Centre for Research in Mathematics

Computing with finite semigroups

James Mitchell (University of St Andrews)
Friday, 5 July 2013, 11:00-12:30

SCEM AccessGrid (KWD-Y.2.39, PTA-EB.1.32, CTN-26.1.50) presented from Parramatta

In this talk I will discuss the problem of how to compute a finite semigroup. What does it mean to 'compute' a finite semigroup? It means to find structural information about that semigroup, for example, calculating their Green's classes, size, elements, group of units, minimal ideal, small generating sets, testing membership, finding the inverses of a regular element, factorizing elements over the generators, and so on.

I will review what is known about computing with finite semigroups and give an overview of recently developed functionality in the computer algebra system GAP. No prior knowledge of computing or of semigroup theory will be assumed. Time permitting, I will demonstrate the Semigroups software package for computing with semigroups of transformations and partial permutations, and describe some recent theoretical advances that will allow the methods in Semigroups to be applied to several other natural types of semigroup including monoids of partitions.

The lattice of subsemigroups of the semigroup of all mappings on an infinite set

James Mitchell (University of St Andrews)
Friday, 5 July 2013, 14:00-13:30

SCEM AccessGrid (KWD-Y.2.39, PTA-EB.1.32, CTN-26.1.50) presented from Parramatta

In this talk I will review some recent results relating to the lattice of subgroups of the symmetric group and its semigroup theoretic counterpart, the lattice of subsemigroups of the full transformation semigroup on an infinite set. As might be expected, these lattices are extremely complicated.

I will discuss several results that make this comment more precise, and shed light on the maximal proper sub(semi)groups in the lattice. I will also discuss a natural related partial order, introduced by Bergman and Shelah, which is obtained by restricting the type of sub(semi)groups and considering classes of, rather than individual, (semi)groups. In the case of the symmetric group, this order is very simple but in the case of the full transformation semigroup it is again very complex.

Association and aggregation for contingency table analysis

Eric Beh (University of Newcastle)
Monday, 15 April 2013, 15:30-16:30

SCEM AccessGrid (KWD-Y.2.39, PTA-EB.1.32, CTN-26.1.50) presented from Parramatta

For reasons of confidentiality, government departments and other agencies often release information to the public that has been aggregated.  When working with aggregated data it is important to ensure that one keeps in mind any loss of information (due to the aggregation process) when making conclusions at the individual level. There are a variety of techniques that exist which analyse aggregate data and these generally lie within the family of ecological inference techniques.

This presentation is not about those techniques.  Rather, we shall consider a very simple illustrative example to highlight the loss of information due to aggregation and show how correspondence analysis can be used as a means of visually identifying the source of such loss.

This presentation will explore the impact of aggregation using the standard statistical techniques of simple linear regression and the chi-squared test of independence.

Adventures with group theory: counting and constructing polynomial invariants for applications in quantum entanglement and molecular phylogenetics [or: the Power of Plethysm]

Peter Jarvis (University of Tasmania)
Monday, 21 January 2013, 15:00-16:00

SCEM AccessGrid (KWD-Y.2.39, PTA-EB.1.32, CTN-26.1.50) presented from Parramatta

In many modelling problems in mathematics and physics, a standard challenge is dealing with several repeated instances of a system under study.  If linear transformations are involved, then the mathematical machinery of tensor products steps in, and it is the job of group theory to control how the relevant symmetries lift from a single system, to having many copies.  At the level of group characters, the construction which does this is called PLETHYSM.

In this talk all this will be contextualised via two case studies: entanglement invariants for multipartite quantum systems, and Markov invariants for tree reconstruction in molecular phylogenetics.  By the end of the talk, listeners will have understood why Alice, Bob and Charlie love Cayley's hyperdeterminant, and they will know why the three squangles -- polynomial beasts of degree 5 in 256 variables, with a modest 50,000 terms or so -- can tell us a lot about quartet trees!

Higher homotopy, higher groupoids

Steve Lack (Macquarie University)
Monday, 15 October 2012, 15:30-16:30

SCEM AccessGrid (KWD-Y.2.39, PTA-EB.1.32, CTN-26.1.50) presented from Parramatta

Topology is a branch of geometry in which objects are regarded as being the same if one can be transformed into the other by bending or stretching (tearing and glueing are not allowed). It is sometimes also called "rubber sheet geometry". Homotopy theory is a part of topology in which the geometric objects (called spaces) are studied via algebraic ones, such
as groups and groupoids.

In this talk, I shall describe groupoids, and how to associate to any space a groupoid, called the fundamental groupoid, which can be seen as measuring the "one-dimensional holes" in the space. I shall then go on to describe higher-dimensional analogues of groupoids, and how they can be used to measure the higher-dimensional holes in spaces.

The Breaking of JN-25

SCEM AccessGrid (KWD-Y.2.39, PTA-EB.1.32, CTN-26.1.50) presented from Kingswood

JN-25 was the name given by the Allies to the main operational code used by the Imperial Japanese Navy (IJN) during World War II. It was broken into almost immediately after its introduction in mid-1939 by John Tiltman, one of the greatest World War II British code breakers. It continued to be broken throughout the war, and yielded by far the majority of useful signals intelligence regarding IJN operations available to the Allies. At the same time, the Imperial Japanese Army (IJA), using the same cryptographic systems, maintained their security until early 1943, when only one important system was broken. All others defied
attack.

The talk will describe the coding system used, why the IJN was insecure, and how this was exploited.