Discrete Geometry & Combinatorics Seminar: Fall 2016

← Back to Current Schedule

---

Braids, Configurations, and Noncrossing Partitions

Michael Dougherty | September 28

• There is a well-known relationship between the n-strand braid group and the complement of a set of hyperplanes in complex n-space. In 1987, Salvetti described a deformation retraction of this space which resulted in a finite cell complex with the same homotopy type. Another homotopy-equivalent complex was obtained by T. Brady in 2001 using a different presentation for the braid group, but without giving an explicit deformation retraction. In this talk I will describe such a homotopy equivalence for n=3 and discuss the situation for more strands.

Sweeping Up Zeta

Nathan Williams | October 5

• Using techniques introduced by H. Thomas and N. Williams in "Cyclic Symmetry of the Scaled Simplex," we prove that modular sweep maps are bijective. We construct the inverse of the modular sweep map by passing through an intermediary set of equitable partitions; motivated by an analogy to stable marriages, we prove that the set of equitable partitions for a fixed word forms a distributive lattice when ordered componentwise. We conclude that the general sweep maps defined by D. Armstrong, N. Loehr, and G. Warrington in "Sweep Maps: A Continuous Family of Sorting Algorithms" are bijective. As a special case of particular interest, this gives the first proof that the zeta map on rational Dyck paths is a bijection. This is joint work with Hugh Thomas.

Hyperplane Complements and Cell Complexes

Jon McCammond | October 12

• For complexified real hyperplane arrangements there is a classic construction due to Salvetti that produces a cell complex in the hyperplane complement whose inclusion is a homotopy equivalence. The first half of the talk will be a review of this construction, and the second half will discuss what can be done when the arrangement under consideration is not a complexification of a real arrangement. The focus will be on one particular concrete situation investigated by Ben Cote in his dissertation.

Eulerian Numbers

T. Kyle Petersen | October 19

• Every student of mathematics is familiar with Pascal's triangle of binomial coefficients. Though not as well known, the classical Eulerian numbers form a triangle that is almost as nice. These numbers have been given many different combinatorial interpretations, as well as algebraic and geometric interpretations. Via the algebraic and/or geometric interpretations, we can generalize the Eulerian numbers in many ways, e.g., to Coxeter groups, or to certain families of polytopes. I will survey some of the history of Eulerian numbers, including current work and open problems.

Life in Hyperbolic Worlds

Steve Trettel | October 26

• We are used to thinking about hyperbolic space using models, partly because they simplify our calculations and partly because we can’t see hyperbolic space from the outside (even the plane doesn’t embed nicely into R^3). However much like stenographically projecting the earth onto a plane ruins the inherent symmetry of a sphere, our models provide us with a highly distorted lens through which to look, and much of how it would actually look or feel to live in a negatively curved world is difficult to recover. In this talk, I will discuss what kind of things need to be computed to be able to piece together an insider’s view of hyperbolic 3-space, and then take you on a tour of nine different hyperbolic spaces with radii of curvature varying from the size of a galaxy to the size of a person. There are no real prerequisites as I will only be presenting the results of computations rather than the process of computing them, and the goal is to try and teach some hyperbolic geometry in an entertaining way.
  
  To get a feel for the style, here’s a few of the things we will discuss:
  
  
  • Would our hyperbolic counterparts believe in aliens?
  • Why does dawn come long after sunrise?
  • Are there solar eclipses?
  • Can you see the ocean from south hall?
  • Is it faster to walk to San Francisco or take a plane?
  • Why don't cars have passenger seats?What does your face look like in a mirror?

Totally Nonnegative Matrices

Siân Fryer | November 2

• A totally positive real matrix is one for which every minor is positive. As with people, total positivity in matrices can get a bit boring after a while, so we turn our attention to something a bit less demanding: total nonnegativity, i.e. each minor is nonnegative. This talk will give an introduction to total nonnegativity in real matrices and the real Grassmannian, and describe how we can get some really nice combinatorics out of this deceptively simple definition.

An Invitation To Lattice Path Combinatorics

David Nguyen | November 9

• A lattice path is a finite sequence of vectors v=(v_1, v_2, ..., v_n) such that each v_j is in the step set S, where S is a given subset of the square lattice Z^2. The case S={(1,1), (1,-1)} corresponds to the classical Dyck paths, for which many ways of getting explicit formulas involving the Catalan numbers are known. This talk will introduce the audience to the wonderful world of lattice paths combinatorics by exploring in details the specific case S = {(1,-2), (1,-1), (1,1), (1,2)}. In particular, the techniques used will be emphasized, rather than the results of the problem, which find applications in, such as, queuing theory and financial pricing options.

The Catenary Degree: A Combinatorial Non-Unique Factorization Invariant

Sherilyn Tamagawa | November 16

• We usually like to talk about unique factorization domains, but what happens when an element is not uniquely factorable? Is there a way to tell how "far" it is from being uniquely factorable?
  
  In this talk, we will define the catenary degree, which is one way of measuring non-uniqueness of factorizations. Along the way, we will define a metric on factorizations of an element. We will focus our attention on arithmetical congruence monoids (ACMs), a multiplicative subset of the natural numbers. Our main result will characterize the catenary degree of a special class of ACM. This talk should be accessible to everyone.
  
  This work was joint with Scott Chapman and Theo McKenzie, at the PURE Math REU 2013.

Posets, Shellability, and the Bruhat Order

Gordon Kirby | November 30

• This talk aims to explore some basic poset topology and combinatorial restrictions on edge labelings of the Hasse diagram in order to define Cohen-Macaulay posets. Specifically we will discuss shellability and EL-shellability. We will also look at the Bruhat order on the symmetric group for some examples. The goal is to convey some appreciation for the interactions of combinatorics, algebraic topology and commutative algebra.