(Recent) Talks Given

Graduate Student Pizza Seminar (Rutgers, Oct. 2006)
Title: Yupana? Quipu? - Incan Abacus and a Portable Planner
Abstract: The Inca led a highly organized and efficient society. Imagine trying to conquer and rule an empire that is over 3000 miles long containing deserts, mountains, and rainforests with no written language and no wheels. How would you do it? And yet, their message delivery system rivals our own. The Inca Empire lasted from about 1438-1533. When the Spanish conquered the region, they destroyed much of the knowledge of Incan mathematics, logic, and record-keeping systems. Hence, much of what I will present is what recent scholars have deduced from a few remaining archeological finds and some colonial chronicles. So, tie a piece of string as a reminder to come; and, for those of you who have trouble sitting still, arm yourselves with maize, paper, pencil, and a clipboard (or other surface).
Constructive Noncommutative Algebra Seminar (Rutgers, Spring 2007)
Title: Irreducible Representations of S_n acting on Q_n
IMR (for incoming graduate students) (Rutgers, Sept. 2007)
Title: Graded Dimensions and Generating Functions
Abstract: The graded dimensions of vector spaces with finite gradings can be expressed in terms of generating functions. We can find these generating functions for various interesting algebras. One reason we would like to find these functions is because by using related techniques we can consider the representations of the automorphism group of an algebra acting on the algebra. In this talk I will describe the basic techniques and apply them to one important example.
Constructive Noncommutative Algebra Seminar (Rutgers, Nov. 2007)
Title: Graded traces and irreducible representations of S_n acting on Q_n
Abstract: The symmetric group S_n acts on the filtered algebra Q_n, and hence on its associated graded algebra. We compute the graded traces of the elements of S_n and then the multiplicities of the irreducible S_n-modules in the homogeneous components of the associated graded algebra.
William Paterson University (Nov. 2007)
Title: Generating Functions and Noncommutative Polynomials
Abstract: Generating functions are used to study sequences and their behavior. Think about them as the next best thing to a closed formula giving the nth term in a sequence. We will first define and give an example of a generating function. Next, we will look at how to factor a noncommutative polynomial. Finally, we will see how generating functions are used to write down the graded dimension of the algebra associated with these factorizations.
Joint Meetings (San Diego, Jan. 2008)
Title: Action of the symmetric group on the universal algebra related to factorization of noncommutative polynomials
Abstract: The algebra Q_n, which arises in the study of factorization of noncommutative polynomials with n independent roots, may be defined in terms of the directed graph Gamma_n=(V_n,E_n) with V_n equal to the subset of all subsets A of {1,...,n} and edges from A to A\{j} for each nonempty subset A in V_n and j in A. To any directed path pi={e_1,...,e_m} in Gamma_n there is a corresponding polynomial P_pi(t)=(1-te_1)...(1-te_m). Then Q_n is the quotient of the free algebra T(E_n) by the relations given by P_{pi_1}(t)=P_{pi_2}(t) where pi_1, pi_2 have the same tail and head. The symmetric group on n elements, S_n, acts naturally on Gamma_n, and so on each of the homogeneous subspaces (Q_n)_{[i]} of Q_n. For sigma in S_n, we compute the graded trace function Tr_sigma(Q_n,t), which equals the sum over i greater than or equal to 0 of the trace of sigma restricted to (Q_n)_{[i]} times t^i, and then use these to obtain the multiplicities of the irreducible S_n-modules in (Q_n)_{[i]}.