Abstract: Kimberling defined a self-generating set  S  of integers as follows: 1 is a member of  S, and if  x  is a member of  S, then 2 x  and 4 x - 1 are in  S. Nothing else is in  S. As an REU project, I investigated several interesting properties of the Kimberling set, and its resulting sequences reduced modulo  m  - particularly the case  m  = 2, which yields the Fibonacci word. Garth and Gouge (2007) proved not only that any of a special class of affinely self-generating sets of integers reduced modulo  m  may be generated by a substitution morphism, but also its characteristic sequence, which turns out to be automatic. The talk will be introductory in nature, and will also discuss some of the open questions posed by Garth and Gouge.

Abstract: Associated to the quantum double  Z(C)  of a spherical fusion category  C  is a representation of the modular group SL(2,Z). The recent paper of Ng and Schauenburg has shown that the kernel of this representation is a congruence subgroup of level  N, where  N  is the Frobenius-Schur exponent of  C. The definition of generalized Frobenius-Schur indicator for spherical fusion categories was introduced and used to study the modular representation for the quantum double  Z(C). The talk will begin with some examples of spherical fusion categories, their quantum doubles, Frobenius-Schur exponents and associated modular representations. We will discuss the definition of generalized Frobenius-Schur indicators and their relation to the modular representations.

Abstract: In a previous talk I presented the general theory developed by George Bergman for computing normal forms in the free semigroup algebra  k< X >  with  k  a commutative ring. This involves choosing a reduction system  S  which translates to a generating set for some ideal  I  of  k< X >. By then choosing a suitable semigroup partial order compatible with  S  we can begin analyzing our reduction system for reduction uniqueness, which will guarantee normal forms for our ring of interest

R = k< X > / I .

The analogous case for commutative rings is Gröbner basis theory which starts with  S  as above and then requires the computation of so-called  S-polynomials which will recursively enlarge  S  until it is indeed a Gröbner basis (i.e. when all  S-polynomials have remainder zero with respect to  S ). In this talk I will present several examples for the noncommutative case, including Hopf algebras, and some practical applications to the theory itself.

Abstract: We consider the Zorn vector matrix algebra construction used by Paige to obtain simple, non-associative, Moufang loops. We use this same construction, but over commutative rings with identity instead of fields, and obtain (not necessarily simple) Moufang loops. Then we describe such loops in terms of a loop extension over an abelian group. Such a description gives us insight into the loop's subloop structure. Particular attention is paid to the example created from the commutative ring  Z/p²Z.

Abstract: Hopf algebras can be considered generalizations of groups, and group algebras are basic examples of Hopf algebras. It is well-known that finite group algebras over a field  k  of characteristic zero are semisimple. However, not all semisimple Hopf algebras are isomorphic to group algebras or the duals of group algebras, and not all finite-dimensional Hopf algebras over  k  are semisimple. A finite-dimensional Hopf algebra  H  is called pointed if all simple  H*-modules are one-dimensional, where  H*  is the dual Hopf algebra of  H. Most of the known examples of finite-dimensional non-semisimple Hopf algebras are either pointed or have a pointed dual. In this talk, I will briefly discuss finite-dimensional Hopf algebras and the current state of the classification problems for Hopf algebras of specific dimensions. In particular, I will discuss recent joint work with Richard Ng that completes the classification of all Hopf algebras of dimension  2p²  over  k, for  p  an odd prime.

Abstract: Many applications of algebra, especially in computer science, lead to algebraic structures with two binary associative operations, one of which distributes over the other (as in rings). These structures are called semirings. I will introduce several such classes of semirings, with examples, some information about their algebraic properties, and some information about the relations between the different classes. The main class to be discussed will be the class of idempotent semirings, where each element is an idempotent with respect to each of the basic operations.

Abstract: A new simple Bol loop of order 96 provides a deeper version of the projective line of order 5, much as the noncommutative geometry of Connes offers new views of classical geometric objects. The existence of similar structures for other Fermat primes remains an open problem.