| Date | Speaker | Title and Abstract |
| Tuesday
Sept 6, 2016 3:00 pm |
Organizational Meeting | Meet at Some Crust Bakery for organizational meeting. |
| Tuesday
Sept 13, 2016 3:00 pm |
Christopher Tuffley Massey University Manawatu |
Title: Intrinsic linking in higher dimensions, and with linking numbers divisible by q.
Abstract: In 1983 Conway and Gordon proved that every embedding of the complete graph K_6 in 3-space contains a pair of disjoint cycles that form a non-separable link --- a fact that is expressed by saying K_6 is intrinsically linked. Since then, a number of authors have shown that embeddings of larger complete graphs necessarily exhibit more complicated linking behaviour, such as links with many components and/or large pairwise linking numbers. With some adaptions to the proofs, similar results can be established for embeddings of large n-complexes in (2n+1)-space. We will look at some of the adaptions required, in the context of proving the existence of two component links with linking number a nonzero multiple of a given integer q. In the course of this we will obtain an improved bound for n greater than or equal to 1 on the number of vertices needed to force a two component link with linking number at least q in absolute value. |
| Tuesday
Sept 20, 2016 3:00 pm |
Catherine Pfaff UC Santa Barbara |
Title: When Outer Space behaves like a hyperbolic space & how we can use this to understand the group Out(F_n). Abstract: A common strategy for studying a group is to study some object that it acts on and how it acts on this object. My favorite group is the outer automorphism group of the free group (or Out(F_n)). I will introduce this group and the object, Culler-Vogtmann Outer Space, that it acts on. I will also relate the study of this group acting on Culler-Vogtmann Outer Space to the study of the group SL(2,Z) acting on the hyperbolic plane. |
| Tuesday
Sept 27, 2016 3:00 pm |
Dave Bachman Pitzer College |
Title: Heegaard Genus is NP-Hard Postponed until next semester! |
| Tuesday
Oct 4, 2016 3:00 pm |
Jieon Kim Osaka City University |
Title: On biquandle cocycle invariants from marked graphs Abstracts: A quandle is a set equipped with a binary operation satisfying certain axioms derived from the Reidemeister moves in knot theory. Quandle homology and cohomology theories have been studied extensively in recent years. L.H. Kauffman and D.E. Radford introduced a generalization of quandles, called biquandles and J.S. Carter, M. Elhamdadi and M. Saito defined a (co)homology theory and cocycle invariants for biquandles. J.S. Carter, S. Kamada and M. Saito defined shadow quandle colored diagrams and shadow quandle cocycle invariants of oriented links and surface-links. Surface-links are represented by broken surface diagrams and marked graph diagrams. In this talk, we'd like to introduce shadow biquandle colorings of oriented broken surface diagrams and those of oriented marked graph diagrams, and describe shadow biquandle cocycle invariants of oriented surface-links via broken surface diagrams and marked graph diagrams. This is a joint work with S. Kamada, A. Kawauchi, and S.Y. Lee. |
| Tuesday
Oct 11, 2016 3:00 pm |
Brittany Fasy Montana State University |
Title: Persistent Local Homology in Road Network Analysis Abstract: Topological data analysis (TDA) has rapidly grown in popularity in recent years. One of the emerging tools is persistent local homology, which can be used to extract local structure from a dataset. In this talk, we provide a definition of this new tool, along with a few applications. In particular, we investigate its use in road network analysis. |
| Tuesday
Oct 18 |
No Meeting | Fall Break |
| Tuesday
Oct 25, 2016 3:00 pm |
Amanda Curtis UC Santa Barbara |
Title: Projectors for SL(3) Abstract: The Temperley-Lieb algebra is known for its usefulness in topological quantum computation and in certain knot invariants. In today's talk, I discuss my work with a similar algebra, the algebra of SL(3) spiders, and a particularly useful set of diagrams within it. As these diagrams have analogous diagrams in the Temperley-Lieb algebra, I offer a summary of the Temperley-Lieb algebra and springboard from there into my discussion of the projectors for the SL(3) spider. |
| Tuesday
Nov 1, 2016 3:00 pm |
Jim Hoste Pitzer College |
Title: Diagramatic Moves on 3-diagrams Abstract: The classical theory of knots and links is often approached via link diagrams and Reidemeister moves. The important result is that two diagrams represent the same link if and only if they are related by a sequence of Reidemeister moves. Recently, several papers have explored the topic of link diagrams with multi-crossings. In these diagrams, n strands are allowed to cross at a single point in the plane, creating what is known as an n-crossing. Many of the obvious results analogous to classical diagrams have been proven. For example, given any n>1, every link has an n-diagram, that is, one with only n-crossings. However, until now, no analog of the Reidemeister moves have yet to be found for multicrossing diagrams. In this talk I will describe a set of 3-diagram moves and prove that they are sufficient to pass between any two 3-diagrams of the same knot. This is joint work with Colin Adams and Martin Palmer. |
| Tuesday
Nov 8, 2016 3:00 pm |
Sam Nelson Claremont McKenna College |
Title: Biquasiles and Dual Graph Diagrams Abstract: Dual graph diagrams are an alternative way to present oriented knots and links with roots in statistical mechanics. Biquasiles are algebraic structures which can be used to color dual graph diagrams analogously to quandle colorings of standard knot diagrams. In this talk we will see examples of biquasiles including a finite biquasile whose counting invariant detects the mirror image of 9_32. |
| Tuesday
Nov 15, 2016 3:00 pm |
Danielle O'Donnol University of Indiana. |
Title: Legendrian theta-graphs Abstract: We will work in three-space with the standard contact structure. An embedded graph is Legendrian if it is everywhere tangent to the contact structure. I will give an overview of the invariants used in this area. Then I will talk about our recent work on classification of planar Legendrian theta-graphs. This is joint with Peter Lambert-Cole (Indiana). |
| Tuesday
Nov 22, 2016 3:00 pm |
Jose Ceniceros Louisiana State Univ. |
Title: Legendrian/Transverse Knots and Knot Floer Homology Abstract: We will give an overview of knot theory supported in a contact 3-manifold with a focus on invariants of Legendrian and transverse knots that take values in knot Floer homology. We will also extend the definition of the BRAID invariant defined by Baldwin, Vela-Vick, Vertesi and define a new invariant that also takes values in knot Floer homology.
|
| Tuesday
Nov 29, 2016 3:00 pm |
Kanako Oshiro Sophia University |
Title: Up-down colorings of virtual-link diagrams and RII-detectors Abstract: In this talk, we introduce an up-down coloring of a virtual-link diagram. For two 2-component virtual-link diagrams D and D', their up-down colorabilities give a lower bound of the minimum number of RII-moves which are needed to transform D to D'. By using the notion of a quandle cocycle invariant, we determine the necessity of RII-moves for a pair of diagrams of the trivial virtual-knot. This implies that for any virtual-knot diagram D, there exists a diagram D' representing the same virtual-knot such that any sequence of generalized Reidemeister moves between D and D' includes at least one RII-move. This is a joint work with Ayaka Shimizu and Yoshiro Yaguchi. |
| Tuesday
Dec 6, 2016 3:00 pm |
Burak Ozbagci Koc University |
Title: Symplectic fillings of contact 3-manifolds Abstract: In the first half of my talk, I will introduce various types of symplectic fillings of three dimensional contact manifolds, and give a brief overview of the results in the literature. In the second half of the talk, I will concentrate on fillings of the unit cotangent bundles of closed surfaces equipped with their canonical contact structures. In particular, I will explain my recent joint work with Youlin Li where the surfaces at hand are non-orientable. |
| Date | Speaker | Title and Abstract |
| Tuesday
Jan 24, 2017 3:00 pm |
Bahar Acu USC |
Title: Foliations of Symplectic Manifolds and the Weinstein Conjecture Abstract: In this talk, we will examine foliations of high dimensional symplectic manifolds by planar curves and show that iterated planar contact manifolds, which we will define in the talk, satisfy "the Weinstein conjecture". The necessary background will be provided in the first half of the talk. |
| Tuesday
Jan 31, 2017 3:00 pm |
Ian Zemke UCLA |
Title: TQFT structures in Heegaard Floer homology. Abstract: To a based 3-manifold, Heegaard Floer homology assigns a chain complex, and to a based link in a 3-manifold, link Floer homology assigns a filtered chain complex. To a 4-dimensional cobordism between two 3-manifolds, Ozsvath and Szabo define a map between the Heegaard Floer homologies of the two ends, though the maps are only defined if the 4-manifold is connected and the two ends are connected. This prevents their maps from satisfying several axioms of Atiyah's definition of a TQFT. In this talk, we describe a TQFT structure on Heegaard Floer homology, and a TQFT structure on link Floer homology, which conjecturally satisfy appropriate analogs of Atiyah's definitions. We discuss applications to computing mapping class group actions on Heegaard Floer homology. |
| Tuesday
Feb 7, 2017 3:00 pm |
Giuseppe Martone USC |
Title: Surfaces: A Mathematical Playground Abstract: The study of surfaces is a rich research area where geometry, topology, algebra and analysis interact deeply, widely and beautifully. Over the centuries, many great mathematicians have made great contributions to the subject: from Mobius and Klein to the Fields medalists Thurston, McMullen and Mirzakhani. In this talk, which will require no background, we will start by defining surfaces (using carthography!) and then venture in this vast field by underlining the role played by hyperbolic geometry. |
| Tuesday
Feb 14, 2017 3:00 pm |
Eleni Panagiotou UCSB |
Title: Quantifying Entanglement in Physical Systems Abstract: Periodic Boundary Conditions (PBC) are often used for the simulation of complex physical systems of open and closed curve models of biopolymers, polymer melts, fluid flows, textiles etc. In this talk we will see methods by which one can measure entanglement in collections of open or closed curves in 3-space and in systems employing Periodic Boundary Conditions. More precisely, 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 and study its properties. We will see how we can apply our entanglement measures to discuss the evolution of the dimensional character of the entanglement as a function of density in Olympic systems which model the behavior of DNA networks. We also apply our measures to investigate how the entanglement of polymeric chains relates to bulk viscoelastic responses in polymeric materials. We study in particular woven polymer configurations having similar polymer densities but very different topologies varying from untangled to strongly entangled conformations. Our approaches provide new mathematical tools for characterizing the origins of the rheological responses of polymeric materials. |
| Tuesday
Feb 21, 2017 3:00 pm |
Matthew Hogancamp USC. |
Title: Torus knots and combinatorics Abstract: The HOMFLY-PT polynomial is a knot invariant that captures many classical invariants (Alexander polynomial, Jones polynomial) as special cases. There is a close connection between the algebra underlying the HOMFLY-PT polynomial and some familiar structures in combinatorics. One aspect of this relationship is that some special knots (torus knots) have very special HOMFLY-PT polynomials (certain generalizations of Catalan numbers). In this talk I will define all the objects above and discuss a new tool for working with these polynomials. Using this tool, the relation to Catalan numbers appears very naturally (though it still looks like magic!). If we have enough time I will discuss a new result, which uses the same trick to compute the categorified HOMFLY-PT polynomial of some torus knots. |
| Tuesday
Feb 28, 2017 3:00 pm |
|
|
| Tuesday
Mar 7, 2017 3:00 pm |
Dave Bachman Pitzer College |
Title: Heegaard Genus is NP-Hard Abstract: In joint work with Ryan Derby-Talbot and Eric Sedgwick, we show that the problem of determining the Heegaard genus of a 3-manifold is NP hard. |
| Tuesday
Mar 14 |
No Meeting | Spring Break |
| Tuesday
Mar 21, 2017 3:00 pm |
|
|
| Tuesday
Mar 28, 2017 3:00 pm |
Lyla Fadali Occidental College |
Title: Exploring categories and topological quantum field theory using surfaces. Abstract:The 2016 Nobel Prize in physics was awarded for work related to topological quantum field theory. I'll talk about topological quantum field theory from a mathematical perspective, and give some examples with knots, graphs, and surfaces. I'll also discuss higher categories and how they relate to topological quantum field theory, including my work on Bar-Natan skein modules. I will assume familiarity with the definitions of knots, manifolds, and their invariants, but will provide background otherwise. |
| Tuesday
Apr 4, 2017 3:00 pm |
Matt Rathbun CSU Fullerton |
Title: Complete classification of generalized crossing changes between Genus One Fibered Knots Abstract: I will define and discuss Genus One Fibered Knots (GOF-knots), and a beautiful method to analyze them that touches on automorphisms of trees, representations of SL(2, Z), and hyperbolic geometry. I will then discuss how this analysis can be used to classify all monodromies of GOF-knots, the ambient manifolds in which they sit, and all generalized crossing changes from one GOF-knot to another within a manifold.
|
| Tuesday
Apr 11, 2017 3:00 pm |
Radmila Sazdanovic North Carolina State Univ. |
Title: Chromatic-type homology theories Abstract: The configuration space of n distinct points in a manifold X is a well-studied object with lots of applications. Eastwood and Huggett define graph configuration spaces M(G,X) by allowing vertices connected by an edge in G to occupy the same point in X. Our work generalizes this construction from graphs to finite simplicial complexes to obtain the space M(S,X). In this talk we will discuss properties of homology of M(S,X) and the polynomial invariant of simplical complexes arising as its Euler characteristic. |
| Tuesday
Apr 18, 2017 3:00 pm |
Neslihan Gugumcu National Technical Univ. of Athens |
Title: The mathematical world of shoelaces: Knotoids and some invariants of them. Abstract: The theory of knotoids that was introduced by V.Tureav in 2012 extends the theory of classical knots. A knotoid is defined like a 1-1 tangle, as an image of the unit interval in a surface with finitely many transversal double points endowed with under/over-data and two distinct endpoints but for knotoids we allow the endpoints to be i n different regions of the diagram. In this talk, we first go through the basic notions of knotoids in the 2-sphere and the plane and remember some notions of the classical and virtual knot theory. We explain how knotoids are related to classical and also to virtual knots. We then introduce some invariants of knotoids including the odd writhe, the affine index polynomial and the arrow polynomial. We show both the odd writhe, the affine index polynomial and the arrow polynomial are used to determine the type of a knotoid. If time permits, we give a geometric interpretation of knotoids and in this direction, discuss on possible applications of knotoids to biology, chemistry and physics. This is a joint work with Prof. Louis Kauffman. |
| Tuesday
Apr 25, 2017 3:00 pm |
Jacob Rooney UCLA |
Title: Cobordism maps in embedded contact homology Abstract: Embedded contact homology is a Z/2-graded homology theory that has been used to prove topological results on the existence of closed Reeb orbits and embeddings of symplectic 4-manifolds. In this talk, we indicate how exact symplectic cobordisms between contact 3-manifolds induce chain maps on embedded contact homology. This is work in progress. |
| Tuesday
May 2, 2017 3:00 pm |
Ramin Naimi Occidental College |
Title: Linear embeddings of graphs and oriented matroids Abstract: This will be a very elementary introduction to oriented matroids, plus a couple of examples of how they have been used to study linking and knotting in linear embeddings of graphs in R^3. |