Claremont Topology Seminar

2013-14 Schedule

Fall, 2013

Date	Speaker	Title and Abstract
Tuesday Sept 10 3:00 pm	Organizational Meeting	Meet at Some Crust Bakery for organizational meeting.
Tuesday Sept 17 3:00 pm
Tuesday Sept 24 3:00 pm
Tuesday Oct 1 3:00 pm
Tuesday Oct 8 3:00 pm
Tuesday Oct 15 4:15 in Mudd 125	Renata Gerecke	Title: Factor Groups of Knot and LOT Groups Abstract: It is difficult to determine whether, given a finite, balanced group presentation, the group is finite or infinite. We study this problem in the context of knot groups and label orientated tree (LOT) groups. More specifically, we are looking at factor groups of knot and LOT groups by powers of meridians. This is in the spirit of Coxeter's work on the factor groups of braid groups. Indeed, our findings generalize Coxeter's work from the three-strand braid groups to knot groups.
Tuesday Oct 22 3:00 pm	No Meeting	Fall Break
Tuesday Oct 29 3:00 pm	Sam Nelson Claremont McKenna College	Title: Augmented Birack Homology Abstract: We introduce a homology theory for augmented biracks. 2-cocyles in the cohomology of a coloring birack define enhancements of the birack counting invariant; we will see some examples.
Tuesday Nov 5 3:00 pm
Tuesday Nov 12 3:00 pm	Henry Segerman University of Melbourne	Title: Regular triangulations and the index of a cusped hyperbolic 3-manifold Abstract: Recent work by Dimofte, Gaiotto and Gukov defines the "index" (a collection of Laurent series) associated to an ideal triangulation of an oriented cusped hyperbolic 3-manifold. "Physics tells us" that this index should be a topological invariant of the manifold, not just of the triangulation of it. The problem is that the index is not well defined on all triangulations. We define a class of triangulations of a 3-manifold, depending only on the topology of the manifold, such that the index is well-defined and has the same value for each triangulation in the class. A key requirement is that the class of triangulations be connected by local moves on the triangulations, since we can prove invariance of the index under these moves. To achieve this requirement we import a result from the theory of regular triangulations of Euclidean point configurations due to Gelfand, Kapranov and Zelevinsky.
Tuesday Nov 19 3:00 pm
Tuesday Nov 26 4:15 at CMC, room TBA	Karly Brint Pitzer College	Title: Stick Number of Torrified Rational Links Abstract: Using the idea of supercoiling of DNA, it is possible to represent knots and links containing large numbers of twists with relatively small numbers of sticks. We use this approach to place an upper bound on stick number for a large class of links we call torrified rational links.
Tuesday Dec 3 3:00 pm	Dave Bachman Pitzer College	Title: Limits on concentrated instability Abstract: Minimal surfaces typically have places where their instability is concentrated. We show that if all of this instability is concentrated in one place, then there is a limit to how unstable the surface is.
Tuesday Dec 10 3:00 pm

Spring 2014 Schedule

Date	Speaker	Title and Abstract
Tuesday Jan 28 3:00 pm	Eleni Panagiotou UCSB	Title: Entanglement in systems of curves with Periodic Boundary Conditions Abstract: Periodic Boundary Conditions (PBC) are often used for the simulation of complex physical systems of open and closed curve models of polymers or vortex lines in a fluid flow. Using the Gauss linking number, we define the periodic linking number as a measure of entanglement for two oriented curves in a system employing PBC. In the case of closed curves in PBC, the periodic linking number is a topological invariant that depends on a finite number of components in the periodic system. For open curves, the periodic linking number depends upon the entire infinite system and we prove that it converges to a real number that varies continuously with the configuration. Finally, we define two cut-offs of the periodic linking number and we compare these measures when applied to a PBC model of polyethylene melts.
Tuesday Feb 4 3:00 pm
Tuesday Feb 11 3:00 pm
Tuesday Feb 18 3:00 pm	Yin Tian USC	Title: a categorification of a Clifford algebra via 3-dimensional contact topology Abstract: Categorification, initiated in the work of Crane and Frenkel, describes richer ``higher-level" structures which aim to clarify objects on the decategorification level and to provide finer invariants. A celebrated example is Khovanov homology which categorifies the Jones polynomial. In this talk we describe a categorifcation of a Clifford algebra. The motivation is from the contact category C(D^2) of a disk D^2 introduced by Honda which describe contact structures on D^2 \times [0,1].
Tuesday Feb 25 3:00 pm
Tuesday Mar 4 3:00 pm	Kristen Hendricks UCLA	Title: Periodic Knots and Heegaard Floer Homology Abstract: We introduce periodic knots and discuss two classical obstructions to periodicity, Murasugi's condition on the Alexander polynomial of a periodic knot and Edmonds' condition on the genus. We then describe a generalization of these two results in the case of doubly-periodic knots which arises from the modern link invariant Heegaard Floer homology. We finish with an example in which our construction gives more information that the two classical theorems.
Tuesday Mar 11 3:00 pm	Hugh Howards Wake Forrest University	Title: Linked spheres in Higher dimensions and how it all shapes up. Abstract: The Borromean Rings are one of the most famous links. A result of Freedman and Skora shows that they cannot be formed out of circles, but they can be formed from two circles and an ellipse. They, however, are the only Brunnian link of 3, 4, or 5 components that can be formed out of convex curves. We look at generalizations of Brunnian Links to higher dimensions and ask if it is possible to form these genralized Brunnian Links out of round spheres or convex components answering the first question in the negative and the second in the positive.
Tuesday Mar 18	No Meeting	Spring Break
Tuesday Mar 25 3:00 pm	Allison Gilmore UCLA	Title: Knot Floer homology and constituent knots of graphs Abstract: Knot Floer homology for singular knots was developed by Ozsvath, Szabo, and Stipsicz using the usual techniques of Heegaard diagrams and counts of holomorphic disks. Much of the talk will be devoted to introducing this theory and discussing its basic properties. The remainder of the talk will discuss joint work in progress with Kristen Hendricks. We will describe spectral sequences from the knot Floer homology of a singular knot to the knot Floer homology of any of its oriented constituent knots. If time permits, some preliminary computations will be presented and potential implications sketched.
Tuesday Apr 1 3:00 pm	Ryan Blair CSU Long Beach	Title: Knots with compressible thin levels. Abstract: Width is an integer invariant of knots that is affected by the number of maxima and minima of a knot as well as the relative heights of these critical points. Width has been a particularly useful invariant due to deep connections between a width minimizing embedding of a knot and the topology of the knot exterior. In particular, Wu showed that a thinnest thin level for a width minimizing embedding is incompressible. In this talk, I will present joint work with Alex Zupan in which we construct the first examples of a width minimizing embedding with compressible thin levels.
Tuesday Apr 8 3:00 pm	Jeremy Pecharich Pomona College	Title: A class of associative algebras arising from quantum mechanics Abstract: Classical mechanics, i.e., all of physics up to 1900, can be described by a Poisson manifold. After 1900, quantum mechanics happened and pandemonium ensues. The pandemonium was caused by a series of experiments which showed that the mathematics of the quantum level is no longer a Poisson manifold. The mathematics of the quantum world is still up for debate, but we will discuss one possibility which constructs an associative algebra from any Poisson manifold. We will then constructs modules over this associative algebra and give applications to intersections of Lagrangian submanifolds. Part of this is joint work with Baranovsky, Ginzburg, and Kaledin.
Tuesday Apr 15 3:00 pm
Tuesday Apr 22 3:00 pm	Ellie Grano Pepperdine University	Title: The pop-switch planar algebra and the Jones-Wenzl projections Abstract: The Temperley-Lieb planar algebra brought us the Jones' polynomial. We will discuss this planar algebra and then define the Jones-Wenzl projections, important elements of the Temperley-Lieb planar algebra. Next we will present the pop-switch planar algebra, a new planar algebra containing the Temperley-Lieb planar algebra. It is motivated by Jones' idea of the "graph planar algebra" of type $A_n$. Complicated calculations using the graph planar algebra can be done pictorially in this new planar algebra. Furthermore, the Jones-Wenzl projections are traditionally very complicated to write down. Viewing the pop-switch planar algebra as a matrix category, the Jones-Wenzl projections are direct sums of very simple diagrams. We will discuss this method of viewing the Jones-Wenzl projections.
Tuesday Apr 29 4:15-5:15 Shanahan B460, HMC	Tim Hsu San Jose State Univ.	Title: Cube Complexes, 3-manifolds, and the Virtual Haken Theorem Abstract: The Virtual Haken Conjecture was, until recently, probably the biggest open problem in 3-manifolds (3-dimensional geometry). Then in March 2012, Ian Agol proved the conjecture by completing a key part of Dani Wise's program of studying nonpositively curved cube complexes. So how did questions in 3-dimensional geometry end up being resolved using spaces made from (very high-dimensional) cubes? We'll give an overview explaining the connection and describe the speaker's joint work with Wise that is part of the emerging and rapidly growing subject of cube complexes.
Tuesday May 6 3:00 pm	Erkao Bao UCLA	Title: Hofer energy and Gromov's Monotonicity of pseudoholomorphic curves Abstract: I will introduce the notion of pseudoholomorphic curve, and the Hofer energy of pseudoholomorphic curves in symplectization of contact manifolds. Then as a direct corollary, we prove the Gromov's Monotonicity with multiplicity. I will make it accessible to general audience.