompson's group V is often described as the group of dyadic homeomorphisms of the Cantor set. Matt Brin has introduced a braided version of this group, BV, whose elements can be realized by braiding subCantor sets. Much of this talk will be geared toward defining this group, but I will also outline how Brin and I have proven that BV is torsion-free, infinite dimensional, and finitely presentable. Further, BV admits a K(\pi,1) that is finite in each dimension. |
| 14 January | Martin Scharlemann | UCSB | Ideal triangulations of punctured torus bundles II | See 5 Nov listing (below) |
| 21 January | Jason Manning | UCSB | FA, Q(uasi)FA, and SL(3,Z) | A group G has property FA if every action of G on a simplicial tree has a global fixed point. A group has property QFA if every quasiaction of G on a simplicial tree has bounded orbits. Serre proved that SL(3,Z) has property FA. I will discuss Serre's proof, give an idea of why it doesn't generalize to show SL(3,Z) has QFA, and then show that SL(3,Z) nonetheless has QFA. |
| 28 January | Jon McCammond | UCSB | A combinatorial Gauss-Bonnet theorem for 2-complexes | Although most geometric group theorists (and low-dimensional topologists) have run across various combinatorial versions of the Gauss-Bonnet theorem for surfaces, fewer are aware of its natural generalization to arbitrary 2-complexes. In this talk I will state and prove this generalization, followed by a complete (and self-contained) proof of the main theorem of small cancelation theory. Other applications will be given as time permits. (Jt. with D.Wise) |
| 4 February | Shelley Harvey | UCSD | Some noncommutative 3-manifold invariants and their applications to 3- and 4-manifolds | We define a sequence of positive integers, delta_n(psi), for a given 3-manifold X and a first cohomology class psi. These are especially important inasmuch as they give lower bounds for the Thurston norm of psi and give obstructions to X fibering over the circle. We show that delta_n(psi) is an nondecreasing function of n. Using results of P. Kronheimer, T. Mrowka and S. Vidussi, this implies that delta_n and Thurston norm agree on some cohomology class whenever X x S^1 admits a symplectic structure. Finally, we give examples of 3-manifolds X such that X x S^1 does not admit a symplectic structure but cannot be distinguished from a symplectic X x S^1 using the Seiberg-Witten invariants. |
| 11 February | Benjamin Martin | Hebrew University | Character varieties of finitely generated groups | Let $F$ be a finitely generated group and let $G$ be either a real Lie group or a linear algebraic group. The set ${\rm R}(F,G)$ of group homomorphisms from $F$ to $G$ has the structure of an analytic or algebraic variety (depending on $G$). If $G$ is reductive then one can form the {\em character variety} ${\rm C}(F,G)={\rm R}(F,G)/G$; roughly speaking, this is the space of conjugacy classes of homomorphisms from $F$ to $G$. I will discuss the geometry of ${\rm R}(F,G)$ and ${\rm C}(F,G)$, and describe some applications to the topology and differential geometry of low-dimensional manifolds. |
| 18 February | No Seminar because of Job Talk | |||
| 25 February | Daryl Cooper | UCSB | F-structures, 3-Manifolds and Voronoi decompositions, Part I | We develop a theory of Voronoi decompositions in Riemannian geometry. This has applications to the topology of 3-manifolds. A sample application: The (infinite) set of diffeomorphism classes of closed orientable 3-manifolds which admit a Riemannian metric of volume at most V > 0 and |sec. curvature| < 1 contains at most finitely many irreducible, atoroidal 3-manifolds which do not satisfy Thurston's geometrization conjecture. This is joint with Michel Boileau. |
| 4 March | Chris Hruska | University of Chicago | Thin triangles and spaces with isolated flats. | The theory of $\delta$-hyperbolic spaces has been enormously fruitful since it was introduced by Gromov in the late 1980s. I will describe a special class of nonpositively curved spaces which share many features with $\delta$-hyperbolic spaces. The spaces in question are those in which the flat Euclidean subspaces are isolated from each other. In fact many results from the $\delta$-hyperbolic setting have natural extensions to the isolated flats setting. In contrast, few methods are currently known for extending results from the ``negatively curved'' setting to arbitrary nonpositively curved spaces. One of the fundamental tools in the theory of $\delta$-hyperbolic spaces is the fact that tringles are $\delta$-thin. I will discuss the analogous notion of a triangle being thin relative to a flat. This property holds in many spaces with isolated flats, and plays a key role in the generalization of results from the hyperbolic setting to the isolated flats setting. |
| 11 March | David Bachman | Cal Poly | Distance and bridge position | Hempel's definition of the "distance" of a Heegaard surface generalizes to a complexity for a knot which is in bridge position with respect to a Heegaard surface. Building on previous work of Hartshorne we show that the distance of a knot in bridge position is bounded above by twice the genus, plus the number of boundary components, of an essential surface in the knot complement. As a corollary we show that for each $n$ there is an $n$-bridge knot for which thin position is bridge position. This is joint work with Saul Schleimer. |
| Date | Speaker | Organization | Title | Abstract |
| 1 October
2002 |
Jon McCammond | UCSB | Combinatorial conditions that imply word-hyperbolicity for 3-manifolds
[+ Organizational discussion] |
Thurston conjectured that a closed triangulated 3-manifold in which every edge degree is either 5 or 6 and no two edges of degree 5 lie in a common 2-cell, has a fundamental group which is word-hyperbolic. In this talk, I'll describe how my coauthors and I establish Thurston's conjecture by proving that such a triangulation admits a piecewise Euclidean metric of non-positive curvature with no isometrically embedded flat planes. The proof involves a mixture of computer computation and techniques from small-cancelation theory. |
| 8 October
2002 |
Stephen Bigelow | UCSB | Reducible representations of braid groups | A representation of a group is a homomorphism into a group of matrices over some field. Put another way, it is an action of a group on a vector space. It is reducible if there is a non-trivial invariant subspace. I will define some families of representations of braid groups, and describe a topological way to "see" them becoming reducible at certain values of the parameter. |
| 15 October 2002 | Stephen Bigelow | UCSB | Reducible representations of braid groups, part II |
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| 22 October 2002 | Darren Long | UCSB | Pseudo-modular surfaces |  |
| 29 October 2002 | Jon McCammond | UCSB | New Eilenberg-Maclane spaces for braid groups | In this talk I will review a relatively recent construction of K(G,1)s for braid groups -- and other finite type Artin groups (due independently to Tom Brady and Daan Krammer). These spaces are the ones used by Krammer to show linearity and they lead to the Birman-Ko-Lee presentations which have proved so useful in algorithmic problems associated to these groups. In addition, the construction is intimately connected with an object from combinatorics called the non-crossing partition lattice, and they belong to a newly defined class of groups called lattice-generated groups (which are also known as Garside groups) |
| 5 November 2002 | Martin Scharlemann | UCSB | Ideal triangulations of punctured torus bundles | Let M be a punctured torus bundle over the circle, with monodromy A chosen so that M is hyperbolic. According to Epstein-Penner, its hyperbolic (finite volume, complete) structure guarantees that it has an ideal hyperbolic triangulation. On the other hand, a beautiful argument given in the early '80's by Hatcher and Thurston, provides M with another natural triangulation by ideal tetrahedra. Might they be the same? Lackenby proves they always are, in an argument exploiting (fairly) normal surface theory and thin position. This talk will give an informal overview. |
| 12 November 2002 | Ben Mann | NSF | On the geometry and topology of certain moduli spaces | Certain spaces of holomorphic maps between complex manifolds can be shown to be equivalent to the moduli spaces of preferred geometric and analytic structures. Examples of such occur naturally in mathematical physics, linear control theory, and complex geometry. This talk will first explain the correspondence between these holomorphic mapping and moduli spaces. Then we will discuss the relationship between these holomorphic mapping spaces and their natural inclusion into the associated spaces of continuous maps between the geometric objects. Finally, using techniques from homotopy theory and configuration spaces, we will describe the topology and geometry of these spaces. |
| 19 November 2002 | Kasra Rafi | UCSB | Continued fraction expansion for foliations on surfaces |  |
| 26 November 2002 | Saul Schleimer | Heegaard splittings vs. surface bundles | We prove that, in fixed genus, the standard Heegaard splitting of a "generic" surface bundle is the unique splitting of minimal genus. The major tools used are Casson and Gordon's notion of strong irreducibility, the Rubinstein-Scharlemann graphic, and the curve complex. This is a joint work with Dave Bachman. | |
| 3 December 2002 | Se-Goo Kim | UCSB | Virtual knot groups and their peripheral structure | Virtual knots, defined by Kauffman in 1996, provide a natural generalization of classical knots. While most invariants of knots extend in a natural way to give invariants of virtual knots, a number of phenomena arise that contrast sharply with what occurs in the classical setting. In particular, I will talk about the fundamental groups of virtual knots, focusing on properties of the peripheral subgroup and on the homology theory of these groups. |
