Claremont Topology Seminar

Fall 2009 Schedule

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	Title: Coxeter groups with Hyperbolic Signature Abstract: The signature of a Coxeter group is the signature of the smallest metric vector space on which it can act as a reflection group. I will define what it means for a Coxeter group to have hyperbolic signature, then present some resent results toward classifying such groups.
Tuesday Dec 8 3:00 pm	Pat Shanahan Loyola Marymount University	Title: Epimorphisms and boundary slopes of 2-bridge knots. Abstract: The question of when there exists an epimorphism from one knot group onto another dates back to Simon in the 1970's but has been the focus of much interest in recent years. In this talk, we will discuss a construction of Ohtsuki, Riley, and Sakuma for producing pairs of 2-bridge knots whose groups surject. Using boundary slopes as our main tool, we present evidence for the conjecture that all epimorphisms between 2-bridge knots arise in this way. This is joint work with Jim Hoste.

Spring 2010 Schedule

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	Title: Representations of the skein algebra Abstract: The skein algebra of a surface is a fundamental object in knot theory and in quantum topology. I will discuss a few conjectures and results on the representation theory of this algebra. This is joint work with Helen Wong.
Tuesday Feb 9 3:00 pm	Daniel Nash UC Irvine	4-manifolds Via Surgery on Tori Abstract: In the study of smooth 4-manifolds, the procedure of surgering along embedded 2-tori has proven continually useful, particularly in the construction/discovery of exotic smooth structures. At the same time, the smooth counterpart to the Poincar?e Conjecture in 4-dimensions has remained unaided by such torus surgery constructions, and classic potentially exotic 4-spheres (such as the examples of Cappell and Shaneson) have gone largely unresolved Ñ until now. I will present recent methods of Gompf that demonstrate an alternate application for torus surgeries in exhibiting entire families of Cappell-Shaneson homotopy 4-spheres as standard. In addition, yet further applications of torus surgery will be demonstrated in conjunction with these homotopy spheres and related model manifolds.
Tuesday Feb 16 3:00 pm	Sam Nelson, CMC	Title: Quandles, Racks and the Fundamental Group Abstract: Quandles and racks are usually defined combinatorially as abstract algebraic structures arising from knot diagrams and Reidemeister moves. In this talk, we will look at the geometric meaning of the knot quandle and the fundamental rack of a framed knot and how these are related to the fundamental group of the knot complement.
Tuesday Feb 23 3:00 pm
Tuesday Mar 2 3:00 pm	Masaaki Suzuki	Title: Epimorphisms between knot groups Abstract: Let G(K) denote the knot group of a knot K. We write K > K' if there exists an epimorphism from G(K) to G(K'). We determine this partial ordering of the set of prime knots with up to 11 crossings. In the latter half of this talk, we especially focus on epimorphisms between two bridge knot groups.
Tuesday Mar 9 3:00 pm	Dave Bachman Pitzer College	Title: Topological, PL, and geometric minimal surfaces Abstract: We discuss a program to show that a topologically minimal surface (of arbitrary index) in a compact 3-manifold can be isotoped to meet a triangulation so that it meets each tetrahedron in precisely the same way that a geometrically minimal surface (of the same index) can meet a ball. We will then discuss the immediate applications to topology, as well as potential applications to geometry.
Tuesday Mar 16 3:00 pm	No Meeting--Spring Break
Tuesday Mar 23 3:00 pm	John Levitt Pomona College	Title: Derived Categories Abstract: Derived categories are a tool from homological algebra that have been of increased interest in recent years to topologists, algebraic geometers, and physicists. In essence, they provide a setting for working with chain complexes rather than (co)homology. In this survey, we will build them from the point of view of triangulated categories, and show how many of the constructions are familiar concepts from topology.
Tuesday Mar 30 3:00 pm	Joel Louwsma Cal Tech	Title: Immersed surfaces in the modular orbifold Abstract: A hyperbolic conjugacy class in the modular group PSL(2,Z) corresponds to a closed geodesic in the modular orbifold. Some of these geodesics virtually bound immersed surfaces, and some do not; the distinction is related to the polyhedral structure in the unit ball of the stable commutator length norm. We prove the following stability theorem: for every hyperbolic element of the modular group, the product of this element with a sufficiently large power of a parabolic element is represented by a geodesic that virtually bounds an immersed surface. This is joint work with Danny Calegari.
Tuesday Apr 6 3:00 pm	Katherine Crowley Washington and Lee University and currently a Congressional Science Fellow	Title: The Geometry of Nonpositively Curved Simplicial Complexes Abstract: Understanding the conditions under which a simplicial complex collapses is a central issue in many problems in topology and combinatorics. Let K be a simplicial complex endowed with the piecewise Euclidean geometry given by declaring edges to have unit length. We will show that if K is nonpositively curved in the sense of CAT(0), then K simplicially collapses to a point. The main tool used in the proof is Forman's discrete Morse theory, a combinatorial analog of the classical smooth theory developed in the 1920s. A key ingredient in the proof is a combinatorial analog of the fact that a minimal surface in R^3 has nonpositive Gauss curvature.
Tuesday Apr 13 3:00 pm
Tuesday Apr 20 3:00 pm	John Huerta UC Riverside	Title: Supersymmetry, division algebras, and Lie n-algebras Abstract: Starting from the four normed division algebras---the real numbers, complex numbers, quaternions and octonions---a systematic procedure gives a 3-cocycle on the Poincare Lie superalgebra in dimensions 3, 4, 6 and 10. A related procedure gives a 4-cocycle on the Poincare Lie superalgebra in dimensions 4, 5, 7 and 11. In general, an (n+1)-cocycle on a Lie superalgebra yields a `Lie n-superalgebra': that is, roughly speaking, an n-term chain complex equipped with a bracket satisfying the axioms of a Lie superalgebra up to chain homotopy. We thus obtain Lie 2-superalgebras extending the Poincare superalgebra in dimensions 3, 4, 6, and 10, and Lie 3-superalgebras extending the Poincare superalgebra in dimensions 4, 5, 7 and 11. As shown in Sati, Schreiber and Stasheff's work on higher gauge theory, Lie 2-superalgebra connections describe the parallel transport of strings, while Lie 3-superalgebra connections describe the parallel transport of 2-branes. Moreover, in the octonionic case, these connections concisely summarize the fields appearing in 10- and 11-dimensional supergravity.
Tuesday Apr 27 3:00 pm	Henry Wilton Cal Tech	Title: Alternating quotients of free groups Abstract: In 1949, Marshall Hall Jr proved that finitely generated subgroups of free groups are closed in the pro-finite topology. In this talk, I will explain how to improve this result to show that these subgroups are, in fact, closed in the pro-alternating topology. The proof combines Stallings's beautiful reinterpretation of Hall's theorem in terms of the topology of graphs with a classical theorem of Jordan on permutation groups.
Tuesday May 4 3:00 pm	Jim Hoste Pitzer College	Title: Epimorphisms Between 2-bridge Knots In his talk at this Seminar earlier this year, Masaaki Suzuki asked if there exists a 2-bridge knot whose fundamental group maps onto the fundamental group of both the trefoil and the figure eight knots. We show that this is not possible if one restricts to epimorphisms of the form described by Ohtsuki, Sakuma and Riely (ORS). By using basic facts about continued fractions, it is relatively easy to decide if two given 2-bridge knots have a common ORS cover.