| Date | Speaker | Title and Abstract |
| Tuesday
Sept 6 3:00 pm |
Organizational Meeting | |
| Tuesday
Sept 13 3:00 pm |
Sara Kalisnik Stanford University |
Title: Topological Persistence and Duality Abstract: I will begin with a brief overview of the classical (i.e. early 21st century) theory of topological persistence, and its applications to data analysis. I will then discuss zigzag persistence, a more recent construction, which among other things gives a way to think about the fiberwise homology of a pair (X,f) where X is a space and f is a real valued function on X. There is an elegant measure theory approach to the zigzag persistence diagram, which I will explain. Finally, I will discuss how the Alexander Duality theorem from classical algebraic topology behaves in this framework. |
| Tuesday
Sept 20 3:00 pm |
Dwayne Chambers Claremont Graduate University |
Title: Topological Symmetry Groups Abstract: I will give a brief history of the concept of Topological Symmetry groups. I will also survey results in the field including my work on Topological Symmetry groups on “small” complete graphs and my $TSG_+(4r+3)$ Theorem which determines the orientation preserving Topological Symmetry groups on complete graphs on $4r+3$ vertices. Lastly I will take a look at a couple of cases of the proof of one direction of my $TSG_+(4r+3)$ Theorem. |
| Tuesday
Sept 27 3:00 pm |
Jim Hoste Pitzer College |
Title: Twisted Alexander Polynomials of 2-bridge Knots Abstract: We show how to compute the twisted Alexander polynomial of 2-bridge knots using the Fox coloring. For genus 1 2-bridge knots, we verify that the polynomial has the form conjectured for all 2-bridge knots by Hirisawa and Murasugi. This is joint work with Pat Shanahan. |
| Tuesday
Oct 4 3:00 pm |
Allison Gilmore Princeton |
Title: An Algebraic Approach to Knot Floer Homology Abstract: Ozsvath and Szabo gave the first completely algebraic description of knot Floer homology via a cube of resolutions construction. Starting with a braid diagram for a knot, one singularizes or smooths each crossing, then associates an algebra to each resulting singular braid. These can be arranged into a chain complex that computes knot Floer homology. After introducing knot Floer homology in general, I will explain this construction, then outline a fully algebraic proof of invariance for knot Floer homology that avoids any mention of holomorphic disks or grid diagrams. I will close by describing some potential applications of this algebraic approach to knot Floer homology, including potential connections with Khovanov-Rozansky's HOMFLY-PT homology. |
| Tuesday
Oct 11 3:00 pm |
Mark Kidwell US Naval Academy |
Title: The Bonahon Metric and Topology Abstract: In his book "Low-Dimensional Geometry: From Euclidean Spaces to Hyperbolic Knots", Francis Bonahon considers no structure more abstract than a metric space. He then needs to define a metric on a quotient space, such as the torus obtained by identifying opposite sides of a rectangle. We explore some quirky consequences of Bonahon's definition of a (pseudo)-metric on a quotient space. We then answer the question: does the topology defined by the Bohahon metric on a quotient space coincide with the quotient topology? Advanced undergraduate math majors should be able to follow most of this talk. |
| Tuesday
Oct 18 3:00 pm |
No Meeting |
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| Tuesday
Oct 25 3:00 pm |
Elena Pavelescu Occidental College |
Title: Legendrian graphs with all cycles unknots of maximal Thurston-Bennequin number. Abstract: A Legendrian graph in a contact structure is a graph embedded in such a way that its edges are everywhere tangent to the contact planes. In this talk we look at Legendrian graphs in R^3 with the standard contact structure. We extend the invariant Thurston-Bennequin number (tb) from Legendrian knots to Legendrian graphs. We prove that a graph can be Legendrian realized with all its cycles Legendrian unknots with tb=-1 if and only if it does not contain K_4 as a minor. |
| Tuesday
Nov 1 3:00 pm |
Yeonhee Jang Hiroshima University, Japan |
Title : Bridge numbers of links and meridian ranks of link groups Abstract : A question posed by Cappell and Shaneson asks whether the bridge number of a given link equals the minimal number of meridian generators of its group. We give a positive answer to this question for certain links containing all arborescent links. |
| Tuesday
Nov 8 3:00 pm |
Ismar Volic Wellesley College |
Title: Integral expressions for Milnor invariants Abstract: The goal of this talk is to describe a cochain map, given by configuration space integrals, from a certain complex of diagrams to the deRham complex of long links in R^n for n>3. When restricted to a particular subcomplex, this gives a cochain map to the deRham complex of homotopy string links in R^n for n>2. In the classical case n=3, one can in this way produce the well-known finite type link invariants and in particular Milnor invariants of homotopy links. Time permitting, I will mention connections to calculus of functors. |
| Tuesday
Nov 15 3:00 pm |
Guillaume Dreyer USC |
Title: An introduction to higher Teichmuller theory Abstract: Let S be a closed connected surface of genus at least two. Such a surface admits hyperbolic structures, namely complete finite area Riemannian metrics of constant curvature equal to −1. The Teichmuller space T(S) is the space which parametrizes isotopy equivalence of hyperbolic structures on the surface S. Teichmuller theory has been a topic of active mathematical research over the last century, which has lead to the development of deep and fruitful interactions between analysis and geometry. About 20 years ago, Hitchin identified other potential Teichmuller spaces, called Hitchin spaces Hit(S). Since then, fundamental work of Choi and Goldman, and more recently, Labourie, Guichard, Wienhard, Fock and Goncharov has revealed beautiful geometric and dynamic properties for the elements of Hit(S), inaugurating the development of higher Teichmuller theory. We will discuss how these other Teichmuller spaces arise, what type of geometric structures they parametrize, how we can use their geometric properties in order to construct invariants and their connections with the usual Teichmuller space T(S). In particular, we will describe how we can extend, to higher Teichmuller theory, several tools and concepts from classic hyperbolic geometry. |
| Tuesday
Nov 22 3:00 pm |
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| Tuesday
Nov 29 3:00 pm |
Michael Yoshizawa UCSB |
Title: Generating Examples of High Distance Heegaard Splittings Abstract: Given a closed orientable 3-manifold M, let S be a Heegaard surface for M and therefore separates the manifold into two handlebodies H_{1} and H_{2}. Then the Hempel distance of this splitting is defined to be the length of the shortest path in the curve complex of S between the disk complexes corresponding to H_{1} and H_{2}. In 2004, Evans developed an iterative process to construct manifolds that admit a Heegaard splitting with arbitrarily high distance. We show that the splittings that Evans proved to have distance n in fact have distance (n+1). |
| Tuesday
Dec 6 3:00 pm |
Kanako Oshiro Japan Women's University |
Title: Minimal numbers of colors and quandle cocycle invariants of surface-knots In this talk, we study the minimal number of colors used for non-trivial Fox colorings of surface-knots. A lower bound for the minimal number is given by using quandle cocycle invariants. In particular, we show that the minimal number of the $2$-twist spinning of $5_2$ knot for Fox 7-colorings is six. This is a joint work with Shin Satoh (Kobe University). |
| Date | Speaker | Title and Abstract |
| Tuesday
Jan 24 |
Sonja Mitchell UCSB |
Title: A Type B version of Thompson's Group F Abstract: Thompson's Group F is a finitely presented infinite group which has proved a rich source of counterexamples for group theorists. We re-consider the group as generated by ``flip-renormalize'' actions on the Farey Tesselation of the hyperbolic plane. Through this lens, the group is linked to the Associahedra, an infinite family of convex polytopes first studied by homotopy theorists in the 1960's. We consider how the Type B Associahedra then allow to us to define a Type B version of Thompson's Group F. A construction of F_B and preliminary results will be discussed. |
| Tuesday
Jan 31 |
Heather Russell USC |
Title: Symmetric group actions coming from spiders Abstract: Spiders are diagrammatic categories that study the representation theory of quantum groups. By specializing the quantum parameter in the spider, we get symmetric group actions on a certain subcollection of morphisms. The morphisms in the spider category, which are called webs, are certain simple planar graphs. The symmetric group action on them is skein theoretic. While this action on webs is known to be irreducible, the relationship between the web basis and other bases for symmetric group representations is not well understood. Using explicit bijections between webs and Young tableaux, we will discuss certain nice combinatorial properties of the web basis. |
| Tuesday
Feb 7 |
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| Tuesday
Feb 14 |
Ayaka Shimizu Osaka City Univ. Advanced Math. Inst. |
Title: Region Select - a game based on knot theory Abstract: A region crossing change at a region of a link diagram is defined to be the crossing changes at all the crossing points on the boundary of the region. In this talk, we show that we can make any crossing change on a knot diagram by some region crossing changes. Using this algorithm, we introduce the game "Region Select" as a joint work with Akio Kawauchi and Kengo Kishimoto. |
| Tuesday
Feb 21 |
Rob Ackerman USCB |
Title: Transition Matrices for pseudo-Anosov Surface Automorphisms Abstract: pseudo-Anosov automorphisms are in a sense the most common type of surface automorphism (up to homotopy), but also the least understood. Once way of examining the action of these maps is to encode it as a Perron-Frobenius matrix, which acts on either the weight space of a train track train track or a Markov partition. The dilatation of the pA map, a real number that encodes a variety of dynamical properties, appears as the largest eigenvalue of the matrix. I've been interested in trying to understand what matrices arise from pA maps, and in particular what numbers can appear as dilatations. I'll outline a result about possible obstructions coming from a symplectic form due to Penner, and possible directions for future work. |
| Tuesday
Feb 28 |
Rena Levitt Pomona College |
Title: The Word Problem for Quandles Abstract. In 1911 Max Dehn stated his now famous word problem: given a finitely generated group G, is there an algorithm to determine if two words in the generators represent the same element in G? The problem remained open for arbitrary groups for almost 40 years, and was one of motivations for the modern study of combinatorial group theory. Recently, we became interested in following question: can the word problem can be generalized to other algebraic constructions, such as quandles? In this talk, I will discuss a natural generalization of Dehn's problem to finitely generated quandles, and show the word problem is solvable for both free and knot-like quandles. The algorithm we define is similar to Dehn's original solution for the fundamental groups of surfaces with genus at least two. This is joint work with Sam Nelson. |
| Tuesday
Mar 6 |
Erica Flapan Pomona College |
Title: Topological Symmetry Groups Abstract: Motivated by considering molecular symmetry groups, we define the topological symmetry group of a graph embedded in a 3-manifold as the subgroup of the automorphism group of the abstract graph induced by homeomorphisms of the embedding of the graph in the manifold. We discuss which groups can occur as topological symmetry groups of graphs embedded in S^3 as well as in other 3-manifolds. |
| Tuesday
Mar 13 |
No Meeting |
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| Tuesday
Mar 20 |
Sam Nelson Claremont McKenna College |
Title: Polynomial Birack Module Invariants Abstract: Birack modules are algebraic structures used to enhance and strengthen the birack counting invariant. Birack modules over polynomial rings can be used to define a customized Alexander polynomial for each birack labeling of of a knot or link, the set of which over of complete set of labelings forms an enhancement of the birack counting invariant. |
| Tuesday
Mar 27 |
Charlie Frohman University of Iowa |
Title: Skeins and Characters Abstract: I will explain how the Cayley-Hamilton Identity for 2 by 2 matrices of determinant 1 leads to the Kauffman bracket skein relation. I then explain the physical significance of a crossing diagram. The talk will be accessible to advanced undergraduate. I will assume that the students are familiar with trace, determinant, eigenvalues, groups, homomorphisms, conjugacy classes in groups, continuity, tangent vectors, and gradient. I will give a quick description of the fundamental group of a space, and it's conjugacy classes during the lecture. |
| Tuesday
Apr 3 |
Helen Wong | Action of Topoisomerases on DNA Knots Certain organisms have DNA that are circular. Under the action of enzymes called topoisomerases, such DNA can become knotted. We classify all possible knots and links that could be products of enzyme actions on a DNA substrate that is the connected sum of torus links. This is joint work with E. Flapan and my former students J. Grevet, Q. Li, and D. Sun. |
| Tuesday
Apr 10 |
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| Tuesday
Apr 17 |
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| Tuesday
Apr 24 |
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| Tuesday
May 1 |
Liam Watson | Title: A conjecture relating Heegaard Floer homology and the fundamental group. Abstract: A group is left-orderable if it admits a strict total order of its elements that is invariant under multiplication on the left. As an immediate consequence (exercise!), left-orderable groups are torsion free. For example, a finite cyclic group cannot be left-ordered; hence the fundamental group of a lens space is not left-orderable. L-spaces provide a generalizations of lens spaces in the context of Heegaard Floer homology. These manifolds have simplest possible Heegaard Floer homology, though they need not have cyclic fundamental group. This talk will describe some evidence supporting the conjecture that L-spaces are equivalent to 3-manifolds with non-left-orderable fundamental group |