Combinatorics/Algebra Seminar
Spring 2017 Archive:

April 24
Christian Roettger: Galois Theory and the Ring of Sauron.

Consider a polynomial f(x) over a field k with degree n and discriminant D. The Galois group G of f(x) can be viewed a a subgroup of the symmetric group Sn via its action on the roots of f(x).

It is a well-known application of Galois Theory that G is contained in the alternating group An exactly if D is a square in the ground field k. Moreover, if this is not the case, then D½ generates the fixed field of the intersection of G with An.

We are looking for formulas that generalize this - i.e., for any transitive subgroup H of Sn, find expressions in the coefficients of a generic polynomial of degree n which are in the ground field exactly if G is contained in H, and which generate the fixed field of the intersection of G and H otherwise. At first, it is not even clear that such expressions have to exist. But they do, and they in principle provide a way to determine G. Practical computations are lengthy and best done by computer, but the theory is very appealing and natural. In particular, we use multivariate polynomial rings and classical representation theory of finite groups (in a very basic way).

This is joint work with John Gillespie.

April 17
Jonas Hartwig: The Heisenberg vertex operator algebra.

In the mid 1980's the mathematical notion of a vertex operator algebra (VOA) coalesced. This gave birth to a branch of mathematical physics displaying extraordinarily rich algebraic structures with deep connections to representation theory, the monster group, modular forms and many other exceptional objects, as well as providing a rigorous framework for developing conformal field theory in physics. In this introductory talk (2MB), my goal is to give the definition of a VOA and then consider an example, the Heisenberg VOA.

April 10
Ken Johnson (Penn State): Random walks which converge after a finite number of steps.
Click here for PDF abstract
April 3
Ken Johnson (Penn State): Gelfand-Tzetlin bases, Jucys-Murphy elements and the relationship to association schemes.
Click here for PDF abstract
March 27
Stefanie Wang: On free quasigroups and quasigroup representations.

The n-th Catalan number gives the number of ways to assign parentheses to a single nonassociative binary operation. A quasigroup is an algebra equipped with three nonassociative binary operations of multiplication, right division, and left division. The n-th peri-Catalan number gives the number of length n inequivalent quasigroup words in a single generator. This talk will provide some background and motivation for the study of peri-Catalan numbers. We will derive a recursive formula for the n-th peri-Catalan number.

March 20
Erich Jauch: Galois embedding.

In the study of finite field extensions, knowledge of Galois extensions is fundamental. We are generally interested in finding which extensions of a given base field K are Galois. In general, if M is Galois over K and L is Galois over M, then L is not necessarily Galois over K. In other words being Galois is not generally a transitive relation. During this seminar, we will focus on the idea of Galois embedding, or extending Galois extensions into new Galois extensions that remain Galois over the base field. We will focus in particular on quadratic extensions, and we will use the field of rational numbers for most of our concrete examples. We will use Kummer Theory to prove most of our results, and we will finish with a nice generalization by Albert [1935] about cyclic extensions.

March 13
Spring break: no seminar.

March 6
Tathagata Basak: Braid-like groups.

Braid-like groups generalize the classical braid group. We will give a set of generators and give some relations for one particular braid-like group. This braid-like group is obtained from the reflection group studied in detail in the previous talk. We study this braid-like group because it seems to be related to a rather large and interesting finite simple group. This is joint work with Daniel Allcock.

February 27
Tathagata Basak: Reflection groups.

We will talk about real and complex reflection groups. Examples of finite real reflection groups are symmetry groups of platonic solids. Many reflection groups appear as symmetry groups of lattices, and we will study them in this form. In particular, we will study in detail one example related to the densest sphere packing in 24 dimensions.

February 20
Dani Szpruch (Howard University): On the Shahidi local coefficients matrix.

The Langlands-Shahidi method is one of the two main approaches for defining and studying automorphic L-functions. The core of this method is a meromorphic invariant associated with representations of quasi-split reductive groups defined over local fields. This meromorphic invariant arises from a uniqueness result known as the uniqueness of the Whittaker model. Among its local applications, one finds irreducibility results and a formula for Plancherel measures. In the context of metaplectic groups, which are non-linear covering groups, uniqueness of the Whittaker model no longer holds. Nevertheless, an analog of the invariant does exist. In the talk we will give a new and simple interpretation to this analog for coverings of p-adic SL(2). We will also give local applications. The talk should be accessible to non-experts.

February 13
Jonathan Smith: Duality for lattices and quasilattices.

The talk will first review the lattice duality of Hartonas and Dunn, which represents a lattice as a Galois connection between its meet and join semilattices. Lattices appear naturally in data analysis as the structures of Hardegree's notion of a natural kind, or Wille's concepts. When extending these ideas to the analysis of complex data arising from systems with distinct levels (as in mathematical biology, for example), one is then led naturally to the idea of a quasilattice as an ordered system of lattices. The second part of the talk will review the duality theory for quasilattices recently developed in joint work with Anna Romanowska.

February 6
Jonathan Smith: Duality for semilattice Galois connections.

Semilattice Galois connections underlie a large class of optimization algorithms. This talk will present the duality for semilattice Galois connections that is due to Hartonas and Dunn. In their version, the dual objects were taken as polarities, i.e., relations between sets. However, although there are times when the relational language of polarities is appropriate, there are other times when an equivalent but more algebraic concept of a pairing is to be preferred. Polarities and pairings are fundamental to Hardegree's notion of a natural kind, or Wille's concepts, which have now become basic tools in the analysis of big data.

January 30
Cliff Bergman: Probabilities of finite algebras.

We present a simple probability measure on the space of finite algebras of a fixed similarity type. We shall use this to describe a striking result of V. L. Murskii, as well as some related results.

January 23
Jiali Li: Congruence n-permutable varieties.

Many experts have been doing research on characterizations of congruence n-permutable varieties in many different ways. In 1973 Hagemann and Mitschke, generalizing Maltsev conditions, provided a simple and nice characterization of congruence n-permutable varieties. We offer our own approach to the characterization of congruence n-permutable varieties, inspired by the Kearnes/Tschantz lemma.

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