Bokhee Im (Chonnam National University):
Number Theory Seminar
Fall 2007 Archive:
Representations of trilinear products in comtrans algebras.
Unlike the set of all Lie algebras, the set of all comtrans algebras on
a given module has a linear structure. Let E be a finite-dimensional
vector space over a field k. Then we want to determine which trilinear
products xyz on E may be represented as linear combinations of the
commutator and translator of a comtrans algebra on E in the manner
of the so-called "bogus product" xyz =
x, y, z
] + (1/6) [
y, z, x
] + (1/6) [
z, x, y
If the underlying field is not of characteristic 3, then we show that the
necessary and sufficient condition for such a representation is
+ (1/3) <
x, y, z
> + (1/3)<
z, x, y
xxy + xyx + yxx = 0 ,
a condition described as strong alternativity. Indeed, if the underlying
field is also not of characteristic 2, then each strongly alternative trilinear product is represented as the bogus product of a comtrans algebra.
An appropriate representation for the case of characteristic 2 will also
Functorial properties of Zorn's construction.
Thanksgiving break: no seminar.
Greco-Latin squares as bijections.
Note: This seminar will form the public part of James Fiedler's Ph.D. defence.
An application of Lie groups to Ramanujan's differential equations for Eisenstein series.
Abstract: We outline a new proof of the parametric representations for the classical Eisenstein series in terms of the complete elliptic integral of the first kind. The derivation given here is distinguished from existing proofs in that we avoid the Jacobi-Ramanujan inversion formula relating theta functions and hypergeometric series. Our approach relies instead on the differential equations satisfied by certain Eisenstein series. We construct a group of transformations under which these equations are invariant. This analysis results in a convenient change of variables, transforming the system into one that can be solved via elementary techniques. If time permits, we will also see how to similarly derive parameterizations from Ramanujan's alternative signatures.
Some existence and construction results for polygonal designs.
Abstract: We revisit the existence and construction problems for polygonal designs (a special class of partially balanced incomplete block designs associated with regular polygons). We present new polygonal designs with various parameter sets by explicit construction. In doing so, we employ several construction methods -- some conventional, and some new. We also establish a link between a class of polygonal designs of block size 3 and the cyclically generated "lambda-fold triple systems." Finally we show that the existence question for a certain class of polygonal designs is equivalent to the existence question for "perfect grouping systems." which we introduce.
Consequences of the Product Rule for derivations (concluded).
Terwilliger algebras in wreath products of association schemes.
Abstract: We will discuss the Terwilliger algebras T of association schemes obtained from wreath powers of the one-class association schemes H(1,q). We compute the dimension of the Terwilliger algebra. For a wreath power of copies of H(1,2), we discuss the irreducible modules, and show that T is isomorphic to a direct sum of complex matrix algebras.
Chris Godsil (University of Waterloo):
Algebras, automorphisms and isomorphisms.
Consequences of the Product Rule for derivations.
Fermat curves and modular curves.
Abstract: For any fixed positive integers n, a Fermat curve of degree n is a complex curve consisting of (x,y,z) in the projective plane satisfying
Invertible and nilpotent elements of quaternion algebras.
Anna Romanowska (Warsaw University of Technology):
xn + yn = zn.
Finite index subgroups of SL(2,Z) act on the upper half complex plane H through linear fractional transformation. The compactified fundamental domains of these groups on H are called modular curves. In this talk, we will study the relations between Fermat curves and modular curves.
Differential modes and embedding into semimodules.
Abstract: It is well known that each idempotent and entropic groupoid, i.e. each groupoid mode, embeds as a subreduct into a semimodule over a commutative semiring. Surprisingly, this is no longer true for modes with operations of larger arity. As shown by M. Stronkowski, a mode embeds as a subreduct into a semimodule over a commutative semiring if and only if it satisfies certain identities pointed out by A. Szendrei. Stronkowski also constructed a mode not satisfying such identities. A simple 3-element example was then provided by D. Stanovsky. We will show that Stanovsky's example belongs to a class of modes that form a ternary counterpart of differential groupoids, containing a broad and interesting class of modes that are not embeddable into semimodules.
Labor Day: no seminar.
Simple Bol loops.
Abstract: Following the classification of finite simple groups, M. Liebeck classified the finite simple Moufang loops. Except for the groups, there is only one family, based on split octonions of norm 1 over finite fields. The original construction is due to L. Paige. Their character tables were determined by S.Y. Song.
After the Moufang identity, the next step from associativity is the Bol identity. For a long time, the existence of finite, simple, non-Moufang Bol loops was an open problem. Then, in Spring 2007, G. Nagy discovered finite simple Bol loops coming from group factorizations.
In this talk (joint work with K.W. Johnson), a more general and conceptual version of G. Nagy's construction will be presented, along with some new finite simple Bol loops. The approach is based on M. Takeuchi's concept of a matched pair of groups, related to matched pairs of Hopf algebras in the sense of W.M. Singer.
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