# An Introduction to Near Vector Spaces

## Presented by Karin-Therese Howell

In [1], the concept of a vector space, i.e., a linear space, is generalized to a structure with some non-linearity, the so-called near-vector space. In this talk I will introduce the basic concepts and discuss some of the most recent developments in the theory of near vector spaces. In particular, I will describe how a result by van der Walt [4] that showed how to construct an arbitrary finite-dimensional near-vector space using a finite number of near-fields, all having isomorphic multiplicative semigroups, can be used to characterize all finite-dimensional near-vector spaces in the case where all the near-fields are equal to a finite field. This talk is based on the joint work [3] with Professor Johan Meyer (UFS).

References:
[1] J. André, Lineare Algebra über Fastkörpern, Math. Z. 136, 1974.
[2] K-T. Howell, Contributions to the Theory of Near-Vector Spaces, Ph.D. Dissertation, University of the Free State, 2008.
[3] K-T. Howell and J. H. Meyer, Near vector spaces determined by finite fields, Journal of Algebra 398, 2014, 55-62.
[4] A. P. J. van der Walt, Matrix near-rings contained in 2-primitive near-rings with minimal subgroups, J. Algebra 148, 1992, 296-304.

## Presented by Zurab Janelidze

By a “form” we mean a functor which is both faithful and amnestic. In this talk we show how a form can be seen, on one hand, as a natural generalization of a poset, and on the there hand, how it generalizes the notion of a category which takes into account not only specifying structures and structure-preserving morphisms, but also the substructures. We then mention some areas of research where this last idea has fruitful applications.

# Majority Categories

## Presented by Michael Hoefnagel

The first part of the talk will consist of a brief overview of universal algebra, up to Mal’tsev conditions. These conditions, broadly speaking, link together certain syntactical aspects of the subject with semantic ones. We then consider how these conditions can be formulated in a general category, and how theorems in universal algebra can translate into theorems in category theory. The remainder of the time will be used to discuss a particular Mal’tsev condition, which eventually leads to the notion of a majority category.

# An Algebraic Perspective on Recognisable Languages

## Presented by Julia Rozanova

The study of recognisable languages has its roots in theoretical computer science, but this subject has deep connections to algebra which allow us to understand its main concept, the notion of recognition, in a purely algebraic way. This has led to the solution of various problems in formal language theory, but from the perspective of algebra perhaps even more interesting is the link with pseudovarieties of finite algebras provided by Eilenberg’s theorem. Pseudovarieties provide an answer to the observation that no Birkhoff variety is comprised only of finite algebras, and Eilenberg’s theorem shows that pseudovarieties of finite monoids are in a one-to-one correspondence with certain families of recognisable languages. Pseudovarieties also have an equational characterisation in the style of Birkhoff variety theorem which depends on the introduction of profinite algebras, since here the “identities” become pairs of profinite words. Any pseudovariety of finite moniods can be associated both with the family of languages recognised by its members, and with a set of profinite identities. A direct connection between the families of languages and these sets of identities has been recently described in terms of extended Stone duality.

In a series of talks, I will expand on the topics mentioned above, gradually showing how they fit together.

In the first talk, I will introduce recognisable/regular languages in the sense of automata theory and discuss the connection to monoids which allows us to talk about recognition in a purely algebraic way.

# Complex Exponentiation

## Presented by Gareth Boxall

I shall discuss the structure whose underlying set is the set of complex numbers and whose structural features consist of addition, multiplication and the exponential function (which sends z to exp(z)). This is a mysterious structure and there are several deep open problems associated with it. I shall discuss what is not known in some detail. I shall also say something about a recent small contribution to what is known. This concerns Zilber’s quasiminimality conjecture which asserts that every set of complex numbers which is first-order definable in this structure is either countable or co-countable. I prove a special case.

# 3-dimensional algebra and topology

## Presented by Bruce Bartlett

In order to study 3-dimensional manifolds algebraically, it is helpful to have a form of “3-dimensional algebra”! In the first hour I will introduce modular categories, which are certain kinds of categories that have a 3-dimensional graphical calculus. In the second hour, I will show that this is no coincedence. Namely, I will show how a modular category is precisely the data which encodes a representation of the algebraic structure formed 3-dimensional manifolds (and their submanifolds, up to codimension 2). Along the way we will encounter knots, Gauss sums, and mapping class groups of surfaces.

# A Brief Description of Constructive Mathematics

## Presented by Mark Chimes

The basic premise of constructive mathematics is that in order to prove the existence of an object satisfying certain properties, it is necessary to provide an algorithm that specifies that object. Most of constructivism is based on intuitionistic logic, in which the concepts of truth are replaced with the concept of constructive provability. In particular, in this setting, the law of excluded middle (LEM) is rejected: one is not allowed to assume that either ‘A’ or ‘not A’ holds until one has constructively shown which of these two is the case. In addition, under the constructive interpretation, ’not not A’ is no longer equivalent to ‘A’.

The motivation behind constructivism is mostly philosophical. However it has applications for computing – since constructive proofs are algorithmic, a proof of the existence of an object can be converted to a computer program actually producing that object.
In this talk, I shall outline various schools of constructivism including Brouwer’s Intuitionism, the Brouwer-Heyting-Kolmogorov interpretation of logic, and Bishop’s Constructive Mathematics. The axiom of choice has an interesting position in constructivism, which will be described in some detail. I shall also mention the position of Martin-Löf type theory in constructivism, which will follow on some of the ideas mentioned by Dr Gareth Boxall in the previous talk on type theory.

# Homotopy Type Theory and Univalent Foundations

## Presented by Gareth Boxall

Homotopy type theory is a new branch of mathematics in which a version of type theory (from mathematical logic and theoretical computer science) is somehow combined with homotopy theory (from algebraic topology). It appears to be an exciting development about which I have heard a few things, enough to make me want to know more. I plan to learn more and to give an introductory talk about it at the seminar.

Here’s what I think I know already. Type theory aims to give foundations for mathematics (or parts of mathematics), as an alternative to set theory. It recognises that mathematical objects come in different types. For example a function isn’t really the same thing as a set, even though we pretend it is when using set-theoretic foundations. Homotopy enters the picture when we think of types as being a bit like topological spaces. The homotopical interpretation of a certain version of type theory, combined with some additional assumptions including so-called univalence, appears to provide a good foundation for much of mathematics. One aspect that’s good about it is it can greatly diminish the gap between a formal and an informal proof. In normal mathematics, we are aware that, when something is proved informally, then it should be possible to construct a completely formal proof (from the axioms of set theory) which a computer could check. In practice, though, we would expect the formal proof to be much longer than the informal one. Sure we can encode our mathematics in set theory, but that encoding is not how we really think and it’s a real pain to actually do. It appears that, in the homotopy type theory approach, the formal proof is much closer to the informal one and using computer proof assistants becomes much more natural and realistic.

I look forward to seeing how much of the above impressions I retain when it comes to giving the talk on Friday 4 September. Possibly many are currently misunderstandings on my part. I am also drawn to this subject for some other reasons. One is that the people working on it seem to be good at communicating to the general public. There is a free book online, which appears to be well written, and there is a blog. I am also impressed that the Institute for Advanced Study at Princeton hosted a “special year” on this subject in 2012-2013 and that one of the leaders of the subject, Vladimir Voevodsky, is a Fields Medal winner. In a subject like mathematics, where it is so difficult to know much about research areas other than one’s own, it is useful to have things like high-level research institutes and prizes as indicators of quality (even if they may not be perfect indicators).

# Second-order structures: ghosts of abstract mathematics or subject of mathematics to come

## Presented by Zurab Janelidze

### Time and Venue: Friday 21 August 14:00-15:30, Room 1006 Mathematics and Industrial Psychology Building.

Students give feedback and discussions.

### Time and Venue: Friday 17 April 14:00-15:00, Room 1006 Mathematics and Industrial Psychology Building.

Students give feedback and discussions.

# Diagram lemmas for group-like structures

## Presented by Zurab Janelidze

The so-called “diagram lemmas” first appeared in the development of homological tools for algebraic topology. Although there these lemmas were on diagrams of abelian groups, the same lemmas also hold for non-abelian groups, and many other group-like structures as well as their generalizations. This gives rise to the problem of finding proofs of the diagram lemmas that are applicable to as wide range of examples as possible. The work that has been done on this problem leads to, among other things, completely new examples where the diagram lemmas hold, and to simplified proofs already in the case of non-abelian groups. It also leads to an open problem of deciding which diagram lemmas hold for (non-abelian) groups based on the combinatorics of the diagram.

# Groups and Categories

## Presented by Prof Hans Porst

Starting from the observation that the notion of category conceptually generalizes the notion of group, we discuss further generalizations of the concept of group using categorical ideas.

# C0-Semigroups Everywhere

## Presented by Retha Heymann (Universität Tübingen, Tübingen, Germany)

Inspired by the scalar- or mstrix-valued exponential function, strongly continuous
semigroups of operators (also called C0-semigroups) emerged in the infinite-dimensional
setting. While retaining many elegant properties of the exponential function, C0-semigroups became an important tool to solve linear partial differential equations. In this talk, we will discuss properties of C0-semigroups, including asymptotics and spectral theory, and mention some applications.

# On the efficiency of code-based steganography

## Presented by Tanjona Fiononana Ralaivaosaona

Steganography is the art of hiding information inside a data host called thecover. The amount of distortion caused by that embedding can influence the security of the steganographic system. By secrecy we mean the detectability of the existence of the secret in the cover, by parties other than the sender and the intended recipient. Crandall (1998) proposed that coding theory (in particular the notion of covering radius) might be used to minimize embedding distortion in steganography. This thesis provides a study of that suggestion.

Firstly a method of constructing a steganographic schemes with small embedding radius is proposed by using a partition of the set of all covers into subsets indexed by the set of embeddable secrets, where embedding a secret s is a maximum likelihood decodingproblem on the subset indexed by s. This converts the problem of finding a stego-scheme with small embedding radius to a coding theoretic problem. Bounds are given on the maximum amount of information that can be embedded. That raises the question of the relationship between perfect codes and perfect steganographic schemes. We define a translation from perfect linear codes to steganographic schemes; the latter belong to the family of matrix embedding schemes, which arise from random linear codes. Finally, the capacity of a steganographic scheme with embedding constraint is investigated, as is the embedding efficiency to evaluate the performance of steganographic schemes.

# The joy of implication

## Presented by Marcel Wild

Pure Horn functions f are particular types of Boolean functions which come up in a variety of situations, both in applied and pure mathematics. As to “applied” they pop up in Relational Database Theory (under the name functional dependencies), Formal Concept Analysis and Association Rule Mining. As to “pure mathematics”, e.g. the problems of enumerating all order ideals of a poset, or all independent vertex sets of a graph, or all transversals of a hypergraph, are special cases of enumerating all models of f.

It turns out that the minimization of a pure Horn function (an often more handy terminology is “implications”) is best addressed within the framework of closure operators and closure systems.

# VII-2. Syntopogenous structures

## Presented by Ando Razafindrakoto (University of Western Cape)

A syntopogenous structure is a suitable family of relations defined on the powerset of a given set. It was introduced by A. Csaszar in 1963 to provide a unified framework for topological, uniform and proximity structures. The basic axioms that give rise to such structures shall be discussed and illustrated with some examples. We will show how syntopogenous structures could be studied in a categorical setting by using neighbourhood operators.

# An Introduction to Bicategories

## Presented by Pierre-Alain Jacqmin (Université catholique de Louvain, Belgium)

The notion of a bicategory can be seen as a gluing of the notions of 2-categories and monoidal categories. The talk will be devoted to the introduction to these three notions, illustrated by many examples throughout the presentation.

# Limits, Colimits and Ultraproducts

## Presented by James Gray

In this talk we will define limits and colimits for various types of structures. We will see that for many algebraic and more general structures limits are very easy to construct while arbitrary colimits are not. It turns out that special kinds of colimits, called directed colimits, are often easier to construct. Using limits and directed colimits we will be able to construct ultraproducts for various types of structures.

# Ultraproducts

## Presented by Gareth Boxall

### Time and Venue: Friday 19 September 14:00-16:00, Room 1006 Mathematics and Industrial Psychology Building.

The ultraproduct construction is a way of creating new mathematical structures out of old ones. It has some remarkable properties and can be used to prove interesting results. I shall discuss the basics and state the main theorem about them before moving on to consider some additional properties and applications.

# Structural and less structural proofs of the Cantor-Bernstein Theorem

## Presented by: Michael Hoefnagel, Francois Van Niekerk, Phillip-Jan Van Zyl, Julia Rozanova, and Thomas Weighill

### Time and Venue: Friday 5 September 14:00-16:00, Room 1006 Mathematics and Industrial Psychology Building.

The speakers will present the proofs of the Cantor-Bernstein Theorem, which they came across during their undergraduate studies at Stellenbosch University. Some of these proofs are original proofs that the students discovered independently and some of them are adaptations of the well-known proofs.

# Exploring the world of Galois connections

## Presented by Zurab Janelidze

### Time and Venue: Friday 22 August 14:00-16:00, Room 1006 Mathematics and Industrial Psychology Building.

This talk will consist of four parts. In the first part we examine in detail the construction of a Galois connection from a binary relation between two sets, which is central in formal concept analysis in the sense of Wille, as explained in the talk by Professor Rewitzky, and we give some well-known (and less well known) examples of Galois connections obtained in this way. In the second part, we switch our attention to the abstract notion of a Galois connection between ordered sets, and discuss some of its fundamental examples. In the third part, we exhibit general and some special properties that all or some Galois connections possess. In the last part of the talk we explain how a systematic study of mathematical structures leads to a more fundamental concept than that of a Galois connection between ordered sets, which is the concept of an adjunction between categories. We will conclude the talk by showing that adjunctions allow to formalize the idea of “duality”, in the sense of this term used in the first part of the talk by Professor Rewitzky.

# Mathematical structures in Formal Concept Analysis

## Presented by Ingrid Rewitzky

### Time and Venue: Friday 8 August 14:00-16:00, Room 1006 Mathematics and Industrial Psychology Building.

Formal concept analysis (introduced by Wille in 1982 and studied further by Ganter and Wille in 1999) is being used extensively in data mining – for example, by Prof Bernd Fischer from Computer Science at Stellenbosch University. After hearing his inaugural lecture on Tuesday 5 August, I decided a presentation on the mathematical structures in formal concept analysis may be a good topic to start off our seminar series. I will show that the fundamental notions of formal concept analysis, namely contexts and concepts, can be studied within the mathematical framework of duality. Thereafter, I will discuss within this framework the characterisation of attribute dependencies and implications and their comparison with the notion of association rule used in data mining.