For more information please contact Evan Houston or Gabor Hetyei
|August 20||First Day of Class||No Meeting|
|August 27||Arindam Roy||UNCC||Elementary Proof of Dirichlet's Theorem|
|September 03||Labor Day||No Meeting|
|September 10||Arindam Roy||UNCC||Elementary Proof of Dirichlet's Theorem (Contd.)|
|October 01||Arindam Roy||UNCC||Elementary Proof of Dirichlet's Theorem (Contd.)|
|October 08||Student Recess||No Meeting|
|October 15||Gabor Hetyei||UNCC||Uniform triangulations of the Legendre polytope|
|October 23||Gabor Hetyei||UNCC||Uniform triangulations of the Legendre polytope (Contd.)|
|October 29||Gabor Hetyei||UNCC||Uniform triangulations of the Legendre polytope (Contd.)|
|November 05||Gabor Hetyei||UNCC|
|November 19||Jim Coykendall||Clemson University||Factorization and the Half-Factorial Property||
The notion of unique factorization is central in commutative algebra and number theory (with very important ramifications spilling over into almost every branch of mathematics). In general, "factorization" may be considered the study of the multiplicative decomposition(s) of elements in a ring (usually an integral domain for our purposes), and generalizations of unique factorization are of much interest. One of the most infamous generalizations of unique factorization is the half-factorial property. Loosely speaking, a half-factorial domain is an integral domain in which elements may not factor uniquely, but any two decompositions of the same nonzero element into irreducibles will have the same number of irreducibles involved (counting possible multiplicities). The aim of this talk will be to give an overview of this generalization of unique factorization from its inception through some very recent results. Along the way, the talk will be seasoned and flavored with many examples comparing and contrasting the half-factorial property with the more familiar notion of unique factorization.
|November 26||Grace Stadnyk||NCSU||Some Combinatorial Results on the Edge-Product Space of Phylogenetic Trees||
I will discuss a topological space that arises in evolutionary biology called the edge-product space of phylogenetic trees. In particular, I will discuss some of the combinatorial properties of a particularly natural CW decomposition of this space. It is known that intervals in the resulting face poset are shellable, but the manner in which the maximal faces intersect is still not well-understood. In particular, the poset in its entirety is not known to be shellable. I will introduce a partial order on the maximal faces of the edge-product space of phylogenetic trees called the enriched Tamari poset, which can be viewed as a generalization of the Tamari lattice. I will use this poset to show that the edge-product space of phylogenetic trees is gallery-connected. I will conclude by discussing the question of whether the face poset of this topological space is shellable. I will define most of the notions that appear in the talk, so it should be accessible to all.