Discrete Geometry & Combinatorics Seminar: Fall 2015
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Topology of Finite Posets
| September 30
- Despite often being considered as topological objects, both simplicial complexes and hyperplane arrangements have intimate connections with the combinatorics of partially ordered sets. In this talk we will explore these connections through the lens of the Möbius function for a poset. It might be useful to know what a simplicial complex is, but no other background is necessary.
Egyptian Fractions, Kellogg's Equation, and Finite Groups
Joe Ricci | October 7
- Kellogg's equation is a certain Diophantine equation involving writing 1 as a sum of fractions with unit numerators. In other words each solution to Kellogg's equation yields an Egyptian fraction expansion for 1. This talk will explore the relationship between Kellogg's equation and the representation theory of finite groups.
| October 14
- After quickly introducing the notion of a noncrossing hypertree (whose definition you might be able to guess), I will spend most of the time explaining a bit of background about noncrossing partitions, braid groups, associahedra and cluster complexes, so that I can state some new results involving noncrossing hypertrees that help explain how these various concepts are closely related.
| October 28
- Every mathematician wonders each night when they go to sleep what sort of isoperimetric inequality they can establish for their favorite group. If your favorite group is one of the three we'll talk about, you'll at long last have this question answered. After defining Dehn functions, we will give some examples. In doing so, we'll examine some pretty pictures (artistic skills permitting). Lastly, we'll summarize some results and try to understand why one should care about Dehn functions.
Knots, Lattices, and SL(2,R)
Steve Trettel | December 2
- I came across some pretty pictures on the internet last week, and this talk grew out of my attempt to understand and recreate them. It turns out that while the space of planar lattices is 4 dimensional, considering them up to scaling results in a space which can be identified with the complement of a trefoil knot in the 3-sphere. A flow on the space of lattices gives rise to a flow on the 3-sphere, and periodic paths of lattices then correspond to knots. In this talk I’ll discuss three essentially different flows that arise on the space of lattices from 1-parameter subgroups of SL(2,R), and show you some pictures of the knots corresponding to (some of) their periodic orbits!