Combinatorics, Algebra,
Number Theory Seminar
Spring 2007 Archive:

April 23

James Fiedler: A certain group from a MAXMOLS, many MAXMOLS from a plane.

Abstract: A maximal set of mutually orthogonal latin squares (MAXMOLS) is equivalent to a projective plane, while a pair of orthogonal latin squares is equivalent to a permutation on the set of n² coordinates. Thus we may generate a group from all permuations of the MAXMOLS and compare this to groups from other MAXMOLS of the same plane. Owens (1992) has given a set of transformations to convert any MAXMOLS to any other of the same or dual plane. Owens and Preece (1995) used these transformations to analyze the MAXMOLS for the four planes of order 9. This talk will present these papers, with some information regarding the groups associated with the MAXMOLS.

April 16

Ted Rice: Are pandiagonal quasigroups a variety of quasigroups?

Abstract: Wythoff quasigroups are constructed so that each element on a diagonal is distinct. Wythoff quasigroups can be seen as infinite examples of pandiagonal latin squares. A natural question is whether such a structure can be realized as an algebra. It seems that there ought to be a way to do this. This talk will introduce some basic results on pandiagonal latin squares to serve as background. I will cover some partial results and comment on difficulties towards a complete solution. At the end of the talk I will solicit ideas for future research towards a complete solution.

April 9

Key One Chung: Topological coalgebras.

Abstract: There is a natural way to turn topological spaces into coalgebras for the filter functor. However, naive coalgebra homomorphisms correspond to open continuous maps. In this talk, we show that a suitably weakened homomorphism concept ensures that coalgebra homomorphisms agree with the correct homomorphisms of topological spaces, namely continuous maps. Based on this appropriate relaxation of the concept of coalgebra homomorphism, we prove the equivalence between the usual category of topological spaces and the category of coalgebras obtained from topological spaces.

April 2

Stefko Miklavic [Štefko Miklavič] (University of Primorska/University of Wisconsin): Leonard triples and hypercubes.

Abstract: Let V denote a finite-dimensional vector space over the field of complex numbers. By a Leonard triple on V we mean an ordered triple of linear operators on V such that for each of these operators there exists a basis of V with respect to which the matrix representing that operator is diagonal, and the matrices representing the other two operators are irreducible tridiagonal. In this talk I will describe a situation in graph theory where Leonard triples arise naturally.

March 26

William Gasarch (University of Minnesota, Duluth):
If you 2-color the lattice points in the plane then ...

Abstract:

1) For all 2-colorings of the lattice points of the plane there there will be a rectangle (parallel to both axes) such that all four corners are the same color. (This could be on a High School Math Competition, though there are some interesting open questions that arise.)

2) For all 2-colorings of the lattice points of the plane there there will be a square (parallel to both axes) such that all four corners are the same color. (This can be derived from Galai's theorem, though it also has a direct proof reminiscent of the proof of van der Waerden's theorem.)

3) For all 2-colorings of the lattice points of the plane there will be a rectangle of dimensions d by d ² for some integer d such that all four corners are the same color. (This can be proven from the Polynomial Hales Jewitt theorem. Elementary, very interesting, but difficult. Harder than (2) above.)

These are all example of theorems in Ramsey theory that get harder and harder. The last one only used to have a proof using ergodic theory, but now has an elementary proof. I will present proofs of the first two and then talk about the proof of the third.

Disclaimer: None of the results are new or mine. However, the fact that the third one can be proven using elementary techniques might be my observation.

March 19

Sungyell Song: Wreath products of association schemes and the subconstituent algebra.

Abstract: We will recall the subconstituent algebra (or Terwilliger algebra) of a symmetric association scheme and the wreath product of symmetric association schemes. Then we will describe the subconstituent algebras T for several association schemes that are obtained as wreath powers of a class of symmetric association schemes, and discuss their properties.

March 12

Spring break: no seminar.

March 5

Bin Zhang (Sichuan University): Multiple zeta values at negative arguments.

Abstract: This is joint work with Li Guo (Rutgers, Newark). Multiple zeta values (MZVs) in the usual sense are the special values of multiple variable zeta functions at positive integers. Their extensive studies are important in both mathematics and physics with broad connections and applications. We define and study multiple zeta functions at integer values by adapting methods of renormalization from quantum field theory, and following the Hopf algebra approach of Connes and Kreimer. This definition of renormalized MZVs agrees with the convergent MZVs and extends the work of Ihara-Kaneko-Zagier on renormalization of MZVs with positive arguments. We further show that the important quasi-shuffle (stuffle) relation for usual MZVs remains true for the renormalized MZVs.

February 26

Ales Drapal (Charles University/University of Wisconsin):
Buchsteiner loops.

Abstract: Buchsteiner loops are defined by the identity

xy.z = x((yz.x)/x).

If Q is a Buchsteiner loop and N is the nucleus of Q, then Q/N is an abelian group of exponent four. If Q really contains an element of order 4 modulo N, then Q has be of order at least 64, and such Buchsteiner loops really exist. All Buchsteiner loops are G-loops, and a conjugacy closed loop is a Buchsteiner loop if and only if it is boolean modulo the nucleus. All Buchsteiner loops of nilpotency class two are conjugacy closed, and 32 is the least order for Buchsteiner loops that are not conjugacy closed. All such loops have been classified. There exists a Buchsteiner loop of order 128 that is of nilpotency class three, and has an abelian inner mapping group.

February 19

Andy Regenscheid: Hashing in SL(2).

Abstract: Hash functions are cryptographic primitives which construct short fingerprints of data. Most hash functions to date operate by performing complex iterations with little mathematical structure. One notable exception is a hashing scheme first proposed by Tillich and Zemor that operates in the group of 2x2 matrices with determinant 1 over a finite field. I will introduce this hashing scheme and describe its desirable and undesirable properties as a hash function. The mathematical structure leads to several attacks that make it possible to find collisions, or two inputs that have the same output.

February 12

Department colloquium: no seminar.

February 5

Tong Liu (University of Pennsylvania):
Finite flat representations and finite flat group schemes.

Abstract: In this talk, we discuss finite flat group schemes over rings of algebraic integers and their associated finite flat representations of Galois groups. Typical examples of such kind are those representations arising from torsion points of elliptic curves or Abelian varieties over number fields. However, these complicated geometric objects can be understood by studying much easier linear algebra data: Kisin modules, which allows us to establish an equivalence between the category of finite flat representations and the category of maximal finite flat group schemes.

January 29

Anna Romanowska (Warsaw University of Technology):
Differential modes.

Abstract: A differential groupoid (G, . ) is a set G with a binary multiplication x . y that is a mode (i.e. the multiplication is idempotent and entropic), and additionally satisfies the identity

x . (x . y) = x.

The main models occur as certain reducts of modules over the dual numbers (differential groups in Mac Lane's terminology). Such groupoids, and some of their generalizations, play an essential role in the representation and classification theory of modes. In particular, a generalization to n-differential groupoids -- groupoid modes defined by the identity

x . (x . (....(x . y)...)) = x

-- provides a basic tool for a classification of all groupoid mode varieties. On the other hand, a generalization to the ternary (and n-ary for n > 2) case provides an answer to the question of representing modes as subreducts (subalgebras of reducts) of semimodules.

January 22

Jonathan Smith: Defining quasigroups (click for slide show).

Abstract: The usual definitions of quasigroups are rather complicated, either by combinatorial conditions amounting to Latin squares, or by four identities on three binary operations. For n-ary quasigroups, matters are even worse. In this talk, a more symmetrical approach to the specification of quasigroups and n-quasigroups is considered.

January 15

King Holiday: no seminar.

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