Claremont Topology-Geometry Seminar

Fall 2004 Schedule

special days, times or locations are in red

Date Speaker Title and Abstract

Tuesday, Sept 14 Ryan Derby-Talbot, Pomona College Title: Stabilizations of Heegaard splittings and incompressible tori.
Abstract: Since the 1930's it has been known that any two Heegaard splittings of a given 3-manifold become isotopic after a certain number of moves called stabilizations. A general statement of how many stabilizations are required, however, remains elusive. After surveying recent progress on this problem, we consider the case that a given 3-manifold, M, contains an incompressible torus. We then analyze the number of required stabilizations for Heegaard splittings of M which differ by a Dehn twist around this torus.

Tuesday, Sept 21 Ryan Derby-Talbot, Pomona College continuation of last week

Tuesday, Sept 28 Dave Bachman, Pitzer College Title: Heegaard splittings and connected sums
Abstract: We will explore some of the history and recent progress in the study of Heegaard splittings of 3-manifolds constructed by connected sum. We will begin by reviewing a classical lemma of Haken, and end with a sketch of the speaker's recent proof of a conjecture of Gordon: The connected sum of unstabilized Heegaard splittings is unstabilized.

Tuesday, Oct 5 Dave Bachman, Pitzer College continuation of last week

Tuesday, Oct 12 Chiung-Ju Liu, UC Irvine Title: The Futaki invariant and Bando-Futaki invariants
Abstract: The Futaki invariant and the Bando-Futaki invariants are defined to be homomorphisms on the lie algebra of the holomorphic vector fields over a Kahler manifold. The Futaki invariant is an obstruction to the existence of a Kahler-Einstein metric on a compact Kahler manifold with positive first Chen class. The Futaki invariant and the Bando-Futaki invariants on hypersurfaces on CP^{n} depend on n, the degree of the defining polynomial, and the given tangent holomorphic vector field.

Tuesday, Oct 19 no seminar (Fall Break)

Wednesday, Oct 27
3-4pm
Dave Gabai, Princeton Univ. In 1935 J. H. C. Whitehead found an example of an open simply connected 3-manifold which was topologically distinct from the standard R^3. In the nearly 70 years since, a theory has developed and more surprises found. E.g. the Artin - Fox example of a 3-manifold whose interior is R^3 and whose boundary is R^2, but is not the standard closed upper half space.
We will survey these developments and discuss the very recent theorem of Agol and Calegari - Gabai which completed a 30 year effort to show that complete hyperbolic 3-manifolds with finitely generated fundamental group are topologically tame.

Tuesday, Nov 2 Tamas Kalman, USC Title: Legendrian invariants of positive knots
Abstract: A Legendrian knot is an embedded closed curve in R^3 which is tangent to the kernel of the 1-form dz - y dx. I will review some recent developments in the study of these knots, most notably their contact homology and ruling invariants. Then I will illustrate some of these constructions on positive knots.

Tuesday, Nov 9 Sam Nelson, UC Riverside Title: Quandles and Generalized Knot Groups
Abstract: Generalized knot groups were discovered independently by Wada and Kelly. The language of quandle theory gives a convenient way of defining these groups, which form an infinite family of groups of which the usual knot group is a special case. We study the question of whether these generalized knot groups contain additional information about knot type not present in the usual knot group.

Tuesday, Nov 16 Alissa Crans, Loyola Marymount Univ. Title: A solution of the Zamolodchikov Tetrahedron Equation
Abstract: The Yang-Baxter equation arises in many contexts in mathematics and physics. All these concepts are related by the fact that this equation is an algebraic distillation of the third Reidemeister move in knot theory. The `Zamolodchikov tetrahedron equation' is the higher-dimensional analogue of the third Reidemeister move, and plays a role in the theory of knotted surfaces in 4-space which is closely analogous to that played by the third Reidemeister move in the theory of ordinary knots in 3-space. We will show that just as any Lie algebra gives a solution of the Yang-Baxter equation, any `Lie 2-algebra', a higher-dimensional analogue of a Lie algebra, gives a solution of the Zamolodchikov tetrahedron equation.

Tuesday, Nov 23 Joseph Maher, Cal Tech Title: Heegaard gradient and virtual fibers.
Abstract: We show that if a hyperbolic 3-manifold has infinitely many covers of bounded Heegaard genus then it is virtually fibered. This is a generalization of a theorem of Lackenby, who showed this in the case that the covers were regular. Although this work is inspired by Lackenby and Lubotzky's work on property tau, our techniques are largely geometric and topological, using sweepouts and hyperbolic geometry. In fact, we prove a slightly stronger result, as we only need the covers to have bounded width, in the sense of Scharlemann-Thompson thin position for Heegaard splittings.

Tuesday, Nov 30 Florence Newberger, CSU Long Beach Title: Some rigidity theorems in Riemannian Geometry and their generalizations
Abstract: Suppose you have an example of a metric space that has a variety of special properties, and you prove a theorem about this metric space. Then in an attempt to generalize your proof to a wider variety of metric spaces, you assume you have a metric space with some but not all of the properties of your original space. We use the word rigid to describe a space for which this strategy does not produce a new theorem: the original metric space M is rigid with respect to a property if any space with that property turns out to be equivalent to M. For example, hyperbolic space and its cousins are rigid with respect to a many of their properties. In this talk, I will give some examples of rigidity theorems for hyperbolic space in Riemannian Geometry with respect to various properties, and then describe some generalizations in Finsler geometry.

Tuesday, Dec 7 Peter Milley, UC Riverside Title: Cellular complexes in small-volume hyperbolic 3-manifolds
Abstract: Although Thurston and Jorgensen proved that any collection of finite-volume hyperbolic 3-manifolds has a minimum-volume element, actually finding such small-volume manifolds and proving their minimality has proven to be quite challenging. In this talk I will review some of the milestones in this area of research and discuss an ongoing project to identify small-volume cusped and closed hyperbolic 3-manifolds.

Spring 2005 Schedule

special days, times or locations are in red

Date Speaker Title and Abstract

Tuesday, Jan 25 no meeting

Tuesday, Feb 1 Aaron Abrams, U of Georgia Title: Braids, graphs, and robots
Abstract: A common technique for solving problems involving lots of moving objects is to introduce a topological space called a ``configuration space'' associated to the problem. This talk will focus on the configuration spaces associated to motions of several particles on a graph. We will play with several examples and hopefully get used to visualizing these spaces. Along the way these spaces will exhibit some neat properties, which are both mathematically interesting and useful for solving related problems in robotics.

Tuesday, Feb 8 Jim Hoste, Pitzer College Title: Boundary Slopes in 2-bridge Link Complements
Abstract: This will be a historical talk first describing the beautiful work of Floyd and Hatcher to determine all incompressible surfaces in the complement of 2-bridge links, and secondly, the upublished work of Lash to compute the boundary slopes of these surfaces. I will include lots of examples and pictures.

Tuesday, Feb 15 Bob Penner, Univ Southern California Title: The structure of arc complexes
Abstract: Riemann's moduli space admits a cell decomposition where cells are indexed by appropriate families of arcs in the underlying surface, and this decomposition has been successfully applied by the speaker and others to elucidate the structure of moduli space. A basic question studied for almost 20 years by the speaker is the extent to which these arc complexes compactify to spheres, manifolds, or orbifolds. Recent work with Dennis Sullivan has resolved this question and characterized compactified arc complexes as a suitable generalization of manifold or orbifold. We shall survey this work, which has consequences for the algebraic topology of Riemann's moduli space, and shall emphasize the combinatorics so as to make this work accessible without assuming extensive background.

Tuesday, Feb 22

Thursday, Mar 3 Hugh Howards, Wake Forrest University Title: The trouble with Brunnian Links
Abstract: A Brunnian link is a link of n>2 components, which is not the unlink, but such that every sub-link is the unlink. Freedman proved that no Brunnian link could be made out of circles. We will generalize this to show that the Borromean Rings are the only Brunnian link of less than 4 components that can be made out of convex curves. We also generalize Freedman's results to higher dimensions.

Tuesday, Mar 8 Constance Leidy, U of Pennsylvania

Tuesday, Mar 15 No Meeting -- Spring Break

Tuesday, Mar 22 Scott Crass, CSU Long Beach Title: The quintic is to the icosahedron as the sextic is to what?
Abstract: To solve a polynomial equation of degree $n$ you need a means of breaking its symmetry. It's enough to work with the alternating group $A_n$. The first task is to find a (low-dimensional) space where the symmetries are realized. In the case of the quintic, the symmetries appear in one complex dimension---the rotations of the icosahedron that preserve the polyhedron's configuration. The geometric and algebraic structure of this action of $A_5$ can be harnessed to produce a dynamcial system that's at the core of a solution to the quintic. For the sixth-degree case, there's an action of $A_6$ on the complex projective plane. We'll explore some of the rich geometry that results as well as the dynamics that can be employed to solve the sextic.

Tuesday, Mar 29

Tuesday, Apr 5

Tuesday, Apr 12 Daniel Groves, Cal Tech Title: A Dehn surgery theorem for relatively hyperbolic groups.
(joint work with Jason Manning)
Abstract: We give a group-theoretic analogue of the Gromov-Thurston 2\pi Theorem for hyperbolic 3-manifolds. Specifically, if G is a torsion-free group which is hyperbolic relative to a free abelian rank 2 subgroup P, then for all but finitely many primitive elements p of P, the group G/Q, where Q is the cyclic generated by p, is infinite, non-elementary and word-hyperbolic.

Tuesday, Apr 19 Francis Bonahon, USC Title: Arborescent knots
Abstract: Arborescent knots and links (also called algebraic by Conway) form a large class of links, particularly prevalent among knots and links with relatively small crossing numbers. I will discuss their classification. This is very old joint work with Larry Siebenmann.

Tuesday, Apr 26

Tuesday, May 3

Date	Speaker	Title and Abstract
Tuesday, Sept 14	Ryan Derby-Talbot, Pomona College	Title: Stabilizations of Heegaard splittings and incompressible tori. Abstract: Since the 1930's it has been known that any two Heegaard splittings of a given 3-manifold become isotopic after a certain number of moves called stabilizations. A general statement of how many stabilizations are required, however, remains elusive. After surveying recent progress on this problem, we consider the case that a given 3-manifold, M, contains an incompressible torus. We then analyze the number of required stabilizations for Heegaard splittings of M which differ by a Dehn twist around this torus.
Tuesday, Sept 21	Ryan Derby-Talbot, Pomona College	continuation of last week
Tuesday, Sept 28	Dave Bachman, Pitzer College	Title: Heegaard splittings and connected sums Abstract: We will explore some of the history and recent progress in the study of Heegaard splittings of 3-manifolds constructed by connected sum. We will begin by reviewing a classical lemma of Haken, and end with a sketch of the speaker's recent proof of a conjecture of Gordon: The connected sum of unstabilized Heegaard splittings is unstabilized.
Tuesday, Oct 5	Dave Bachman, Pitzer College	continuation of last week
Tuesday, Oct 12	Chiung-Ju Liu, UC Irvine	Title: The Futaki invariant and Bando-Futaki invariants Abstract: The Futaki invariant and the Bando-Futaki invariants are defined to be homomorphisms on the lie algebra of the holomorphic vector fields over a Kahler manifold. The Futaki invariant is an obstruction to the existence of a Kahler-Einstein metric on a compact Kahler manifold with positive first Chen class. The Futaki invariant and the Bando-Futaki invariants on hypersurfaces on CP^{n} depend on n, the degree of the defining polynomial, and the given tangent holomorphic vector field.
Tuesday, Oct 19	no seminar (Fall Break)
Wednesday, Oct 27 3-4pm	Dave Gabai, Princeton Univ.	In 1935 J. H. C. Whitehead found an example of an open simply connected 3-manifold which was topologically distinct from the standard R^3. In the nearly 70 years since, a theory has developed and more surprises found. E.g. the Artin - Fox example of a 3-manifold whose interior is R^3 and whose boundary is R^2, but is not the standard closed upper half space. We will survey these developments and discuss the very recent theorem of Agol and Calegari - Gabai which completed a 30 year effort to show that complete hyperbolic 3-manifolds with finitely generated fundamental group are topologically tame.
Tuesday, Nov 2	Tamas Kalman, USC	Title: Legendrian invariants of positive knots Abstract: A Legendrian knot is an embedded closed curve in R^3 which is tangent to the kernel of the 1-form dz - y dx. I will review some recent developments in the study of these knots, most notably their contact homology and ruling invariants. Then I will illustrate some of these constructions on positive knots.
Tuesday, Nov 9	Sam Nelson, UC Riverside	Title: Quandles and Generalized Knot Groups Abstract: Generalized knot groups were discovered independently by Wada and Kelly. The language of quandle theory gives a convenient way of defining these groups, which form an infinite family of groups of which the usual knot group is a special case. We study the question of whether these generalized knot groups contain additional information about knot type not present in the usual knot group.
Tuesday, Nov 16	Alissa Crans, Loyola Marymount Univ.	Title: A solution of the Zamolodchikov Tetrahedron Equation Abstract: The Yang-Baxter equation arises in many contexts in mathematics and physics. All these concepts are related by the fact that this equation is an algebraic distillation of the third Reidemeister move in knot theory. The `Zamolodchikov tetrahedron equation' is the higher-dimensional analogue of the third Reidemeister move, and plays a role in the theory of knotted surfaces in 4-space which is closely analogous to that played by the third Reidemeister move in the theory of ordinary knots in 3-space. We will show that just as any Lie algebra gives a solution of the Yang-Baxter equation, any `Lie 2-algebra', a higher-dimensional analogue of a Lie algebra, gives a solution of the Zamolodchikov tetrahedron equation.
Tuesday, Nov 23	Joseph Maher, Cal Tech	Title: Heegaard gradient and virtual fibers. Abstract: We show that if a hyperbolic 3-manifold has infinitely many covers of bounded Heegaard genus then it is virtually fibered. This is a generalization of a theorem of Lackenby, who showed this in the case that the covers were regular. Although this work is inspired by Lackenby and Lubotzky's work on property tau, our techniques are largely geometric and topological, using sweepouts and hyperbolic geometry. In fact, we prove a slightly stronger result, as we only need the covers to have bounded width, in the sense of Scharlemann-Thompson thin position for Heegaard splittings.
Tuesday, Nov 30	Florence Newberger, CSU Long Beach	Title: Some rigidity theorems in Riemannian Geometry and their generalizations Abstract: Suppose you have an example of a metric space that has a variety of special properties, and you prove a theorem about this metric space. Then in an attempt to generalize your proof to a wider variety of metric spaces, you assume you have a metric space with some but not all of the properties of your original space. We use the word rigid to describe a space for which this strategy does not produce a new theorem: the original metric space M is rigid with respect to a property if any space with that property turns out to be equivalent to M. For example, hyperbolic space and its cousins are rigid with respect to a many of their properties. In this talk, I will give some examples of rigidity theorems for hyperbolic space in Riemannian Geometry with respect to various properties, and then describe some generalizations in Finsler geometry.
Tuesday, Dec 7	Peter Milley, UC Riverside	Title: Cellular complexes in small-volume hyperbolic 3-manifolds Abstract: Although Thurston and Jorgensen proved that any collection of finite-volume hyperbolic 3-manifolds has a minimum-volume element, actually finding such small-volume manifolds and proving their minimality has proven to be quite challenging. In this talk I will review some of the milestones in this area of research and discuss an ongoing project to identify small-volume cusped and closed hyperbolic 3-manifolds.