Combinatorics/Algebra Seminar
Spring 2018 Archive:

April 23
Alex Nowak: Mendelsohn triple systems and quasigroup representation theory. Abstract:

Quasigroups satisfying both the semisymmetric and idempotent identities, (yx)y = x and x2 = x, respectively, are the algebraic avatars of a class of balanced incomplete block designs known as Mendelsohn triple systems. We describe representations of these Mendelsohn quasigroups via modules over quotients of free group rings. The module theory provides construction techniques, and prompts an extension problem for Mendelsohn quasigroups.

April 16
Shuvra Gupta: Parameter spaces of Galois extensions.

We'll discuss how the notion of a Galois extension of fields can be generalized to a notion of a Galois extension of rings. We then recall the work of Saltman to define a generic Galois extension, which can be thought of as a parameter space of Galois extensions. If we have time, we will then discuss some examples and methods to construct such parameter spaces.

April 9
Josh Zelinsky: On the total number of prime factors of an odd perfect number.

Let N be an odd perfect number. Ochem and Rao showed that if ω(N) is the number of distinct prime factors of N, and that Ω(N) is the number of prime factors of N counting multiplicity then Ω(N) is no less than (18 ω(N) - 31) / 7. We discuss improvements of this inequality, as well as related open problems concerning the behavior of cyclotomic polynomials.

April 2
Christian Roettger: Balanced numbers and symmetric numbers.

In a recent talk on "integer complexity", Josh Zelinsky looked at the binary expansion of simple fractions 1/n, where n is odd. Depending on whether the number of 1's in the period of that expansion is less than, greater than, or equal to the number of 0's, he calls n an efficient, inefficient, or balanced number. While not central to the topic of integer complexity, these concepts raise interesting questions themselves:

  • What is the density of these sets of numbers?
  • What is the density of prime numbers with these properties?
  • Are there easy criteria to determine whether a number is efficient/inefficient or balanced?

One easy criterion is that an odd number n is balanced if - 1 equals a power of 2 modulo n. The converse is true for the case of n prime, but false in general. This motivates us to call an odd number n symmetric if - 1 equals a power of 2 modulo n. Then we can state that every symmetric number is balanced. Again, we can ask the questions above about symmetric numbers and primes. We have some answers to these questions, relying on some all-time classics (Fermat's Little Theorem, Quadratic Reciprocity, and the structure of the multiplicative group modulo n).

March 26
Joanna Meinel (University of Bonn): Computing decomposition numbers using Catalan numbers.

In this talk we consider the Lie algebra sl2(C) and tensor powers of its natural representation: We obtain a formula for the decomposition numbers of such tensor powers that involves Catalan numbers.

March 19
Gerrit Smith: Maximal green sequences of quiver mutations.

The quiver mutations used to define some cluster algebras can be classified as red or green with respect to an initial seed. Some combinatorial aspects of the resulting maximal green sequences (MGS) will be described. One surprising application is a relationship between MGS and (quantum) dilogarithm identities.

March 12
Spring break: no seminar.

March 5
Gene B. Kim (University of Southern California): A central limit theorem for descents in conjugacy classes of  Sn.

The distribution of descents in certain conjugacy classes of  Sn  has been studied previously, and it is shown that its moments have interesting properties. This talk provides a bijective proof of the symmetry of the descents and major indices of matchings and uses a generating function approach to prove a central limit theorem for the number of descents in matchings. We extend this result to all conjugacy classes of  Sn.

February 26
Jonathan Smith: Sylow theory for quasigroups.

We extend Wielandt's treatment of Sylow theory from finite groups to quasigroups. Because of the non-associativity inherent to general quasigroups, the quasigroup version comes in two flavors, based either on the action of the full group generated by the set of left multiplications, or just based on the actions of the left multiplications themselves. The theory offers powerful and readily computed isomorphism invariants, capable, for example, of distinguishing all eighty Steiner triple systems of order fifteen. Part of this work is joint with Michael Kinyon and Petr Vojtechovsky.

January 22, 29; February 5, 12, 19
Candidate talks: no seminar.

January 15
King Day: no seminar.

January 8
Jorge Chavez-Salas: Fuzzy logic fundamentals.

The fundamental concepts and results of fuzzy logic theory will be introduced from an algebraic and logical point of view, and a very simple but original application to human behavior (mathematical psychology or cognitive science in this case) will be presented.

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