Discrete Geometry & Combinatorics Seminar: Winter 2016
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Noncrossing Partitions and Parking Functions with Undesired Spaces
| January 20
- Thanks to its ever-growing list of pleasing properties, the lattice of noncrossing partitions has made several appearances in combinatorics, group theory, and topology in recent decades. Among its many interesting features is Stanley's bijection between the maximal chains of this lattice and the set of parking functions, another versatile combinatorial object. Given subsets of parking functions, we obtain new posets by taking the union of the Hasse diagrams for the corresponding maximal chains. In this talk, we present a decomposition of parking functions which yields a collection of surprisingly nice subposets of the noncrossing partition lattice.
This talk features joint work with Melody Bruce, Max Hlavacek, Ryo Kudo, and Ian Nicolas, who were participants of the 2015 UCSB Mathematics REU.
Random 2-bridge Chebyshev Billiard Table Diagrams
(Technion - Israel Institute of Technology) | January 27
- The study of random knotting is mostly experimental with difficulty in defining good probability spaces.
Koseleff and Pecker show that all knots can be parametrized by Chebyshev polynomials in three dimensions. These long knots can be realized as trajectories on billiard table diagrams. We use this knot diagram model to study random knot diagrams by flipping a coin at each 4-valent vertex of the trajectory.
We truncate this model to study 2-bridge knots together with the unknot. We give the exact probability of a knot arising in this model. Furthermore, we give the exact probability of obtaining a knot with crossing number c.
This is joint work with Sunder Ram Krishnan and Chaim Even-Zohar.
Hands-on Examples of 3-Manifolds
Steve Trettel | February 3
- In this talk I'll tell you about a cool procedure for producing manifolds whose points consist of (loosely put) "the ways to place a shape X inside a shape Y". We will look at spaces associated to various tessellations of the plane, and the space of ways to place a tetrahedron / dodecahedron inside of a 2-sphere. These give examples of compact 3-manifolds, and their description in terms of configurations of 2-dimensional objects gives us a nice hands-on way to understand them.
There shouldn't be any real necessary prerequisites (besides knowing what a manifold and a group are!) so hope to see you tomorrow!
The Congruence Subgroup Problem and B3
| February 17
- Thanks to a theorem of Bass, Lazard, and Serre, for d at least 3, the finite index subgroups of SL(d,Z) come in one flavor. Indeed, they are all congruence subgroups. However, when d=2, this fails meaning there are “non-obvious “ subgroups of SL(2,Z) with finite index. Motivated by a result of Ng and Schauenburg, I will introduce the congruence subgroup problem and discuss some examples of noncongruence subgroups coming from representations of the 3-strand braid group. The talk will be introductory and is intended for a general audience.
Introduction to Coxeter Groups
| February 24
This talk will roughly serve as an informal "part one" of my Master's thesis defense, so that next quarter I can start with some work already developed. After I provide the basic definitions and constructions for Coxeter groups, I will go over a couple of illustrating examples and outline some important theorems. This will ultimately be aimed at understanding the limiting distribution of roots in the Lorentzian case, but today we will focus on general theory. The talk should be comfortable for any attendee, albeit a bit repetitive for those of you who are already familiar with Coxeter stuff.
A Combinatorial Approach to Quaternionic Structures
Justin Kelz | March 2
- A quaternionic structure is a elementary abelian group of exponent 2, with distinguished
element -1, together with a set Q with distinguished element 0, and a surjective function q : GXG -> Q
that satisfies four axioms.
My talk will show that when 1 = -1, that these structures give rise to a family of complete
graphs with certain edge colorings, which turn out to be Steiner Triple Systems with
analogous block colorings, and ultimately this allows us to construct the Fano Plane
in a novel way.
Generating Random Polygons Using Orthoschemes
Kyle Chapman | March 9
- One useful part of knot theory applications is being able to randomly generate polygons, piecewise linear equilateral knots in R^3. A nice coordinate system for such things allows for the proper measure using symplectic geometry, given by Cantarella and Shonkweiler. This leads us to wanting to sample a convex polytope, which it turns out can be subdivided nicely into particular simplicies called orthoschemes.