
Combinatorics, Algebra, Number Theory Seminar Fall 2008 Archive:
December 8
Chris Kurth: Modular
forms for some noncongruence subgroups of SL(2,Z) .
December 1
David Failing: Affinely selfgenerating sets and substitution sequences
Abstract: Kimberling defined a selfgenerating 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 selfgenerating 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.
November 24
Thanksgiving break: no seminar.
November 3, 17
SiuHung (Richard) Ng: Congruence subgroups and generalized FrobeniusSchur indicators.
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 FrobeniusSchur exponent of C. The definition of generalized FrobeniusSchur 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, FrobeniusSchur exponents and associated modular representations. We will discuss the definition of generalized FrobeniusSchur indicators and their relation to the modular representations.
October 27
Ryan Martin: The edit distance in graphs.
October 20
Andrew Wells and Jonathan Smith: Report from the AMS Sectional Meeting, Kalamazoo, Michigan.
October 13
John Gillespie:
Applying the Diamond Lemma in ring theory.
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 socalled Spolynomials which will recursively enlarge S until it is indeed a Gröbner basis (i.e. when all Spolynomials 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.
October 6
Andrew Wells: The structure of Moufang loops arising from Zorn vector matrix algebras.
Abstract: We consider the Zorn vector matrix algebra construction used by Paige to obtain simple, nonassociative, 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/p^{2}Z.
September 29
Mike Hilgemann: Classifying finitedimensional Hopf algebras.
Abstract: Hopf algebras can be considered generalizations of groups, and group algebras are basic examples of Hopf algebras. It is wellknown 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 finitedimensional Hopf algebras over k are semisimple. A finitedimensional Hopf algebra H is called pointed if all simple H*modules are onedimensional, where H* is the dual Hopf algebra of H. Most of the known examples of finitedimensional nonsemisimple Hopf algebras are either pointed or have a pointed dual. In this talk, I will briefly discuss finitedimensional 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 2p^{2} over k, for p an odd prime.
September 22
Anna Romanowska (ISU/Warsaw University of Technology): Idempotent semirings.
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.
September 8, 15
Jonathan Smith:
The nonassociative projective line of order 5.
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.
September 1
Labor Day: no seminar.
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