Discrete Geometry & Combinatorics Seminar: Winter 2015
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Large Scale Geometry
| January 21
- This is an expository talk. Large scale geometry ignores small scale phenomena.
We will explore some highlights. Curvature is an infinitesimal property of Riemannian metrics but
it has large scale implications. Negative curvature implies geodesics fly apart quickly.
This idea can be generalized to metric spaces that are not even connected: Gromov hyperbolicity
and delta-thin triangles.
Fractals, Graphs and Fields
| February 4
- This talk is on F. Mendivil's Hasse-prize-winning paper "Fractals, Graphs and Fields." In it, we'll walk through the concepts of an iterated function system, and talk about how graphs and fields can be used to help us visualize the limit points of such systems.
Jones-Wenzl Projectors for the Temperley-Lieb Algebra
| February 11
- This is an expository talk based on the first half of the Temperley-Lieb-Jones Theories chapter of Zhenghan Wang's monograph "Topological Quantum Computation". In it, we will define the Temperley-Lieb algebra, become familiar with its properties as a diagrammatic algebra and ultimately define the Jones-Wenzl projectors (or idempotents). Understanding this algebra, and in particular, these projectors, is essential for topological quantum computation.
Steve Trettel | February 18
- A square can easily be drawn in the plane so that its vertices have integer coordinates, but a short argument shows this to be impossible for the equilateral triangle (which however, can be drawn in R^3 with integer points for vertices). In this talk we will see which polytopes are able to be embedded in some R^k so that all of their vertices are integer points.
Associahedra, Cyclohedra and Thompson's Group T
| February 25
- The first half of this talk is a gentle
introduction to the families of polytopes known as
Associahedra and Cyclohedra. The second half
considers the infinite versions of these polytopes
and their somewhat surprising connection with the
infinite finite-presented simple group known as
Thompson's group T. This connection is
facilitated by the triangulation of the hyperbolic
plane known as the Farey tesselation.
Representations of the Braid Group
Joe Ricci | March 4
- Solutions to the Yang-Baxter Equation are invertible linear maps on the tensor square of some vector space. These operators lead naturally to representations of the braid groups. This talk will explain the YBE in the setting of braid group representations and how the problem can be discretized.
Artin Groups, Hyperplane Arrangements, and Salvetti Complexes
| March 11
- There is a natural relationship between the symmetric group and the braid group which can be seen through the construction of an arrangement of hyperplanes in complex space. A similar construction can be used for any finite Coxeter group and its corresponding Artin group, although it is a little less easy to see how the hyperplanes are arranged in this setting. The Salvetti complex gives us a (relatively) simple way to introduce a finite cell structure on the complement of the arrangement, easing our computations. In today’s talk we will assume none of the above definitions - awareness of the fundamental group and cell complexes are the only prerequisites.