Jordan Tirrell
Topic Areas:
Lattices, NT/Algebra
Prepared Talks:
- Nested Chain Partitions of Normalized Matching Posets
- In what ways can we partition a partially ordered set (poset) into linearly ordered subsets (chains)? We will report on recent progress made on a thirty year old conjecture. In particular, two chains in a finite ranked poset P (a finite poset is ranked if all maximal chains have the same size) are said to be nested if the levels occurring in one are a subset of the levels occurring in the other. A thirty-year old conjecture of Griggs gives a sufficient condition--the so-called normalized matching condition, also known as the LYM property--for guaranteeing a decomposition of a poset into pairwise nested chains. In this talk, we present our results in support of the conjecture. As a consequence of our main theorem, the conjecture is true for rank 3 posets of width (size of the largest collection of incomparable elements) less than 12.
- Matrix Generation of Integer Length Integer Vectors for Dimensions 2 < n < 10
- It is well known that Pythagorean Triples, or integer length integer vectors of dimension 2, may be generated parametrically. It is somewhat less well known that they may also be generated via matrices (the resulting structure is known as the Barning Tree). We describe how to generalize this to produce all integer length integer vectors in /n/ dimensions (also known as Diophantine solutions to a sum of /n/ squares that is square) for 2 < n < 10. Up to dimension 8, all solutions may be obtained via matrix multiplication from a single type of initial solution. For dimension 9, two types of initial solution are required.
Contact Information:
- Institution: Lafayette College
- Phone Number: (484) 356-6220
- e-Mail Address: tirrellj@lafayette.edu
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