A weekly seminar on topics in discrete geometry with a combinatorial
We meet on Wednesdays 2:00-3:00pm in South Hall, 4607B.
more information, contact the organizers: Gordon
Dougherty, and Jon McCammond.
Subgroups of C, Knots and Tori
| April 18
In this talk, we'll discuss the following beautiful result of Hubbard & Pourezza:
The space of closed subgroups of the complex plane is homeomorphic to the four-sphere, with the collection of lattices forming an open dense subset with complement a two-sphere. Furthermore, this two-sphere is not locally-flat embedded, and intersections with three-spheres of constant latitude form trefoil knots.
We will begin by introducing the Chabauty topology on the space of closed subgroups of a topological group, and as a warm-up compute the Chabauty space of R. We'll take a detour into Seifert fibered spaces and discuss SL(2,R)/SL(2,Z) before tackling the actual problem of computing the Chabauty space of C. We will then discuss an interpretation of this space that's relevant when considering moduli spaces of tori.
This talk is meant to be elementary with the main goal of introducing the Hausdorff metric, Chabauty topology, Seifert fibrations and suspensions to younger graduate students, so first and second years are especially encouraged to stop by!
Convexity conjecture and the word problem for Artin groups
Gordon Kirby | May 2
- In this talk, I'll give an introduction to Artin groups and discuss some of the open problems related to Artin groups of infinite type.
We will begin by defining Artin groups and their corresponding Coxeter systems. We'll focus on defining and giving examples of a combinatorially defined cell complex introduced by Mike Davis, which is associated with an Artin group with a specified presentation. Then we'll introduce the concept of convexity and irreducibility of subcomplexes the Davis complex so that we'll be able to state a new problem whose solution would solve the word problem and resolve other outstanding conjectures.
Some combinatorial analogs of homogeneous spaces
Daryl Cooper | May 9
- A space is called homogeneous if every point has a neighborhood that looks the same.
For example Euclidean space or projective space. We will define some combinatorial analogs
of this idea and explore the question of what classes of spaces have combinatorial characterizations.
Joint with Priyam Patel.
What does it look like inside geometric 3 manifolds?
| May 30 (1:30 pm start time in SH 6635)
- If you were an ant, and you lived on a flat torus, what would you see? How would the experience be qualitatively different if you instead lived on a punctured torus with a complete hyperbolic metric? Could you tell which of those two spaces you lived on if you weren't allowed to walk around and were armed only with a telescope?
I encourage you to think a bit about this question before tomorrow's talk.
In the talk, we will discuss the relevant mathematics for answering questions such as these, and then take a field trip over to the Allosphere to visualize the 3-dimensional case in virtual reality. We will go on a tour of a couple of Euclidean 3-manifolds (compact and noncompact, orientable and non-orientable) to warm up our visual faculties, and after briefly reviewing how curvature distorts our view, we will explore the intrinsic geometry of some spherical 3 manifolds (quotients of SU(2) by double covers of the symmetries of platonic solids) and finally some hyperbolic manifolds (compact, one-cusped, and infinite volume).
| June 6