Combinatorics, Algebra,
Number Theory Seminar
Spring 2008 Archive:

April 28
Bokhee Im (ISU/Chonnam National University): Directed graphs, intercalate changes, and sharply transitive quasigroup actions.
Abstract: We construct two nontrivial sharply transitive sets of quasigroup actions. The constructions take place within the context of certain double-cover quasigroups for the group of symmetries of the equilateral triangle. These quasigroups are specified by directed graphs using the group as vertex set. Directed edges in the graph determine intercalate changes within the Latin square multiplication table of the quasigroup.
April 21
George Andrews (Pennsylvania State University): Old and new thoughts on the Rogers-Ramanujan identities.
Abstract: The Rogers-Ramanujan identities first came to prominence in the early part of the 20th century, when Ramanujan conjectured them, and later they were found fully proved in an old and very much neglected paper by L.J. Rogers. In the talk, we shall provide an account of this history and what has happened subsequently, and will conclude with a look at recent discoveries and ongoing research.
April 14
John Gillespie: The Diamond Lemma in ring theory.
April 7
Zhu Cao (University of Illinois Urbana-Champaign): Product identities for theta functions.
Abstract: We derive a general method for establishing  q-series identities. Using this method, products of theta functions can be written as linear combinations of other products of theta functions . Many known identities can be shown to be special cases of this general formula. Several entries in Ramanujan's notebooks, as well as new identities, are obtained as applications. Next we give several generalized forms of Schröter's formula. A general identity by W. Chu and Q.Yan, and the Blecksmith-Brillhart-Gerst theorem, are both special cases of our generalized Schröter formula.
March 31
Ae Ja Yee (Pennsylvania State University): Lecture hall partitions.
Abstract: Lecture hall partitions are partitions whose parts satisfy a certain ratio condition. Their enumeration by Bousquet-Melou and Eriksson gives a finite version of a theorem of Euler on strict partitions. In this talk, I will discuss a generalization of Euler's theorem and some open questions. This is joint work with Carla Savage from North Carolina State University.
March 24
Sungyell Song: Normally regular digraphs, Cayley graphs, and their automorphism groups.
Abstract: A digraph is called normally regular if its adjacency matrix A is normal, (i.e., AA' = A'A), and the product AA' can be expressed as a linear combination of A, A', and J - I - A - A'. (Here J is all-one matrix and I is the identity matrix.) Many normally regular digraphs arise from Cayley graphs and group rings. In this talk I will explain the necessary conditions on the connection set S under which the Cayley graph
Cay(G, S)
of a group G becomes a normally regular digraph. I then reformulate the conditions in terms of group rings. If time permits, I will discuss a few Cayley graphs and their generating sets that are isomorphic to a special type of normally regular tournament. Some of the generating sets for such Cayley graphs give rise to a certain class of relative difference sets.
March 17
Spring break: no seminar.
March 10
Chris Kurth: Farey symbols and modular symbols.
Abstract: PSL2(Z) is the group of integer-valued 2x2 matrices of determinant 1, modulo {I,-I}. Farey symbols are a method of doing computations in subgroups of PSL2(Z). In this talk we discuss Farey symbols and an algorithm for computing modular symbols from them, allowing one to calculate Fourier series for modular forms of finite-index subgroups of PSL2(Z).
March 3
Tim Huber: New  q-Euler numbers arising from an expansion in Ramanujan's Lost Notebook.
Abstract: The Hadamard product for a generalization of the Rogers-Ramanujan series appears on page 57 of Ramanujan's Lost Notebook. Coefficients appearing in series expansions for the zeros of Ramanujan's function have interesting analytic and combinatorial properties. This lecture will demonstrate connections between these coefficients and a new class of  q-tangent numbers associated with geometrically distributed random variables.
February 25
Mike Hilgemann: Quasitriangular bialgebras and solutions of the Yang-Baxter Equation.
Abstract: This talk examines the notion of quasitriangular bialgebras and their relation to the Quantum Yang-Baxter Equation. In particular, we show how the universal  R-matrix of a quasitriangular bialgebra  B  gives rise to a braiding on its category of modules, and hence provides a solution to the Quantum Yang-Baxter Equation in each  B-module.
February 18
Siu-Hung (Richard) Ng: Group cohomology and quasi-Hopf algebras.
Abstract: We discuss a construction of quasi-Hopf algebras using finite groups and their 3-cocycles. In particular, if  G  is a finite abelian group of group-like elememts in the center of a Hopf algebra  H, one can use a 3-cocycle on  G  to turn  H  into a quasi-Hopf algebra. We show that each of the two eight-dimensional noncommutative group algebras over  C  can be obtained from the other in this way.
February 11
Anna Romanowska (ISU/Warsaw University of Technology): Differential modes embeddable into semimodules.
Abstract: In my talk last September, I discussed the problem of embedding modes (idempotent and entropic algebras) into semimodules over commutative semirings. In particular, I showed that the class of ternary differential modes contains:
  • a subclass of algebras that embed into modules;
  • a larger class of algebras that embed into semimodules, and finally
  • a subclass of non-embeddable modes.
This talk will focus on differential modes embeddable into semimodules, and will provide more information about their structure and varieties.
February 4
Jonathan Smith: Groups and hyperquasigroups.
Abstract: A hyperquasigroup  ( Q . G )  is a structure consisting of a reflexion-inversion space  G, a set equipped with an involutive reflexion map  S  and an involutive inversion map  T , together with a binary action  x y g of each point  g  of  G  on  Q , such that the hypercommutativity identity
x y Sg = y x g
and hypercancellativity identity
x ( x y g ) Tg= y
hold for all  x, y  in  Q  and  g  in  G . Hyperquasigroups provide a more symmetrical approach to the theory of quasigroups. This talk examines some connections between groups and hyperquasigroups. In particular, it is shown that certain linear hyperquasigroups are equivalent to group representations. In other words, the theory of group representations is subsumed in the theory of hyperquasigroups.
January 28
Mehmet Haluk Sengun (University of Wisconsin-Madison):
The abstract Galois group of quadratic fields.
Abstract: Let  K  be a quadratic field. We prove that there is no irreducible continuous mod p  representation of the absolute Galois group of  K  that is unramified away from  p  and infinity. We discuss Serre's conjecture and its generalizations as a motivation of our result.
January 21
King holiday: no seminar.

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