 ## Combinatorics/Algebra Seminar:Spring 2004 Archive

April 26
Jonathan Smith: Counting conjugacy classes.

Abstract: To what extent can counts of conjugacy classes provide structural information about groups and quasigroups? The number of conjugacy classes is called the rank of the quasigroup. A random quasigroup has rank two. For further studies, the most appropriate counting of conjugacy classes is provided by the entropy
H(Q) = - S1 ≤ ir (ni / n) log (ni / n)
of the partition of the set of n elements of a quasigroup or group Q into conjugacy classes of size ni , i = 1, ..., r. Now
log n - (1 - n - 1) log (n - 1) H(Q) log n .
Equality holds on the left if and only if Q has rank two. Equality holds on the right if and only if Q is abelian. For groups, it is conjectured that a simple group minimizes the entropy. For quasigroups, there is a basic structural trichotomy:
abelian / central non-abelian / non-central .
To separate the three classes of this trichotomy, one needs the asymptotic entropy h(Q), the lim sup of 1/t times the entropy of the power Qt as t tends to infinity. Then Q is central if and only if the equality
h(Q) = log n
holds.

April 19
Mandi Maxwell: A classification of all nonlinear power maps over GF(2n) with low uniformity for certain values of n.

Abstract: A power map f (x) = xd over GF( pn ) is said to be differentially k-uniform if k is the maximum number of solutions x e GF( pn ) of f (x + a) - f (x) where a, b e GF( pn ) and a is non-zero. A 1-uniform map is called perfect nonlinear, a 2-uniform map is called APN (almost perfect nonlinear). We collect and classify all binary k-uniform power maps up to n = 11 and we will discuss other infinite families of functions with low uniformity and some open problems in this area.

April 12
Faculty meeting: no seminar.
April 5
Ted Rice: Greedy quasigroups and combinatorial games.

Abstract: Greedy quasigroups arose out of a desire to better understand certain combinatorial games. I will discuss some basic combinatorial game theory to provide motivation, and quasigroup theory as background information. Greedy quasigroups have remarkable algebraic properties. In particular, I will answer the question of the existence of subquasigroups and isomorphism classes of greedy quasigroups.

March 22, 29
Benard Kivunge: Sedenion subloops.
March 15
Spring Break: no seminar.
March 8
Richard Ng: Algebraic structures of linearly recursive sequences.

Abstract: For each primitive n-th root q of unity in a field k, one can construct an algebra Lq on the space of linearly recursive sequences L of k. These algebras are indeed Hopf algebras in a certain context. We will discuss the construction of these algebras and their relations to Lq. We will also talk about the characterization of the units of these algebras.

March 1
Ling Long: Representing natural numbers as sums of integer squares.

Abstract: Representing natural numbers as sums of integer squares is a question with a long history. For example, it is known by Jacobi that for any odd prime number p there are
8 ( p + 1 )
ways to represent p as the sum of 4 integer squares. In this talk we are going to present simple proofs of some theorems in this direction (like Jacobi's result) by using Hecke operators, a standard tool in the theory of modular forms.

February 23
Christian Roettger: Periodic points in Markov shifts, II.

Abstract: In the last talk on this topic, we considered the space X of doubly-infinite, doubly-indexed sequences over a finite abelian group G, subject to the condition that any entry equals the sum of the one below and below-right of it. We considered the conjecture that G is determined up to isomorphism by the ensemble of U-periodic points, where U ranges over all subgroups of Z2 of finite index ( Z2 operates on the space X by shifting left and upward). Now we can prove this conjecture without any restriction on G. We will review the setup and then explain the new ingredient, namely Teichmüller systems.

February 9, 16
Cliff Bergman: Stream ciphers in general and the alternating-step generator in particular.

February 2
Anna Romanowska: Congruences on dyadic simplices.

Abstract: We study the geometry of simplices defined over dyadic rationals instead of over the reals. The resulting combinatorial topology involves number-theoretical issues in parallel with the geometry.

January 26
Jonathan Smith: New developments with octonions and sedenions. Extended abstract.

January 19
King Holiday: no seminar.

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