Claremont Topology-Geometry Seminar

Spring 2004 Schedule

special times and locations are in red

Date Speaker Title and Abstract

Jan 27

Tuesday, Feb 3 Jason Manning, CIT Title: Bounded generation and quasi-actions on trees
Abstract: A quasi-action by a group is the coarse analogue of an isometric action. We show that for a certain class of boundedly generated groups every quasi-action on a tree is trivial. Our method involves working with the coarse geometry of various Cayley graphs (with infinite generating sets) of these groups.

Tuesday, Feb 10 Dan Knopf, U of Iowa
Pitzer Job candidate colloquium
Broad Hall 210, Pitzer College, 4-5pm; Tea at 3:30, Scott Hall Courtyard
Title: The Ricci Flow Program for Geometrization of 3-manifolds
Abstract: One of the triumphs of nineteenth-century mathematics was the Uniformization Theorem, which implies that every smooth surface admits an essentially unique conformal metric of constant curvature. This result provides a classification of 2-dimensional manifolds into three families --- those of constant positive, zero, or negative curvature.
William Thurston proposed an analogous classification of 3-dimensional manifolds, known as the Geometrization Conjecture. It says that each closed 3-dimensional manifold should be canonically decomposable into pieces each of which admits exactly one of eight standard geometric structures. The Geometrization Conjecture is vastly more difficult than the Uniformization Theorem, and subsumes the Poincare' Conjecture (one of the seven million-dollar Millennium Problems selected by the Clay Institute).
The Ricci Flow Program is a strategy designed by Richard Hamilton for proving Geometrization by evolving a Riemannian metric --- possibly after performing topological surgeries on the underlying manifold --- toward one which is canonical. Many mathematicians have contributed to this program; and today, Grisha Perelman is claiming to have completed it and thereby proved both the Geometrization and Poincare' Conjectures. Although experts have not yet verified his complete argument, the results obtained thus far are very promising and confirm that flows are among the most powerful techniques known for studying geometry and low-dimensional topology.
My talk will introduce the Ricci Flow Program, and will describe in particular how singularities of the flow yield geometric and topological information about the underlying manifold. The talk will be accessible to a general mathematical audience.

Tuesday, Feb 17 Inna Korchagina, Rutgers University
Pitzer Job candidate colloquium
Broad Hall 210, Pitzer College, 4-5pm; Tea at 3:30, Scott Hall Courtyard
Title: The Classification of Finite Simple Groups: Aspects of the Second Generation Proof
Abstract: The classification of finite simple groups is widely acknowledged to be one of the major results in modern mathematics. The successful completion of its proof was announced in the early 1980's by Daniel Gorenstein. The original proof occupied somewhere around 15,000 journal pages spread across more than 500 separate articles written by more than 100 mathematicians. Shortly thereafter, a "revision" project has been started, whose goal was to produce a new unified correct proof of the Classification Theorem of less than 5,000 pages in length. The strategy of the revision proof differs from the original one.
In this talk we will outline the "Generation 2"-proof of the Classification, and discuss a specific part of it, in which the speaker is involved.

Thursday, Feb 19, NOTE SPECIAL DAY Dave Bachman, Cal Poly SLO
Pitzer Job candidate colloquium
Broad Hall 210, Pitzer College, 4-5pm; Tea at 3:30, Scott Hall Courtyard
Title: A survey of algorithms in 3-manifold topology
Abstract: The unknotting problem is to give an algorithm which will determine if a loop of string can be deformed (without cutting!) into a round circle. The history of this problem illustrates many of the connections between the algebra, geometry, and topology of 3-manifolds. In 1961 the unknotting problem was solved by W. Haken. Since then, Haken's algorithm has been modified to solve many other interesting problems in 3-manifold topology. For example, in 1991 Rubinstein gave an algorithm which determines if a given 3-manifold is homeomorphic to the 3-sphere.
In this talk we will review Haken's original algorithm and some of the more recent ones that are based on it. We will then discuss some of the open problems that seem likely to fall to similar techniques. Along the way we will mention some of the questions the algorithms themselves have raised, in regards to computational complexity issues.

Tuesday, Feb 24 Erica Flapan, Pomona Topological symmetry groups of complete graphs in $S^3$
For a graph $\Gamma $ which is embedded in $S^{3}$, the symmetries of $\Gamma $ are those automorphisms of the graph that are induced by a diffeomorphism of $S^{3}$. The topological symmetry group, $\roman {TSG}_{+}(\Gamma )$, is the subgroup of the automorphism group of $\Gamma $ consisting of those automorphisms induced by some orientation preserving diffeomorphism of $S^{3}$.
In general, we are interested in which groups can occur as $\roman {TSG}_{+}(\Gamma )$ for some graph $\Gamma $ embedded in $S^{3}$. In 1938, Frucht showed that every finite group is the automorphism group of some graph. In a previous paper we showed that this is not true for the topological symmetry groups. In particular, while every finite abelian group and every symmetric group can occur as $\roman {TSG}_{+}(\Gamma )$ for some embedded graph $\Gamma $, the alternating group $A_{5}$ and the cyclic groups of prime order are the only simple groups which can occur. In this talk, we characterize those groups which can occur as $\roman {TSG}_{+}(\Gamma )$ for some complete graph $\Gamma $ embedded in $S^{3}$.

Tuesday, Mar 2 Sam Nelson, UCR Title: Quandles and the 2-cocycle invariant
Abstract: In 2001, an infinite family of knot invariants was described by Carter, Jelsovsky, Kamada, Langford and Saito, one for each element of the second cohomology group of any finite quandle. In this talk, we will define the quandle 2-cocycle invariant and give an example of its computation.

Tuesday, Mar 9 Berit Givens, Cal Poly Pomona Title: The Bohr Topology and a Hypergraph Topology
Abstract: The Bohr topology on an infinite Abelian group is the coarsest topology such that all characters from the group into the unit circle are continuous. In this talk, I will give some examples of groups with the Bohr topology. Then I will define a topology on a graph space and show how the graph space can answer a question about the Bohr topology.

Tuesday, Mar 16 no seminar (Spring Break)

Tuesday, Mar 23 no seminar

Tuesday, Mar 30 no seminar

Tuesday, Apr 6 no seminar

Tuesday, Apr 13 Francis Su, HMC Title: A combinatorial fixed point theorem for trees
Abstract: We prove a fixed point theorem for n-leaf trees, via a constructive combinatorial lemma about labeled triangulations (segmentations) of trees. We also show the equivalence of these theorems with a KKM-like result for trees. Time permitting, we extend these results to other graphs and give applications (voting, civil engineering). This is joint work with Andrew Niedermaier (HMC '04).

Tuesday, Apr 20 Rob Gaebler, HMC CANCELLED Alexander Polys of 2-bridge knots and links

Tuesday, Apr 27 Rollie Trapp, Cal State Univ, San Bernardino Title: Polygonal Cable Links
Abstract: We describe an algorithm that begins with a polygonal companion knot and constructs polygonal cables of the knot. In some cases the construction is seen to produce minimal polygonal cables, and applications to stick number of torus links are considered. Time permitting, we will discuss applications to Kauffman's notion of minimal flat knotted ribbons.

Tuesday, May 4 John Alongi, Pomona

Date	Speaker	Title and Abstract
Jan 27
Tuesday, Feb 3	Jason Manning, CIT	Title: Bounded generation and quasi-actions on trees Abstract: A quasi-action by a group is the coarse analogue of an isometric action. We show that for a certain class of boundedly generated groups every quasi-action on a tree is trivial. Our method involves working with the coarse geometry of various Cayley graphs (with infinite generating sets) of these groups.
Tuesday, Feb 10	Dan Knopf, U of Iowa Pitzer Job candidate colloquium Broad Hall 210, Pitzer College, 4-5pm; Tea at 3:30, Scott Hall Courtyard	Title: The Ricci Flow Program for Geometrization of 3-manifolds Abstract: One of the triumphs of nineteenth-century mathematics was the Uniformization Theorem, which implies that every smooth surface admits an essentially unique conformal metric of constant curvature. This result provides a classification of 2-dimensional manifolds into three families --- those of constant positive, zero, or negative curvature. William Thurston proposed an analogous classification of 3-dimensional manifolds, known as the Geometrization Conjecture. It says that each closed 3-dimensional manifold should be canonically decomposable into pieces each of which admits exactly one of eight standard geometric structures. The Geometrization Conjecture is vastly more difficult than the Uniformization Theorem, and subsumes the Poincare' Conjecture (one of the seven million-dollar Millennium Problems selected by the Clay Institute). The Ricci Flow Program is a strategy designed by Richard Hamilton for proving Geometrization by evolving a Riemannian metric --- possibly after performing topological surgeries on the underlying manifold --- toward one which is canonical. Many mathematicians have contributed to this program; and today, Grisha Perelman is claiming to have completed it and thereby proved both the Geometrization and Poincare' Conjectures. Although experts have not yet verified his complete argument, the results obtained thus far are very promising and confirm that flows are among the most powerful techniques known for studying geometry and low-dimensional topology. My talk will introduce the Ricci Flow Program, and will describe in particular how singularities of the flow yield geometric and topological information about the underlying manifold. The talk will be accessible to a general mathematical audience.
Tuesday, Feb 17	Inna Korchagina, Rutgers University Pitzer Job candidate colloquium Broad Hall 210, Pitzer College, 4-5pm; Tea at 3:30, Scott Hall Courtyard	Title: The Classification of Finite Simple Groups: Aspects of the Second Generation Proof Abstract: The classification of finite simple groups is widely acknowledged to be one of the major results in modern mathematics. The successful completion of its proof was announced in the early 1980's by Daniel Gorenstein. The original proof occupied somewhere around 15,000 journal pages spread across more than 500 separate articles written by more than 100 mathematicians. Shortly thereafter, a "revision" project has been started, whose goal was to produce a new unified correct proof of the Classification Theorem of less than 5,000 pages in length. The strategy of the revision proof differs from the original one. In this talk we will outline the "Generation 2"-proof of the Classification, and discuss a specific part of it, in which the speaker is involved.
Thursday, Feb 19, NOTE SPECIAL DAY	Dave Bachman, Cal Poly SLO Pitzer Job candidate colloquium Broad Hall 210, Pitzer College, 4-5pm; Tea at 3:30, Scott Hall Courtyard	Title: A survey of algorithms in 3-manifold topology Abstract: The unknotting problem is to give an algorithm which will determine if a loop of string can be deformed (without cutting!) into a round circle. The history of this problem illustrates many of the connections between the algebra, geometry, and topology of 3-manifolds. In 1961 the unknotting problem was solved by W. Haken. Since then, Haken's algorithm has been modified to solve many other interesting problems in 3-manifold topology. For example, in 1991 Rubinstein gave an algorithm which determines if a given 3-manifold is homeomorphic to the 3-sphere. In this talk we will review Haken's original algorithm and some of the more recent ones that are based on it. We will then discuss some of the open problems that seem likely to fall to similar techniques. Along the way we will mention some of the questions the algorithms themselves have raised, in regards to computational complexity issues.
Tuesday, Feb 24	Erica Flapan, Pomona	Topological symmetry groups of complete graphs in $S^3$ For a graph $\Gamma $ which is embedded in $S^{3}$, the symmetries of $\Gamma $ are those automorphisms of the graph that are induced by a diffeomorphism of $S^{3}$. The topological symmetry group, $\roman {TSG}_{+}(\Gamma )$, is the subgroup of the automorphism group of $\Gamma $ consisting of those automorphisms induced by some orientation preserving diffeomorphism of $S^{3}$. In general, we are interested in which groups can occur as $\roman {TSG}_{+}(\Gamma )$ for some graph $\Gamma $ embedded in $S^{3}$. In 1938, Frucht showed that every finite group is the automorphism group of some graph. In a previous paper we showed that this is not true for the topological symmetry groups. In particular, while every finite abelian group and every symmetric group can occur as $\roman {TSG}_{+}(\Gamma )$ for some embedded graph $\Gamma $, the alternating group $A_{5}$ and the cyclic groups of prime order are the only simple groups which can occur. In this talk, we characterize those groups which can occur as $\roman {TSG}_{+}(\Gamma )$ for some complete graph $\Gamma $ embedded in $S^{3}$.
Tuesday, Mar 2	Sam Nelson, UCR	Title: Quandles and the 2-cocycle invariant Abstract: In 2001, an infinite family of knot invariants was described by Carter, Jelsovsky, Kamada, Langford and Saito, one for each element of the second cohomology group of any finite quandle. In this talk, we will define the quandle 2-cocycle invariant and give an example of its computation.
Tuesday, Mar 9	Berit Givens, Cal Poly Pomona	Title: The Bohr Topology and a Hypergraph Topology Abstract: The Bohr topology on an infinite Abelian group is the coarsest topology such that all characters from the group into the unit circle are continuous. In this talk, I will give some examples of groups with the Bohr topology. Then I will define a topology on a graph space and show how the graph space can answer a question about the Bohr topology.
Tuesday, Mar 16		no seminar (Spring Break)
Tuesday, Mar 23		no seminar
Tuesday, Mar 30		no seminar
Tuesday, Apr 6		no seminar
Tuesday, Apr 13	Francis Su, HMC	Title: A combinatorial fixed point theorem for trees Abstract: We prove a fixed point theorem for n-leaf trees, via a constructive combinatorial lemma about labeled triangulations (segmentations) of trees. We also show the equivalence of these theorems with a KKM-like result for trees. Time permitting, we extend these results to other graphs and give applications (voting, civil engineering). This is joint work with Andrew Niedermaier (HMC '04).
Tuesday, Apr 20	Rob Gaebler, HMC	CANCELLED Alexander Polys of 2-bridge knots and links
Tuesday, Apr 27	Rollie Trapp, Cal State Univ, San Bernardino	Title: Polygonal Cable Links Abstract: We describe an algorithm that begins with a polygonal companion knot and constructs polygonal cables of the knot. In some cases the construction is seen to produce minimal polygonal cables, and applications to stick number of torus links are considered. Time permitting, we will discuss applications to Kauffman's notion of minimal flat knotted ribbons.
Tuesday, May 4	John Alongi, Pomona