| Date | Speaker | Title and Abstract |
| Tuesday
Sept 9 3:00 pm |
Organizational Meeting | Meet at Some Crust Bakery for organizational meeting. |
| Tuesday
Sept 16 3:00 pm |
Sam Nelson Claremont McKenna College |
Title: Finite type enhancements of biquandle counting invariants Abstract: Finite type invariants, also known as Vasiliev invariants, are integer-valued knot invariants satisfying a certain skein relation. Many of the coefficients of the Jones and Alexander polynomials, for example, are known to be Vassiliev invariants, and the set of all Vassiliev invariants dtermines a powerful invariant known as the Kontsevich integral. We adapt a scheme for computing finite type invariants due to Goussarov, Polyak and Viro to enhance the biquandle counting invariant. The simplest nontrivial case has connections to the concept of parity in virtual knot theory. This is joint work with Pomona student Selma Paketci. |
| Tuesday
Sept 23 3:00 pm |
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| Tuesday
Sept 30 3:00 pm |
Matt Rathbun CSU Fullerton |
Title: Monodromy action on unknotting tunnels in fiber surfaces and applications for DNA Abstract: DNA encodes the instructions used in the development and functioning of all living organisms. The DNA molecule, however, often becomes knotted, linked, and generally entangled during normal biological processes like replication and recombination. The subject of Knot Theory, correspondingly, can inform our understanding of these processes. I will introduce Knot Theory, and some of the myriad of tools that mathematicians use to understand knots and links. In particular, I will focus on a special class of link called fibered links. I will explain some recent results, joint with Dorothy Buck, Kai Ishihara, and Koya Shimokawa, about transformations from one fibered link to another, and explain how these results are relevant to microbiology. |
| Tuesday
Oct 7 3:00 pm |
Arielle Leitner UCSB |
Title: Geometric Transitions of the Cartan Subgroup in SL(n,R) Abstract: A geometric transition is a continuous path of geometries which abruptly changes type in the limit. We explore geometric transitions of the Cartan subgroup in SL(n,R))$. For n=3, it turns out the Cartan subgroup has precisely 5 limits, and for n=4, there are 15 limits, which give rise to generalized cusps on convex projective 3-manifolds. When n>6, there is a continuum of non conjugate limits of the Cartan subgroup, distinguished by projective invariants. To prove these results, we use some new techniques of working over the hyperreal numbers. |
| Tuesday
Oct 14 3:00 pm |
Leyda Almodovar University of Iowa |
Title: Clustering and topological analysis of brain structuresp> Abstract: Topological data analysis is a relatively new area that uses several disciplines in conjunction such as topology, statistics and computational geometry. The idea behind topological data analysis is to describe the “shape” of data by recovering the topology of the sampled space. Also, it is useful to find topological attributes that persist in the data, helping us gain a better understanding of how different properties of the data interact. Data was collected from MRI experiments with 96 subjects between the ages of 0 and 18, some of them predisposed to Huntington’s disease. Data analysis is performed via different topological approaches including clustering and persistent homology with the goal of identifying whole networks of points in the brain. The main purpose of this work is to compare the structure of brain networks of healthy subjects versus subjects predisposed to Huntington’s disease. |
| Tuesday
Oct 21 3:00 pm |
No Meeting | Fall Break |
| Tuesday
Oct 28 3:00 pm |
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| Tuesday
Nov 4 3:00 pm |
Cornelia Van Cott University of San Francisco |
Title: Surfaces and iterated Bing doubles Abstract: We consider the classical problem of finding minimal genus Seifert surfaces for knots and links. In particular, we will consider the problem in the case of iterated Bing doubles, and we will describe a construction of minimal genus surfaces for these links. Next, we will broaden our perspective and consider the problem of finding minimal genus surfaces for iterated Bing doubles in the four-ball. |
| Tuesday
Nov 11 3:00 pm |
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| Tuesday
Nov 18 3:00 pm |
Faramarz Vafaee Cal Tech |
Title: Heegaard Floer homology and L-space knots Abstract: Heegaard Floer theory consists of a set of invariants of three- and four-dimensional manifolds. Three-manifolds with the simplest Heegaard Floer invariants are called L-spaces and the name stems from the fact that lens spaces are L-spaces. The primary focus of this talk will be on the question of which knots in the three-sphere admit L-space surgeries. We will also discuss about possible characterizations of L-spaces that do not reference Heegaard Floer homology. |
| Tuesday
Nov 25 3:00 pm |
Subhojoy Gupta Cal Tech |
Title: Riemann surfaces, projective structures and grafting Abstract: A complex projective surface is obtained by gluing pieces of the Riemann sphere using Mbius maps. In this talk I'll introduce these, and discuss ways of deforming such a geometric structure by the cut-and-paste operation of "grafting". Recent joint work with Shinpei Baba shows that complex projective surfaces with any fixed holonomy are dense in the moduli space of Riemann surfaces. I shall talk of some ingredients of the proof, including Thurston's notion of "train-tracks on surfaces". This talk will be aimed towards a general audience. |
| Tuesday
Dec 2 3:00 pm |
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| Tuesday
Dec 9 3:00 pm |
Maria Trnkova Cal Tech |
Title: Thurston's spinning construction. Abstract: Thurston defined the spinning construction for hyperbolic 3-manifolds. It is a way how to get an ideal triangulation from a regular one. In this talk I will demonstrate different points of view on this construction and explain what is its importance. The talk is accessible to general audience. |
| Date | Speaker | Title and Abstract |
| Tuesday
Jan 27 |
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| Tuesday
Feb 3 |
Vin de Silva Pomona College |
I will present Chapter 1 of Conway's book, "The (Sensual) Quadtratic Form." |
| Tuesday
Feb 10 |
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| Tuesday
Feb 17 |
Jim Hoste Pitzer College |
I will discuss the Appendix to Chapter 1 of "The (Sensual) Quadratic Form." |
| Tuesday
Feb 24 |
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| Tuesday
Mar 3 |
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| Tuesday
Mar 10 |
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| Tuesday
Mar 17 3:00 pm |
No Meeting | Spring Break |
| Tuesday
Mar 24 |
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| Tuesday
Mar 31 |
Sam Nelson Claremont McKenna College |
TItle: Quadratic Forms and the Arf Invariant Abstract: I will describe multiple ways to view the Arf invariant of a quadratic form and how the Arrf invariant arises in knot theory. |
| Tuesday
Apr 7 |
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| Tuesday
Apr 14 |
Jim Hoste Pitzer College |
Title: Knots with finite n-quandles Abstract: Associated to every knot or link is its fundamental quandle and it's quotients, the n-quandles. An obvious conjecture is that the n-quandle of a link L is finite if and only if the fundamaental group of the n-fold cyclic branched cover of the 3-sphere branched over L is finite. One direction of this statement, finite n-quandle imples finite fundamental group, is known to be true. It follows from results in Joyce's unpublished Ph.D. thesis combined with results in Winker's unpublished Ph.D. thesis. A strategy for proving the opposite direction is to examine all links with n-fold cyclic covers having finite fundamental group (a known list using the Orbifold Theorem) adn prove their n-quandles are finite. In this talk I will do this for all torus links, 2-bridge links, and 3-pretzel links. This leaves the more general Montesino's links to check, which is work in proggress. This is joint work with Patrick Shanahan. |
| Tuesday
Apr 21 |
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| Tuesday
Apr 28 |
Yewon Joung Pusan National University |
Title: State-sum invariants of surface-links via marked graph diagrams Abstract: A surface-link of n components is n mutually disjoint connected and closed 2-manifolds smoothly embedded in Euclidean 4-space. A marked graph diagram is a link diagram on $\mathbb R^2$ possibly with some 4-valent vertices equipped with markers. In 2009, Lee defined a polynomial $[[D]]$ for marked graph diagrams D of surface-links in 4-space by using a state-sum model involving a given classical link invariant. In this talk, I would like to discuss some obstructions to obtain an invariant for surface-links represented by marked graph diagrams D by using the polynomial $[[D]]$ and introduce an ideal coset invariant for surface-links, which is defined to be the coset of the polynomial $[[D]]$ in a quotient ring of a certain polynomial ring modulo some ideal and represented by a unique normal form, i.e. a unique representative for the coset of $[[D]]$ that can be calculated from $[[D]]$ with the help of a Gr\"obner basis package on computer. In addition, A. S. Lipson constructed two state models yielding the same classical link invariant obtained from the Kauffman polynomial $F(a;z)$. In this talk, I apply Lipson's state models to marked graph diagrams of surface-links, and observe when they induce surface-link invariants. |
| Tuesday
May 5 |
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