Claremont Topology Seminar

The next meeting of the

Southern California Topology Colloquium

will be held in Claremont, Saturday, February 13, 2010. Click here for more details.

The Claremont Topology Seminar meets on Tuesday's from 3:00-4:00 pm in Millikan 211, Pomona College. Millikan is located on the NE corner of College Avenue and 6th Street. Following the talk we go for coffee/tea in downtown Claremont. Sometimes we meet a little later and then go to dinner. (Look for special times on the calendar.) Click here for a map of the Pomona Campus.

To reach Pomona College from the 10 freeway, exit at Indian Hill, go north, turn right (east) on 6th street and then turn left (north) on College. To reach Pomona College from the 210 freeway, if traveling East, exit at Towne Avenue, turn right (south) on Towne, turn left (east) on Foothill Blvd, and turn right (south) onto College Avenue. If traveling West on the 210 freeway, exit at Baseline/Padua, turn right (west) onto Baseline, turn left (south) onto Padua at the first light, turn right (west) onto Foothill Blvd at the third light, turn left (south) onto College Avenue.

Parking on College Avenue is free.

For more information about the Seminar, or to suggest speakers, contact Jim Hoste , Dave Bachman , Sam Nelson , Erica Flapan or Vin de Silva.

Fall 2009 Schedule

special days, times or locations are in red

Date Speaker Title and Abstract

Tuesday
Sept 8

3:00 pm
Organizational Meeting

Tuesday
Sept 15

3:00 pm
Rollie Trapp
Cal State University San Bernardino
Title: Ropelength of Alternating and Paired Conformations
Abstract: The ropelength of a curve in space is the ratio of its arclength to the radius of the thickest tube around it. The ropelength of a knot is the minimum ropelength of all its conformations. Various bounds have been shown to relate ropelength to crossing number. In particular it is conjectured that the ropelength of alternating knots is at least linear in the crossing number. In this talk we review known bounds on ropelength, and prove that the ropelength of a conformation admitting an alternating projection has length at least four times the crossing number. Similar bounds are given for "paired" (to be defined) and almost alternating conformations. This is joint work with Shahla Sadjadi, Jessica Alley, and Charley Mathes.

Tuesday
Sept 22

3:30-4:30
Dave Bachman
Pitzer College
Title: Generic Dehn filling preserves the set of essential surfaces.
Abstract: We show that after generic filling along a torus boundary component of a 3-manifold, no two closed, essential, 2-sided surfaces become isotopic, and no essential surfaces becomes compressible. That is, the set of essential surfaces (considered up to isotopy) survives unchanged in all suitably generic Dehn fillings. Furthermore, for all but finitely many non-generic fillings, we show that two essential surfaces can only become isotopic in a very constrained way. This is joint work with Ryan Derby-Talbot and Eric Sedgwick.

Tuesday
Sept 29

3:00 pm
Mikael Vejdemo-Johansson
Stanford University
Title: Persistent cohomology and circular coordinates
Abstract: Dimension reduction methods deal with finding new coordinates to describe a dataset - conceptually described as a set of vectors in some Euclidean space, or a point cloud - that condenses the intrinsic information. The idea is that if we can reduce the important information in a dataset, as faithfully as possible, to maybe even 2-, 3- or 4-dimensional descriptions, we can start processing the data in familiar terms and with clear visualizations. In joint work with Vin de Silva (Pomona) and Dmitriy Morozov (Stanford), we propose to extend the expected coordinate functions from the classical linear coordinates to include circle-valued coordinatizations as well. This allows us, for instance, to parametrize circles and tori with the intrinsic dimension instead of being forced to embed them in higher dimensional spaces. For the computation of such circular coordinates, we propose using persistent cohomology, acquired by adapting the persistence algorithm to work on cocycles, combined with a smoothing step. In this talk, I will sketch out the work we have done so far, including the methods we have in place to compute such circular coordinates, and show some examples of the method at work. Given time, I will finish with some of the application areas we have been able to identify so far.

Tuesday
Oct 6

3:00 pm
Qingtao Chen
Univ. of Southern California
Title: Quantum invariants of links
Abstract: The colored HOMFLY polynomial is a quantum invariant of oriented links in S associated with a collection of irreducible representations of each quantum group U_q(sl_N) for each component of the link. We will discuss in detail how to construct these polynomials and their general structure. Then we will discuss the new progress, Labastida-Marino-Ooguri-Vafa conjecture. LMOV also gives the application of Lichorish-Millet type formula for links. The corresponding theory of colored Kauffman polynomial and orthogonal LMOV conjecture could also be developed in a same fashion by using more complicated algebra structures.

Tuesday
Oct 13

3:00 pm
Yi Ni
California Institute of Technology
Title: Dehn surgeries that reduce the Thurston norm of a fibered manifold
Abstract: Suppose K is a knot on the fiber of a surface bundle over the circle. If we do surgery on K with slope specied by the fiber, then the Thurston norm of the homology class of the fiber will decrease in the new manifold. We will show that the converse is also true. Namely, if a Dehn surgery on a winding number 0 knot in a fibered manifold reduces the Thurston norm of the homology class of the fiber, then the knot must lie on the fiber and the slope is the natural one.

Tuesday
Oct 20

3:00 pm
No Meeting-Fall Break

Tuesday
Oct 27

3:00 pm
Brie Feingold
UCSB
Title: A simply connected complex on which SL(2,Z[i]) acts.
Abstract: When a group acts on a simply connected CW complex, there is a way to recover a presentation (generators and relations) for the group in terms of stabilizers of cells. Given any commutative ring R with one such that SL(2,R) is generated by transvections, I will describe the construction of the 2nd Torus Complex over that ring, which is a connected simplicial complex. For certain rings (Z[sqrt -n] n=1,3,5,11). I can prove that the complex is also simply connected.

Tuesday
Nov 3

3:00 pm
Paul Melvin
Bryn Mawr
Title: Degree formulas for higher order linking
Abstract: The linking number of a pair of closed curves in 3-space can be expressed as the degree of a map from the 2-torus to the 2-sphere, by means of the linking integral Gauss wrote down in 1833. In the early 1950's, John Milnor introduced a family of higher order linking numbers (the "mu-bar invariants"). In this talk I will describe a formula for Milnor's triple linking number as the "degree" of a map from the 3-torus to the 2-sphere. This is joint work with DeTurck, Gluck, Komendarczyk, Shonkwiler and Vela-Vick.

Tuesday
Nov 10

3:00 pm
Radmila Sazdanovic
George Washington University
Title: Torsion in chromatic graph cohomology and Khovanov link homology
Abstract: Chromatic graph cohomology was introduced by Helme-Guizon, Rong as a comultiplication free version of the Khovanov cohomology of alternating links, where alternating link diagrams are translated to plane (Tait) graphs. Being free of topological restrictions, chromatic graph cohomology can be extended to any commutative algebra. Przytycki shows that the chromatic graph cohomology can be viewed as a generalization of the Hochschild homology of a polygon. Motivated by this interpretation, we compute torsion in the chromatic graph cohomology and the corresponding torsion in Khovanov homology, partially answering the conjecture by A. Shumakovitch about existence of 2-torsion. Moreover, we analyze chromatic graph cohomology over a few algebras of truncated polynomials and relations to various properties of graphs.

Tuesday
Nov 17

3:00 pm
Chris Hruska
Univ. Wisconsin-Milwaukee
Title: Relatively hyperbolic groups
Abstract: Word hyperbolic groups have played a central role in geometric group theory since they were introduced by Gromov in the 1980s. These groups have a large-scale geometry similar to the fundamental group of a compact negatively curved manifold. Gromov also introduced the notion of a relatively hyperbolic group, which has a geometry similar to the fundamental group of a finite volume negatively curved manifold with cusps. Relatively hyperbolic groups are, roughly speaking, negatively curved outside of certain ``peripheral subgroups'' that play the role of the cusps in the finite volume manifold case. Relatively hyperbolic groups arise in many contexts throughout group theory and low dimensional topology. I will give an introduction to the theory of relatively hyperbolic groups with an emphasis on examples.

Tuesday
Nov 24

3:00 pm
Lenny Fukshansky
Claremont McKenna College
Title: On minimal lattice spherical configurations in three dimensions
Abstract: The kissing number problem asks for the maximal number of non-overlapping unit balls in R^n that touch another unit ball. The answer is only known in dimensions 1,2,3,4,8,24. In fact, in dimension 3 this was the subject of a famous argument between Isaac Newton and David Gregory, which was only settled in 1953. The kissing number problem can be reformulated as follows: find the maximal configuration of points on the unit sphere in R^n such that the angular separation between any pair of these points is at least pi/3. Such configurations are usually expected to come from sets of minimal vectors of lattices, at least this is the case in all known dimensions. This consideration raises the following natural related question: given a spherical lattice triangle, what is the minimal possible spherical area (i.e. measure of the corresponding solid angle) it can have? In this talk, I will give at least a partial answer to this question in dimension three. This is joint work with Sinai Robins.

Tuesday
Dec 1

3:00 pm
Ryan Ottman
UCSB

Tuesday
Dec 8

3:00 pm
Pat Shanahan
Loyola Marymount University

Spring 2010 Schedule

special days, times or locations are in red

Date Speaker Title and Abstract

Tuesday
Jan 26

3:00 pm
Rena Levitt
Pomona College
Title: Algorithmic Properties of Groups Acting on Nonpositively Curved Simplicial 3-Complexes
Abstract: In a paper in 1912, Max Dehn posed three seminal problems in combinatorial group theory, now known as the word problem, the conjugacy problem, and the isomorphism problem. While stated in terms of finitely presented groups, each problem arose naturally in Dehn's study of fundamental groups of $2$-dimensional surfaces. In this talk, I will discuss these three problems, and how they may be solved by looking at the the connection between finitely presented groups and the metric spaces they act on, focusing on the example of groups acting on nonpositively curved piecewise euclidean complexes. Specifically, I will discuss recent work in which I use metric conditions to show that groups acting on non positively curved simplicial $3$-complexes are biautomatic, a condition that gives a positive solution to both the word problem and the conjugacy problem for these groups.

Tuesday
Feb 2

3:00 pm
Francis Bonahon
University of Southern California

Tuesday
Feb 9

3:00 pm

Tuesday
Feb 16

3:00 pm

Tuesday
Feb 23

3:00 pm

Tuesday
Mar 2

3:00 pm

Tuesday
Mar 9

3:00 pm

Tuesday
Mar 16

3:00 pm
No Meeting--Spring Break

Tuesday
Mar 23

3:00 pm

Tuesday
Mar 30

3:00 pm

Tuesday
Apr 6

3:00 pm

Tuesday
Apr 13

3:00 pm

Tuesday
Apr 20

3:00 pm

Tuesday
Apr 27

3:00 pm

Tuesday
May 4

3:00 pm

Archived Schedules

Date	Speaker	Title and Abstract
Tuesday Sept 8 3:00 pm	Organizational Meeting
Tuesday Sept 15 3:00 pm	Rollie Trapp Cal State University San Bernardino	Title: Ropelength of Alternating and Paired Conformations Abstract: The ropelength of a curve in space is the ratio of its arclength to the radius of the thickest tube around it. The ropelength of a knot is the minimum ropelength of all its conformations. Various bounds have been shown to relate ropelength to crossing number. In particular it is conjectured that the ropelength of alternating knots is at least linear in the crossing number. In this talk we review known bounds on ropelength, and prove that the ropelength of a conformation admitting an alternating projection has length at least four times the crossing number. Similar bounds are given for "paired" (to be defined) and almost alternating conformations. This is joint work with Shahla Sadjadi, Jessica Alley, and Charley Mathes.
Tuesday Sept 22 3:30-4:30	Dave Bachman Pitzer College	Title: Generic Dehn filling preserves the set of essential surfaces. Abstract: We show that after generic filling along a torus boundary component of a 3-manifold, no two closed, essential, 2-sided surfaces become isotopic, and no essential surfaces becomes compressible. That is, the set of essential surfaces (considered up to isotopy) survives unchanged in all suitably generic Dehn fillings. Furthermore, for all but finitely many non-generic fillings, we show that two essential surfaces can only become isotopic in a very constrained way. This is joint work with Ryan Derby-Talbot and Eric Sedgwick.
Tuesday Sept 29 3:00 pm	Mikael Vejdemo-Johansson Stanford University	Title: Persistent cohomology and circular coordinates Abstract: Dimension reduction methods deal with finding new coordinates to describe a dataset - conceptually described as a set of vectors in some Euclidean space, or a point cloud - that condenses the intrinsic information. The idea is that if we can reduce the important information in a dataset, as faithfully as possible, to maybe even 2-, 3- or 4-dimensional descriptions, we can start processing the data in familiar terms and with clear visualizations. In joint work with Vin de Silva (Pomona) and Dmitriy Morozov (Stanford), we propose to extend the expected coordinate functions from the classical linear coordinates to include circle-valued coordinatizations as well. This allows us, for instance, to parametrize circles and tori with the intrinsic dimension instead of being forced to embed them in higher dimensional spaces. For the computation of such circular coordinates, we propose using persistent cohomology, acquired by adapting the persistence algorithm to work on cocycles, combined with a smoothing step. In this talk, I will sketch out the work we have done so far, including the methods we have in place to compute such circular coordinates, and show some examples of the method at work. Given time, I will finish with some of the application areas we have been able to identify so far.
Tuesday Oct 6 3:00 pm	Qingtao Chen Univ. of Southern California	Title: Quantum invariants of links Abstract: The colored HOMFLY polynomial is a quantum invariant of oriented links in S associated with a collection of irreducible representations of each quantum group U_q(sl_N) for each component of the link. We will discuss in detail how to construct these polynomials and their general structure. Then we will discuss the new progress, Labastida-Marino-Ooguri-Vafa conjecture. LMOV also gives the application of Lichorish-Millet type formula for links. The corresponding theory of colored Kauffman polynomial and orthogonal LMOV conjecture could also be developed in a same fashion by using more complicated algebra structures.
Tuesday Oct 13 3:00 pm	Yi Ni California Institute of Technology	Title: Dehn surgeries that reduce the Thurston norm of a fibered manifold Abstract: Suppose K is a knot on the fiber of a surface bundle over the circle. If we do surgery on K with slope specied by the fiber, then the Thurston norm of the homology class of the fiber will decrease in the new manifold. We will show that the converse is also true. Namely, if a Dehn surgery on a winding number 0 knot in a fibered manifold reduces the Thurston norm of the homology class of the fiber, then the knot must lie on the fiber and the slope is the natural one.
Tuesday Oct 20 3:00 pm	No Meeting-Fall Break
Tuesday Oct 27 3:00 pm	Brie Feingold UCSB	Title: A simply connected complex on which SL(2,Z[i]) acts. Abstract: When a group acts on a simply connected CW complex, there is a way to recover a presentation (generators and relations) for the group in terms of stabilizers of cells. Given any commutative ring R with one such that SL(2,R) is generated by transvections, I will describe the construction of the 2nd Torus Complex over that ring, which is a connected simplicial complex. For certain rings (Z[sqrt -n] n=1,3,5,11). I can prove that the complex is also simply connected.
Tuesday Nov 3 3:00 pm	Paul Melvin Bryn Mawr	Title: Degree formulas for higher order linking Abstract: The linking number of a pair of closed curves in 3-space can be expressed as the degree of a map from the 2-torus to the 2-sphere, by means of the linking integral Gauss wrote down in 1833. In the early 1950's, John Milnor introduced a family of higher order linking numbers (the "mu-bar invariants"). In this talk I will describe a formula for Milnor's triple linking number as the "degree" of a map from the 3-torus to the 2-sphere. This is joint work with DeTurck, Gluck, Komendarczyk, Shonkwiler and Vela-Vick.
Tuesday Nov 10 3:00 pm	Radmila Sazdanovic George Washington University	Title: Torsion in chromatic graph cohomology and Khovanov link homology Abstract: Chromatic graph cohomology was introduced by Helme-Guizon, Rong as a comultiplication free version of the Khovanov cohomology of alternating links, where alternating link diagrams are translated to plane (Tait) graphs. Being free of topological restrictions, chromatic graph cohomology can be extended to any commutative algebra. Przytycki shows that the chromatic graph cohomology can be viewed as a generalization of the Hochschild homology of a polygon. Motivated by this interpretation, we compute torsion in the chromatic graph cohomology and the corresponding torsion in Khovanov homology, partially answering the conjecture by A. Shumakovitch about existence of 2-torsion. Moreover, we analyze chromatic graph cohomology over a few algebras of truncated polynomials and relations to various properties of graphs.
Tuesday Nov 17 3:00 pm	Chris Hruska Univ. Wisconsin-Milwaukee	Title: Relatively hyperbolic groups Abstract: Word hyperbolic groups have played a central role in geometric group theory since they were introduced by Gromov in the 1980s. These groups have a large-scale geometry similar to the fundamental group of a compact negatively curved manifold. Gromov also introduced the notion of a relatively hyperbolic group, which has a geometry similar to the fundamental group of a finite volume negatively curved manifold with cusps. Relatively hyperbolic groups are, roughly speaking, negatively curved outside of certain ``peripheral subgroups'' that play the role of the cusps in the finite volume manifold case. Relatively hyperbolic groups arise in many contexts throughout group theory and low dimensional topology. I will give an introduction to the theory of relatively hyperbolic groups with an emphasis on examples.
Tuesday Nov 24 3:00 pm	Lenny Fukshansky Claremont McKenna College	Title: On minimal lattice spherical configurations in three dimensions Abstract: The kissing number problem asks for the maximal number of non-overlapping unit balls in R^n that touch another unit ball. The answer is only known in dimensions 1,2,3,4,8,24. In fact, in dimension 3 this was the subject of a famous argument between Isaac Newton and David Gregory, which was only settled in 1953. The kissing number problem can be reformulated as follows: find the maximal configuration of points on the unit sphere in R^n such that the angular separation between any pair of these points is at least pi/3. Such configurations are usually expected to come from sets of minimal vectors of lattices, at least this is the case in all known dimensions. This consideration raises the following natural related question: given a spherical lattice triangle, what is the minimal possible spherical area (i.e. measure of the corresponding solid angle) it can have? In this talk, I will give at least a partial answer to this question in dimension three. This is joint work with Sinai Robins.
Tuesday Dec 1 3:00 pm	Ryan Ottman UCSB
Tuesday Dec 8 3:00 pm	Pat Shanahan Loyola Marymount University