seminarsp99.html

Spring 1999 seminars

Topology / Diff. Geom.

Tues. Feb. 2 2:10pm RH 211

Michael Tsau
Saint Louis University

Bracket and Regular Isotopy of Singular links

Math Seminar

Thurs. Feb. 4 1:00pm RH 320

Dr. W. Wistar Comfort
Wesleyan University

Subgroups of Compact groups

Topology / Diff. Geom.

Tues. Feb. 9 2:10pm RH 211

Michael Tsau
Saint Louis University

Bracket and Regular Isotopy of Singular links . (Cont'd)

Topology / Diff. Geom.

Tues. Feb. 16 2:10pm RH 211

Michael Tsau
Saint Louis University

Bracket and Regular Isotopy of Singular links . (Cont'd)

Math Colloquium

Thurs. Feb. 18 1:10pm RH237

Kevin Scannell
Saint Louis University

The Sullivan Dictionary

This is the first in a series of three colloquia which will introduce some of the work of 1998 Fields Medal winners Curtis T. McMullen, W. Timothy Gowers and Richard Borcherds respectively. Each talk will be geared to a general mathematical audience and all are invited to attend.
Abstract: Curt McMullen's work involves two beautiful and closely related subjects: the theory of Kleinian groups and hyperbolic 3-manifolds, and the study of iterates of holomorphic functions of \C^ ("complex dynamics"). In the 1980's, Dennis Sullivan proposed a dictionary by means of which one can try to understand complex dynamics in terms of powerful analytic and geometric techniques in Kleinian groups (eg. quasiconformal mappings) and vice versa. I will discuss this dictionary, mention some of McMullen's contributions and show some pictures.

Topology / Diff. Geom.

Tues. Feb. 23 2:10pm RH 211

John Kalliongis
Saint Louis University

Properties of Lensspaces . (Cont'd)

Topology / Diff. Geom.

Tues. Mar. 2 2:10pm RH 211

John Kalliongis
Saint Louis University

Properties of Lensspaces . (Cont'd)

Math Colloquium

Thurs. Mar. 4 2:10pm RH237

Darrin Speegle
Saint Louis University

(Almost) Arbitrarily Strange Banach Spaces

Abstract: From the beginning of the theory in Banach spaces, people have tried to find analogues of orthonormal bases in Hilbert spaces. (Orthonormal bases themselves are, from the point of view of an analyst, natural analogues of bases in vector spaces.) I will present the two most natural concepts of bases in Banach spaces, and I will discuss the question:''Do bases exist for all Banach spaces?'' I will also explain the contribution that W.T.Gowers (jointly with B.Maurey) made in this area, which was that he was able to give a way of constructing (almost) arbitrarily strange and fabulous Banach spaces. In particular, I will explain how Gowers and Maurey answer the fourth of the four most natural questions relating to bases in Banach spaces. If time permits, I will discuss how his construction can be used to solve several other questions dating back to Banach.

Topology / Diff. Geom.

Tues. Mar. 16 2:10pm RH 211

John Kalliongis
Saint Louis University

Properties of Lensspaces . (Cont'd)

Topology / Diff. Geom.

Tues. Mar. 23 2:10pm RH 211

John Kalliongis
Saint Louis University

Properties of Lensspaces . (Cont'd)

Math Colloquium

Thurs. Mar. 25 1:10pm RH237

Charles Ford
Saint Louis University

Monstrous Moonshine

Abstract: The title of Dr. Ford's talk is taken from the title of a 1979 paper of Conway and Norton concerning the representation theory of the famous "Monster" group (a simple group of order approximately 10⁵⁴) first constructed by R. Griess around 1980. Particularly interesting are the many similarities between genus zero subgroups of the modular group SL(2,Z) and the subgroups of the Monster. Some examples of these connections will be discussed, along with (time permitting) a brief introduction to Borcherd's techniques. The great majority of the talk will be accessible to undergraduate students - anyone wishing to know more about this exciting area of modern mathematical research is encouraged to attend.

Topology / Diff. Geom.

Tues. Mar. 30 2:10pm RH 211

John Kalliongis
Saint Louis University

Properties of Lensspaces . (Cont'd)

Topology / Diff. Geom.

Tues. Apr. 6 2:10pm RH 211

Anneke Bart
Saint Louis University

On: "Some surface subgroups survive surgery", a paper by Cooper and Long.

Math Colloquium

Thur. Apr. 8 4:00pm RH 128

Dr. D. Darren Long
Univ. of California, Santa Barbara

A hitchiker's guide to the Geometrization Conjecture.

Topology / Diff. Geom.

Fri. Apr.9 11:00am RH 237

Dr. D. Darren Long
Univ. of California, Santa Barbara

The subgroup separability of the Bianchi groups.

Topology / Diff. Geom.

Tues. Apr. 13 2:10pm RH 211

Anneke Bart
Saint Louis University

On: "Some surface subgroups survive surgery", a paper by Cooper and Long. (Cont'd)

Math Colloquium

Thur. Apr. 15 4:00pm RH 128

Dr. Jonathan Simon Univ. of Iowa

Critical Shapes of Knots: From Pretty Pictures to DNA Gel-Electrophoresis.

Abstract: Given a knot K in space, we can compute "energies", numbers E(K), that somehow measure how complicated or tangled the given knot is. The motivation for people developing this idea has been a combination of basic mathematics together with a desire to model "physical knots", such as flux tubes in plasmas, DNA loops, or just plain ropes.

Definitions are based on the knot trying to avoid or exclude itself, some kind of self-repelling or rope thickness. We can see in a brief video that the energies are suprisingly effective in simplifying tangled figures.

For smooth knots, energies and crossing numbers are connected: Ignoring multiplicative constants, we have
(crossing number) < (repelling energies) < (4/3) power of "rope length energy"
The latter being an energy based on self-exclusion. The exponent (4/3) is the right one.

Computer experiments flowing knots to minimum energy levels reveal:
(1) Critical conformations often are beautiful, typically the kinds of elegant pictures that in the past were drawn for knot tables.
(2) With polygons, it is possible to "get stuck"; that is, there may be several local minima, even for the unknot!
(3) The minimum energy numbers observed for the different knot types correlate well with the relative velocities of those knot types in gel-electrophoresis experiments for DNA knots.

When DNA of equal length are tied in different types of knots, and run in gel-electrophoresis experiments, different knots are separated.We do not yet fully understand this phenomenon, just that it happens in reliable and predictable ways. We can offer the beginning of a model to account for the correlation between gel velocity and knnot energy, but it is just a beginning.

Topology / Diff. Geom.

Fri. Apr.16 3:00pm RH 128

Dr. Jonathan Simon Univ. of Iowa

Critical Shapes of Knots II

Topology / Diff. Geom.

Tues. Apr. 20 2:10pm RH 211

Jon Short

When is a metrizable, separable, finite diml.topological group arc-connected? (After a paper by F.B.Jones)

Topology / Diff. Geom.

Tues. Apr. 27 2:10pm RH 211

Jon Short

When is a metrizable, separable, finite diml.topological group arc-connected? (After a paper by F.B.Jones) (Cont'd)

Topology / Diff. Geom.

Thursd. May 6 11:00am RH102

Anthony Bedenikovic Univ. of Illinois, Urbana-Champaign

Knot Complement Cone Complexes and the Property P Conjecture.

Abstract: A tame knot K in the 3-sphere is said to have Property P if no non-trivial surgery along K yields a homotopy 3-sphere. It was conjectured in the early 1970's that every non-trivial knot has Property P. The conjecture is known to hold for a large class of knots, a class that includes non-trivial torus knots, satellite knots, and alternating knots. If it holds in general, then surgery on a knot cannot produce a counterexample to the Poincare Conjecture.
We study the Property P Conjecture via knot complement cone complexes (kccc's). These are 3-complexes which consist of a knot complement in a cube with handles, together with the cone over the outer boundary component. Given a knot K and a prescribed surgery, we produce a kccc which is (up to 3-deformation) a spine of the surgered manifold. Thus, kccc's account for all homotopy 3-spheres produced by a surgery on a knot. We hope, with this new view, to get information about these homotopy 3-spheres. In particular we ask: Must every such homotopy 3-sphere have a spine which 3-deforms to a point?

Topology / Diff. Geom.	Tues. Feb. 2 2:10pm RH 211	Michael Tsau Saint Louis University	Bracket and Regular Isotopy of Singular links
Math Seminar	Thurs. Feb. 4 1:00pm RH 320	Dr. W. Wistar Comfort Wesleyan University	Subgroups of Compact groups
Topology / Diff. Geom.	Tues. Feb. 9 2:10pm RH 211	Michael Tsau Saint Louis University	Bracket and Regular Isotopy of Singular links . (Cont'd)
Topology / Diff. Geom.	Tues. Feb. 16 2:10pm RH 211	Michael Tsau Saint Louis University	Bracket and Regular Isotopy of Singular links . (Cont'd)
Math Colloquium	Thurs. Feb. 18 1:10pm RH237	Kevin Scannell Saint Louis University	The Sullivan Dictionary
This is the first in a series of three colloquia which will introduce some of the work of 1998 Fields Medal winners Curtis T. McMullen, W. Timothy Gowers and Richard Borcherds respectively. Each talk will be geared to a general mathematical audience and all are invited to attend. Abstract: Curt McMullen's work involves two beautiful and closely related subjects: the theory of Kleinian groups and hyperbolic 3-manifolds, and the study of iterates of holomorphic functions of \C^ ("complex dynamics"). In the 1980's, Dennis Sullivan proposed a dictionary by means of which one can try to understand complex dynamics in terms of powerful analytic and geometric techniques in Kleinian groups (eg. quasiconformal mappings) and vice versa. I will discuss this dictionary, mention some of McMullen's contributions and show some pictures.
Topology / Diff. Geom.	Tues. Feb. 23 2:10pm RH 211	John Kalliongis Saint Louis University	Properties of Lensspaces . (Cont'd)
Topology / Diff. Geom.	Tues. Mar. 2 2:10pm RH 211	John Kalliongis Saint Louis University	Properties of Lensspaces . (Cont'd)
Math Colloquium	Thurs. Mar. 4 2:10pm RH237	Darrin Speegle Saint Louis University	(Almost) Arbitrarily Strange Banach Spaces
Abstract: From the beginning of the theory in Banach spaces, people have tried to find analogues of orthonormal bases in Hilbert spaces. (Orthonormal bases themselves are, from the point of view of an analyst, natural analogues of bases in vector spaces.) I will present the two most natural concepts of bases in Banach spaces, and I will discuss the question:''Do bases exist for all Banach spaces?'' I will also explain the contribution that W.T.Gowers (jointly with B.Maurey) made in this area, which was that he was able to give a way of constructing (almost) arbitrarily strange and fabulous Banach spaces. In particular, I will explain how Gowers and Maurey answer the fourth of the four most natural questions relating to bases in Banach spaces. If time permits, I will discuss how his construction can be used to solve several other questions dating back to Banach.
Topology / Diff. Geom.	Tues. Mar. 16 2:10pm RH 211	John Kalliongis Saint Louis University	Properties of Lensspaces . (Cont'd)
Topology / Diff. Geom.	Tues. Mar. 23 2:10pm RH 211	John Kalliongis Saint Louis University	Properties of Lensspaces . (Cont'd)
Math Colloquium	Thurs. Mar. 25 1:10pm RH237	Charles Ford Saint Louis University	Monstrous Moonshine
Abstract: The title of Dr. Ford's talk is taken from the title of a 1979 paper of Conway and Norton concerning the representation theory of the famous "Monster" group (a simple group of order approximately 10⁵⁴) first constructed by R. Griess around 1980. Particularly interesting are the many similarities between genus zero subgroups of the modular group SL(2,Z) and the subgroups of the Monster. Some examples of these connections will be discussed, along with (time permitting) a brief introduction to Borcherd's techniques. The great majority of the talk will be accessible to undergraduate students - anyone wishing to know more about this exciting area of modern mathematical research is encouraged to attend.
Topology / Diff. Geom.	Tues. Mar. 30 2:10pm RH 211	John Kalliongis Saint Louis University	Properties of Lensspaces . (Cont'd)
Topology / Diff. Geom.	Tues. Apr. 6 2:10pm RH 211	Anneke Bart Saint Louis University	On: "Some surface subgroups survive surgery", a paper by Cooper and Long.
Math Colloquium	Thur. Apr. 8 4:00pm RH 128	Dr. D. Darren Long Univ. of California, Santa Barbara	A hitchiker's guide to the Geometrization Conjecture.
Topology / Diff. Geom.	Fri. Apr.9 11:00am RH 237	Dr. D. Darren Long Univ. of California, Santa Barbara	The subgroup separability of the Bianchi groups.
Topology / Diff. Geom.	Tues. Apr. 13 2:10pm RH 211	Anneke Bart Saint Louis University	On: "Some surface subgroups survive surgery", a paper by Cooper and Long. (Cont'd)
Math Colloquium	Thur. Apr. 15 4:00pm RH 128	Dr. Jonathan Simon Univ. of Iowa	Critical Shapes of Knots: From Pretty Pictures to DNA Gel-Electrophoresis.
Abstract: Given a knot K in space, we can compute "energies", numbers E(K), that somehow measure how complicated or tangled the given knot is. The motivation for people developing this idea has been a combination of basic mathematics together with a desire to model "physical knots", such as flux tubes in plasmas, DNA loops, or just plain ropes. Definitions are based on the knot trying to avoid or exclude itself, some kind of self-repelling or rope thickness. We can see in a brief video that the energies are suprisingly effective in simplifying tangled figures. For smooth knots, energies and crossing numbers are connected: Ignoring multiplicative constants, we have (crossing number) < (repelling energies) < (4/3) power of "rope length energy" The latter being an energy based on self-exclusion. The exponent (4/3) is the right one. Computer experiments flowing knots to minimum energy levels reveal: (1) Critical conformations often are beautiful, typically the kinds of elegant pictures that in the past were drawn for knot tables. (2) With polygons, it is possible to "get stuck"; that is, there may be several local minima, even for the unknot! (3) The minimum energy numbers observed for the different knot types correlate well with the relative velocities of those knot types in gel-electrophoresis experiments for DNA knots. When DNA of equal length are tied in different types of knots, and run in gel-electrophoresis experiments, different knots are separated.We do not yet fully understand this phenomenon, just that it happens in reliable and predictable ways. We can offer the beginning of a model to account for the correlation between gel velocity and knnot energy, but it is just a beginning.
Topology / Diff. Geom.	Fri. Apr.16 3:00pm RH 128	Dr. Jonathan Simon Univ. of Iowa	Critical Shapes of Knots II
Topology / Diff. Geom.	Tues. Apr. 20 2:10pm RH 211	Jon Short	When is a metrizable, separable, finite diml.topological group arc-connected? (After a paper by F.B.Jones)
Topology / Diff. Geom.	Tues. Apr. 27 2:10pm RH 211	Jon Short	When is a metrizable, separable, finite diml.topological group arc-connected? (After a paper by F.B.Jones) (Cont'd)
Topology / Diff. Geom.	Thursd. May 6 11:00am RH102	Anthony Bedenikovic Univ. of Illinois, Urbana-Champaign	Knot Complement Cone Complexes and the Property P Conjecture.
Abstract: A tame knot K in the 3-sphere is said to have Property P if no non-trivial surgery along K yields a homotopy 3-sphere. It was conjectured in the early 1970's that every non-trivial knot has Property P. The conjecture is known to hold for a large class of knots, a class that includes non-trivial torus knots, satellite knots, and alternating knots. If it holds in general, then surgery on a knot cannot produce a counterexample to the Poincare Conjecture. We study the Property P Conjecture via knot complement cone complexes (kccc's). These are 3-complexes which consist of a knot complement in a cube with handles, together with the cone over the outer boundary component. Given a knot K and a prescribed surgery, we produce a kccc which is (up to 3-deformation) a spine of the surgered manifold. Thus, kccc's account for all homotopy 3-spheres produced by a surgery on a knot. We hope, with this new view, to get information about these homotopy 3-spheres. In particular we ask: Must every such homotopy 3-sphere have a spine which 3-deforms to a point?