The Algebra, Cryptology, Geometry Seminar meets on selected Fridays, dates indicated below. Everybody interested is welcome to join.The seminar is co-organized by Zach Teitler and Marion Scheepers. If you wish to present in the seminar or need information, please contact firstname.lastname@example.org.
Marion Scheepers (Boise State University)
The parity of permutations and quadratic reciprocity.
In this talk we explain a clever proof, discovered in the 1870’s by Zolotarev, of the quadratic reciprocity law for the integers. Zolotarev’s proof allows generalization to various other rings.
Samuel Coskey (Boise State University)
The set splittability problem.
The set splittability problem asks whether, given a collection of finite sets, there exists a single set that selects exactly half the elements from each set in the collection. If a set has odd size, we may select either the floor or the ceiling of half its elements. The question is naturally a part of combinatorial discrepancy theory, since a collection is splittable if and only if its discrepancy is at most 1. In this talk we will show that the set splittability problem is NP complete. On the other hand, we will give several partial solutions to the problem for small collections and other special collections. This work was completed during our REU program in collaboration with P. Bernstein, C. Bortner, S. Li, and C. Simpson.
Liljana Babinkostova (Boise State University)
Anomalous Primes and the Elliptic Korselt Criterion
Although there now are polynomial time primality tests that correctly identify numbers as prime or composite, these tests run slower than tests that sometimes incorrectly classify certain composite numbers as prime.
The classical test based on Fermat’s Little Theorem is an example of this second class of tests. Carmichael numbers, described by Korselt before their discovery, are such composite numbers misclassified by Fermat’s Little Theorem-based test.
Primality tests based on elliptic curves were introduced in the 1980’s. Some of these tests, like the classical Fermat’s Little Theorem based tests, misclassify some composite numbers as prime. This gave rise to the notion of elliptic Carmichael numbers. Silverman formulated criteria for identifying these numbers, and called one class the Type I elliptic Korselt numbers. In this talk we define anomalous prime numbers and explain their connection to the Type I elliptic Korselt numbers, generalizing previous results. This is joint work with J. Bahr, Y. Kim, E. Neyman and G. Taylor.
Jellyfish Brauer and partition algebras
Let O(n) denote the group of orthogonal nxn matrices and V the vector space of length n column vectors. There is a natural action of O(n) on tensor powers of V. The corresponding centralizer algebra was described via stacking certain diagrams by Brauer in 1937. The resulting diagram algebras (now called Brauer algebras) have been a hot topic in representation theory over the past 30 years. In Brauer’s 1937 paper he also describes some additional diagrams in order to model the centralizer algebra for the action of the special orthogonal group SO(n). However, Brauer does not explain how to multiply these new diagrams. Instead, he writes “The rule for the multiplication . . . can also be formulated. It is, however, more complicated and shall not be given here.” These diagrams received little to no attention until 1999 when Grood finally formulates a multiplication algorithm. Grood’s algorithm is indeed complicated, involving many steps. Moreover, as Grood points out, her multiplication rule is not even associative!
In this talk I will describe an alternative method for drawing and multiplying Brauer’s diagrams related to SO(n). In some sense, my diagrams are obtained from Brauer’s by “adding jellyfish”. The resulting “Jellyfish Brauer algebra” turns out to be an associative quotient of Grood’s algebra. I will also describe “Jellyfish partition algebras” which model the centralizer algebras for the action of alternating groups.
Jason Smith (Boise State University)
The Fibonacci Number of the Tadpole Graph.
An independent set of vertices of a graph G contains only nonadjacent vertices. Since the number of independent sets of the path Pn is the Fibonacci number Fn+2, we call the number of independent sets of a graph G its Fibonacci Number. Since the Fibonacci Number of the cycle graph Cn is the Lucas number Ln, it is natural to look for other graphs whose Fibonacci Numbers form a Gibonacci sequence. The Fibonacci Numbers of Tadpole Graphs not only form a Gibonacci sequence but can also be organized into a symmetric Pascal-like triangle that has many combinatorial properties.