Date: Tuesday Sept 23rd and Sept 30th 2014
Speaker: Seungwon Kim
Title: Heegaard diagrams corresponding to Turaev surfaces
Abstract: I will talk about Cody Armond, Nathan Druivenga, Thomas
Kindred's recent paper. From the link diagram on S^2, we can construct
the Turaev surface by splitting each crossing. Turaev surface is a
splitting surface(Heegaard surface) of S^3. I will talk about how we
can find specific Heegaard diagram of Turaev surface from the link
diagram on S^2.
Date: Tuesday Oct 7th 2014
Speaker: Joe Quinn
Title: Arithmetic hyperbolic 3-manifolds
Abstract: I will give the definition of "arithmetic" in this context,
and introduce the various ingredients involved. I will explain what
this definition has to do with arithmetic groups in the language of
Borel and Harish-Chandra, casting light on the motivation for using a
quaternion algebra with conditions on its ramification set. I will
present some general results on arithmetic hyperbolic 3-manifolds,
especially the role of the Bianchi orbifolds, and I will give several
examples.
Date: Tuesday Oct 14th 2014
Speaker: Joe Quinn
Title: The arithmetic knot
Abstract: First I will show that the figure-8 knot complement is arithmetic. Then I will explain Alan Reid's proof from 1991 that the figure-8 knot is the only arithmetic knot. I will briefly explain the number theory facts that he references, and focus more on his general strategy of case-by-case analysis and elimination. We will gain an appreciation for the possibilities in exploiting arithmetic results and combining them with topological ideas and other calculations, via this example.
Date: Tuesday Oct 21st 2014
Speaker: Matt Sunderland
Title: Random Walks, Ideal Boundary, and Hitting Measure
Abstract: A random walk in H^3 induces a probability measure on S^2. Similarly, a random walk on any group or metric space induces a measure on its ideal boundary. In this way, random walks give us a natural way to associate probabilities to mathematical objects, including 3-manifolds. In this talk we develop the theory of random walks starting from the definition of random process.
Date: Tuesday Nov 18th 2014
Speaker: Rochy Flint
Title: Geometry of Fully Augmented Links
Abstract: I will talk about a class of links called Augmented Links.
We will look at the polyhedral decomposition of the link complement
into two isometric totally geodesic ideal hyperbolic polyhedra, relate
their circle packings and triangulation of S^2. There is a very nice
way to determine the cusp shape directly from the circle packing. We
will begin with these topological objects and end up with geometric
objects with easy to determine geometric structure.
Date: Tuesday Nov 25th 2014
Speaker: Rochy Flint
Title: Volume bounds on fully augmented links
Abstract: I will talk about volume bounds on fully augmented link
complements. We will see that the volume bound is sharp precisely when the
ideal polyhedra are obtained from a gluing of regular ideal octahedra.
I will describe how to see this from the nerve of the circle
packing associated to the augmented link.
Date: Tuesday Dec 2nd 2014
Speaker: Seungwon Kim
Title: Link diagrams with Turaev genus one
Abstract : Every link diagram is cellularly embedded on its Turaev
surfaces and is alternating on it. However, not every cellularly
embedded alternating diagram on a surface comes from Turaev's
construction. I will talk about a necessary condition for such a
diagram to arise from Turaev's construction, and use it to classify
link diagrams with Turaev genus one.