Combinatorics/Algebra Seminar
Fall 2018 Archive:

November 26

Josh Zelinsky: On the smallest and largest prime divisors of an odd perfect number.

Abstract:

Let N be an odd perfect number with N = p₁^a₁p₂^a₂ ... p_k^a_k with the p_i distinct primes. Assume further that p_i < p₂ ... < p_k. We discuss inequalities relating N, k and p_i when either i is small or when i is close to k. We will discuss related avenues for future research including open problems which may be suitable for investigation by advanced undergraduates or beginning graduate students.

November 19

Thanksgiving Break: no seminar.

November 12

Cliff Bergman: From Plonka sums to Maltsev products.

Abstract:

The Maltsev product is a construction first explored for groups by Hanna Neumann, and then generalized to universal algebra by A. I. Maltsev. This is an intriguing object that has largely resisted analysis. We know few necessary or sufficient conditions for the product to be closed under homomorphic images. We know nothing about the axiomatization, etc. As a starting point, we will consider a special case: the Maltsev product of varieties A and B, in which A is defined by a single identity of the form t(x,y) = x, and B is the variety of semilattices. The Maltsev product will contain all Plonka sums of algebras from A, but of course, much more. We prove that the product is again a variety, and we provide an equational base. Joint work with Anna Romanowska.

November 5

Erich Jauch: Galois orders.

Abstract:

Galois rings and Galois orders form a class of algebras that contain many important examples including generalized Weyl algebras, U(gl_n), and shifted yangians. I will discuss ways to construct such objects in skew monoid rings, and time permitting prove that U(gl_n) is a Galois order.

October 29

Gavin Nop: Concepts of quasilattice completeness.

Abstract:

We will examine a type of generalization of completeness from lattices to quasilattices. First, we will start with some introductory material on lattices, joins, meets, and various theorems for completeness on lattices. Then we will introduce quasilattices starting with an algebraic definition, and outlining the intuitions behind it with Plonka's Theorem. Finally, we will introduce the notions of local completeness and global completeness for quasilattices. Some results and examples will be given, tying these new notions of completeness back to completeness on lattices.

October 15, 22

Joey Iverson: Doubly transitive lines and roux.

Abstract:

Lines (one-dimensional subspaces) with wide angles between them are desirable in many applications. In an "optimal" line packing, the minimum angle is made as large as possible. Optimal line packings often display remarkable symmetry. In particular, lines with a 2-transitive automorphism group are automatically optimal. In this talk, we develop machinery to construct such doubly transitive lines directly from 2-transitive permutation groups. This leads to a partial classification of doubly transitive lines.

In addition, we introduce a combinatorial generalization of doubly transitive lines called roux. These can be viewed variously as matrices over a group ring, coverings of the complete graph, or commutative association schemes. Every roux produces an optimal line packing, and the resulting class includes all doubly transitive lines, all real equiangular tight frames, and all lines made by abelian distance-regular antipodal covers of the complete graph (DRACKNs).

This is joint work with Dustin Mixon.

October 8

Alex Nowak: Representation theory for certain combinatorial designs.

Abstract:

In combinatorial design theory, Mendelsohn triple systems (sometimes referred to as cyclic triple systems) generalize Steiner triple systems, and in universal algebra, quasigroups are the so-called nonassociative groups. Mendelsohn triple systems may be realized as idempotent, semisymmetric quasigroups, which are quasigroups satisfying the identities x² = x and y(xy) = x. Exploiting this universal algebraic description, I will present a theory of modules over Mendelsohn triple systems. As in the group case, the module theory for these nonassociative algebras prompts questions regarding extension. I will address some of these extension problems, and if time permits, I will also discuss the interplay between representations of Steiner and Mendelsohn triple systems.

October 1

Hal Schenck: Wachspress varieties: algebraic geometry meets numerical analysis.

Abstract:

Let P_d be a convex polygon with d vertices. The associated Wachspress surface W_d is a fundamental object in approximation theory, defined as the image of the rational map w_d from P² to P^{d-1}, determined by the Wachspress barycentric coordinates for P_d. We show w_d is a regular map on a blowup X_d of P², and if d > 4 is given by a very ample divisor on X_d, so has a smooth image W_d. We determine generators for the ideal of W_d, and prove that in graded lex order, the initial ideal of I(W_d) is given by a Stanley-Reisner ideal. As a consequence, we show that the associated surface is arithmetically Cohen-Macaulay, of Castelnuovo-Mumford regularity two, and determine all the graded Betti numbers of I(W_d).

September 24

Anna Romanowska (Warsaw University of Technology): Extensions of convex sets.

Abstract:

Convex sets may be viewed as abstract algebras, sets closed under the set of binary convex combinations indexed by the open unit interval of real numbers. Such algebras appear inside affine real spaces viewed as abstract algebras, sets equipped with binary affine combinations. In this talk, I will discuss certain algebraic extensions of the concept of a convex set. Such extensions take (ordered) subfields (or subrings) of the field of real numbers, and consider different intervals within such rings.

September 10, 17

Jonathan Smith: Character groups and quasigroups.

Abstract:

Duality for finite abelian groups, which includes discrete and fast Fourier transforms, provides a group structure for the multiplication of characters of a finite abelian group. In these talks, we will discuss a new approach to aspects of the representation theory of finite nonabelian groups, introducing character groups and quasigroups which encode the decomposition of the product of irreducible characters as sums of irreducible characters.

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