In this seminar, mathematics postgraduate students (Honours, Masters and PhD) to give talks to their peers, generally on aspects their own work, though it is also fine to simply give an introduction to a topic of common interest. Constructive feedback from the audience is collated, anonymized and sent to the speaker after their talk. Staff are welcome to attend if they wish.

From time to time, students may volunteer to write a “What is…” article, a two-page introduction to a mathematical topic in the style of the “What is…” column of the Notices of the AMS. These write-ups are recorded on this page.

The archives of talks from previous years (2009-2011) are also available.

- Monday 22 July

Speaker: Yann Ranorovelonalohotsy

Title: The Riemann -Roch Theorem for Global Function Fields.

Abstract: Among the problems in mathematics, the Riemann hypothesis for number fields is still unsolved. But, the analogue of this for the function fields over finite field was solved first by Andre Weil in 1948. In this case of the Global Function Fields, the Riemann- Roch theorem plays a key role.

The talk will introduce some basic facts of Global function fields, and state the Riemann- Roch theorem for this. Our main goal is to use the Riemann- Roch theorem in order to prove some important consequences. Among these, we will discuss a necessary and sufficient condition for a Global function field to be a rational function field.

- Monday 29 July

Speaker: Andry Rabenantoandro

Title: Profinite groups

Abstract: The talk will be an introduction to profinite groups, that is groups that arise as projective limits of finite ones. We will restrict to the case of topological groups even if everything (almost?) could be done over objects of categories that have infinite products. We will particularly be interested in Haussdorf compact topological structures and will discuss few examples: the ring of p-adic integers, the profinite completion of the ring of integers and the absolute Galois group of a field.

A good reference is the book of Ribes and Zalesski entitled Profinite Groups.

- Monday 5 August

Speaker: Isaac Owino

Title: Counting labelled trees by sources, sinks and indegree sequences

Abstract: Du and Yin, Shin and Zeng, and Wagner proved a formula for the number of labelled trees with respect to a given indegree sequence, where each edge is oriented from a vertex of lower label towards a vertex of higher label. In this talk, we present some of our recent results on the enumeration of these trees by the number of their sources, sinks and indegree sequences. Some of the known results will follow as corollaries of our main theorems. We also obtain a differential equation and a functional equation satisfied by the generating function for these trees.

- Monday 12 August

Speaker: Eric Andriantiana

Title: Expander Graphs

Abstract: Loosely speaking, one can say that there is an expansion in a graph if any ‘not too big’ subset of its set of vertices has relatively large set of neighbours. In this talk, we will discuss and compare several expansion parameters, including the so called spectral gap. We aim to introduce the notion of expander graphs. I also hope to have time to state and prove the Expander Mixing Lemma, which shows that regular graphs are ‘good’ approximation for complete graphs.

- Monday 19 August

Speaker: Hatson John Njagarah

Title: Avian Malaria: effect of vector preference for host influenced by Parazitaemia

Abstract: Avian malaria is a parasitic disease that predominantly affects wild birds. The birds also serve as the major hosts for the parasite. The etiologic agent is Plasmodium relictum and reaches the host through bites by the mosquito (the vector) of Culex spp. mainly Culex quinquefasciatus. In this talk, I will give three different host biting preference rates (three models ) and and indicate how the affect severity of avian malaria. The modelling work is based on systems of deterministic ODEs. Disease threshold values for each of the three models will be given and indicate how it can be used to analyse stability of the corresponding system. Numerical results will be showed to compare the effect of different host preferences on severity of the disease.

- Monday 26 August

Speaker: Morenikeji Deborah Akinlotan

Title: Modeling the dynamics of HIV-related malignancies

Abstract:

In recent years, HIV-associated malignancies have proven to be the bane of our time, since HIV is decimating humanity across the globe, even in the twilight of the last century. Cancer rates continue to rise in developing countries, where 95% of the world’s HIV-infected population lives, yet less than 1% have access to antiretroviral therapy. HIV-infected individuals have a higher proclivity to develop cancers, mainly from immunosuppression. An under-standing of the immunopathogenesis of HIV-related cancers is therefore a major prerequisite for rationally developing and/or improving therapeutic strategies, developing immunotherapeutics and prophylatic vaccines. In this study, we explore the pathology of HIV-related cancers, taking into account the pathogenic mechanisms and their potential for improving the treatment of management of these malignancies especially in developing

countries. We mathematically model the dynamics of malignant tumours in an HIV-free environment, investigate the impact of cancers on HIV-positive patients and explore the benefits of a therapeutic intervention strategy in the management of HIV-related cancers. We use a deterministic model of infectious diseases to implement these. We use HIV-related lym-phomas in the Western Cape of South Africa as a case study. The proposed model have been validated using real data from the Tygerberg Lymphoma Study Group (TLSG), Tygerberg Hospital, Western Cape, South Africa. We show that the increasing prevalence of HIV increases lymphoma cases, and thus, other HIV-related cancers. Our model also suggests that an increase in the roll-out of the HAART program can reduce the number of lymphoma cases in the nearest future, while it averts many deaths.

- Monday 2 September

Speaker: Thomas Weighill

Title: Duality in the category of groups via Grothendieck bifibrations

Abstract: In his 1950 paper, Mac Lane proposed the axiomatic study of the category of groups and the category of abelian groups, a study which led to the development of the notions of semi-abelian and abelian categories respectively. One of the most striking differences between the category of groups and the category of abelian groups (and between semi-abelian and abelian categories in general) is that there are many statements that hold in both, but whose duals (i.e. the statements obtained by “reversing the arrows”) hold only in the abelian case. In this talk I will describe how duality in the category of groups can be restored by considering the Grothendieck bifibration of subgroups (this idea is due to Z. Janelidze). In particular, I will define this bifibration and show how two isomorphism theorems in groups follow from an axiom on the bifibration whose dual also holds.

- Monday 16 September

Speaker: Jacques Masuret

Title: Sequences and other weak compact notions in Frames

Abstract: Sequences in general topology can be used to describe not only a number of weaker compactness notions and how they relate, e.g. countable compactness and sequential compactness, but also certain kind of closedness. The question is if the same can be done in the point-free setting. In this talk, we will show that with the “right” notion of sequence, we can obtain results similar to those characterised by sequences.

- Monday 23 September

Speaker: Gerrit Goosen

Title: Fermionic Levin-Wen invariants, string nets and super fusion categories

Abstract: Categorical methods have seen remarkable application in recent decades, ranging from 3-dimensional models of quantum gravity to topological quantum computation. In particular, it was discovered that the notion of a Topological Quantum Field Theory could be elegantly summed up as a symmetric monoidal functor Z : nCob –> Vect. Apart from providing new insights in theoretical physics, they also provide a powerful means of computing numerical invariants of manifolds. In this talk we compare the two so-called state-sum models of Kitaev [1] and Levin-Wen [2], following the approach of Balsam and Kirillov [3] [4]. We also discuss briefly the generalisation of these models to the Z_{2} graded case, which is the subject of ongoing research.

- Monday 30 September

Speaker: Chama Bensmail

Title: Extracting meaningful information from Medline database

Abstract: The growth of the use of high throughput biology methods has created the need to better model and understand extant biological knowledge in order to better interpret the systems-based results that are commonly generated in biological laboratories. With this in mind we propose to develop new ways with which to model and extract knowledge from the scientific literature.

- Monday 7 October

Speaker: Ronalda Benjamin

Title: Fredholm, Weyl and Browder elements in a Banach algebra

Abstract: Fredholm operators were originally defined in an algebra of bounded linear operators on a Banach space as follows: A bounded linear operator $T$ on a Banach space $X$ is Fredholm if it has closed range and the dimension of its null space as well as the dimension of the quotient $X/T(X)$ are finite.

In 1982, R. E. Harte used Atkinson’s theorem to generalize this definition to Fredholm elements in a Banach algebra relative to a fixed Banach algebra homomorphism. The definitions of Weyl and Browder elements in a Banach algebra will also be given and afew results regarding elements with these properties will be discussed.

The spectra corresponding to the sets of Fredholm, Weyl and Browder elements, introduced by Harte, will be defined and several interesting inclusion results of these spectra will be given.

Lastly, we will discuss to which extent these newly introduced spectra obey a spectral mapping theorem.

- Monday 14 October

Speaker: Joubert Oosthuizen

Title: Random walks on graphs and electrical networks

Abstract: Any graph can be associated to an electrical network by replacing vertices by nodes and edges by electrical resistances. The effective resistance between any two nodes x and y is defined as the voltage between x and y, when a unit current enters x and leaves y. On the other hand, the hitting time between two vertices x and y is the expected number of steps to reach y, starting from x. In this talk we aim to give a formula for the hitting time in terms of a weighted sum of effective resistances, due to Tetali.

- Monday 21 October

Speaker: Abdalla Abdurhman

Title: Closure operators on special finite lattices

Abstract: In this talk I will give an overview of closure operators on special finite lattices and discuss the relation between the number of idempotent closure operators and the sub lattices that contain the top element. Lastly, I will introduce and discuss closure operators on a form.

The following graph shows the attendance of the postgraduate seminar by the various students during 2012.

Congratulations to **Evans Ocansey** who attended the most seminars (17, not counting the one day attendance wasn’t taken) during 2012, only missing one – the day he went to another seminar on the NASA mission to Mars!

- Monday 20 February

Speaker: Eric Andriantiana

Title: The Estrada index of graphs (write-up).

Abstract: Let G be a simple n-vertex graph whose eigenvalues are λ_1, … ,λ_n. The Estrada index of G is defined as EE(G) = exp(λ_1) + … + exp(λ_n). The importance of this topological index extends much further than just pure graph theory. For example, it has been used to quantify the degree of folding of proteins and to measure centrality of complex networks. The talk aims to give an introduction to the Estrada index and to discuss some techniques used to study it. Selected results on extremal graphs with respect to the Estrada index will also be reviewed.

- Monday 27 February

Speaker: Ronalda Benjamin

Title: Group inverses in a Banach algebra (write-up).

Abstract: Let A be a Banach algebra. An element a in A is group invertible if there exists an element b in A such that ab=ba, bab=b and aba=a. We will discuss properties like existence and uniqueness of group inverses in a Banach algebra, the spectrum of a group invertible element and conditions under which continuity of group inversion is achieved. Examples and applications of group inverses will be given.

- Monday 5 March

Speaker: Dirk Basson

Title: Lattices, Eisenstein Series and Modular Forms (write-up)

Abstract: The nineteenth century is, among other things, known for the huge development in complex function theory that took place. The properties of many fundamental functions like the exponential, gamma and zeta functions were explored. In this talk I would like to concentrate on functions that have some relation with lattices in the complex plane, in particular, on modular forms. To relate this to my research, I would like to draw some parallels between the classical theory over the complex numbers, and a characteristic p theory.

- Monday 12 March

Speaker: Kelvin Muzundu

Title: Positive operators on Banach lattices (write-up)

Abstract: A vector lattice is a real vector space E endowed with a reflexive, antisymmetric and transitive ordering <= such that (E, <= ) has a lattice structure compatible with the algebraic structure of E. A Banach lattice is a real Banach space E with an ordering such that (E, <= ) is a vector lattice and the norm on E is compatible with the lattice structure of E. An element x in E is called positive if 0 <= x. A linear operator T: E -> F between Banach lattices is said to be positive if it maps positive elements of E to positive elements of F. In this talk we will discuss positive operators on Banach lattices

- Monday 19 March

Speaker: Alessandro Rossetti

Title: Global differential geometry: The Bonnet Theorem

Abstract: The Bonnet Theorem states the following. Let the Gaussian curvature K of a complete surface S satisfy the condition

K>0. Then S is compact.

One of the most interesting aspect of Bonnet’s Theorem is that it expresses the union of three mathematical fields: Differential Geometry of Surfaces, Topology and Calculus of Variations. I would like to explain the tools needed to understand the proof, like Christoffel symbols, the covariant derivative, geodesics and the First and Second Variations of the ArcLength. At the end I would like to give the idea of the proof.

- Monday 26 March

Speaker: Bruce Bartlett

Title: The cohomological proof of Brouwer’s fixed point theorem (write-up)

Abstract: The simplest version of Brouwer’s fixed point theorem (discovered by others before him, of course) is as follows: Every continuous map from the unit disc in the plane to itself has a fixed point. I will explain the “cohomological” proof of this theorem, using divergence and curl of vector fields. The generalization of this technique to higher dimensions is called De Rham cohomology.

- Monday 16 April

Speaker: Savannah Nuwagaba

Title: The Architecture of Antagonistic networks (write-up)

Abstract: Antagonistic networks (e.g. predator-prey networks, host-parasite networks and plant-herbivore networks) are characterised by the compartmentalised architecture, where species often form modules and interact more with other species in the same module than with those from other modules. To date, the generative mechanisms of the development of this distinctive architecture are lacking. It is commonly observed that in antagonistic networks species tend to switch their interactive partners as a result of environmental changes or enhancing their fitness. We develop a consumer-resource model that incorporates the interaction switch to simulate the development of this network architecture. This model allows species to switch their interactive partners and organise an initial random network asymptotically into a compartmentalised network. The predictions of the level of compartmentalisation fit well with the observations from 61 real antagonistic networks. This implies that the interaction switch may be an important adaptive process that gives rise to the distinctive architecture of many ecological networks.

- Monday 23 April

Speaker: Darlison Nyirenda

Title: Algebraic Codes (write-up)

Abstract: We present basic ideas about algebraic codes. The concept of distance of a code and its relationship with error detection/correction is emphasized. We then narrow down to binary linear codes and their decoding.

- Monday 30 April

Speaker: Evans Ocansey

Title: The Matrix-Tree theorem (write-up)

Abstract: The matrix-tree theorem is one of the classical theorems in algebraic graph theory. It provides a formula for the number of spanning trees of a connected labelled graph in terms of eigenvalues or determinants of associated matrices. It was first proved in 1847 by Gustav Kirchhoff — a German physicist in his study of electrical networks. There are other known proves as well. In this presentation, an idea to the prove the matrix tree theorem

for any connected labelled graph will be given and we will use it to derive explicit formulas for the number of spanning trees in various important classes of graphs. An interesting connection to the theory of electrical networks and spanning trees will also be shown.

- Monday 11 May

Speaker: Maryke Van der Walt

Title: Interpolatory ternary subdivision (write-up)

Abstract: Subdivision methods provide a simple but effective tool for rendering smooth curves and surfaces in computer graphics. In the literature, attention has been mostly restricted to developing binary subdivision schemes. The primary emphasis of this talk is on ternary subdivision, and in particular on the interpolatory case. Explicit construction methods, as well as a corresponding convergence analysis, will be presented. Graphical illustrations of the results will be provided.

- Monday 6 August

Speaker: Alex Bamunoba

Title: Carlitz cyclotomic polynomials

Abstract: In this talk, I will describe two basic problems concerning the arithmetic and computation of Carlitz polynomials in my PhD project. The first is one is arithmetic in nature and is a conjecture which asserts that: A=Fr[T] is the set of coefficients of all Carlitz cyclotomic polynomials over k:=Fr(T) . The second problem deals with designing and implementing of fast and efficient algorithms for computing Carlitz cyclotomic polynomials with the aid of discrete Fourier transforms.

- Monday 13 August

Speaker: Ando Razafindrakoto

Title: Extension property in metric spaces

Abstract: The Hahn-Banach theorem says that a linear functional f: X

—> |R admits a nice extension F: Y —-> |R, with X a subset of Y, under suitable conditions. We usually focus our interest on the properties of X and Y. However, the real line has a property, namely injectivity, which implies the existence of F. In this talk we shall define this property and discuss some of its consequences. We shall see in particular that treating the Hahn-Banach theorem on the other hand, and the Brouwer fixed-point theorem on the other in a “diagrammatic” way, reduces to studying this extension property.

- Monday 20 August

Speaker: Jacques Masuret

Title: The point of pointless topology

Abstract: What is pointless (point-free) topology? Why is research done in this setting? How is it connected to classical topology? In this talk a brief outline of the history and development of point-free topology will be given.

- Monday 27 August

Speaker: Abey Kelil

Title: Interpolatory refinable functions and surface subdivision (write-up, slides)

Abstract: In this talk, interpolatory bivariate refinable functions especially box splines (especially interpolatory ones) will be introduced. We will particularly focus on the refinable box splines, which are the basis functions for subdivision surfaces, and we integrate their application in the construction of bivariate schemes, especially the Butterfly subdivision scheme. A special focus will be given to the algebraic derivation of the Butterfly subdivision scheme from the normalized box spline symbols, and graphical illustrations of Butterfly surfaces will also be considered.

- Monday 3 September

Speaker: Andry Rabenantoandro

Title: Fundamental theorems of Galois theory (write-up).

Abstract: Let L be an algebraic Galois extension of K . The fundamental theorem of Galois theory for finite extensions describes nicely a bijective correspondence between the subgroups of Gal(L/K) and the intermediate fields extensions of K contained in L.

On the other hand, for infinite Galois extensions it turns out that there are more groups than intermediate fields extensions. However, there exists a one to one correspondence between the intermediate fields and the subgroups of Gal(L/K) that arise as closed sets of a canonical topology defined on Gal(L/K): the Krull topology. In this talk I would like to give a description of these fundamental theorems with a particular focus on the infinite.

- Monday 17 September

Speaker: Tovohery Randrianarisoa

Title: Title: The number of matrices over F_q with irreducible characteristic polynomial (write-up)

Abstract: Let F_q be a finite field with q elements. M. Gerstenhaber and Irving Reiner has given two different methods to show the number of matrices with a given characteristic polynomial. In this talk, we will give another proof for the particular case where the characteristic polynomial is irreducible. The number of such matrices is important to know the efficiency of an algorithm to factor polynomials using Drinfeld modules.

- Monday 1 October

Speaker: Marie Brilland Yann Ranorovelonalohotsy

Title: Vector fields and meromorphic functions

Abstract: In order to study two-dimensional fluid flow, we need many tools such as complex analysis.

In this talk, we introduce some basic knowledge about incompressible two-dimensional fluid flow, and then we study two extensions of the Gauss-Lucas theorem which states that the zeros of the derivative of a polynomial f(z) in C[z] , where C is the field of complex number, lie in the convex hull generated by the roots of that polynomial. We will precisely look to the case of the entire function f(z)=b*exp(a*z)*(z-z_1)*…(z-z_m), and the case of the rational functions.

- Monday 8 October

Speaker: Hatson John Njagarah

Title: Evolution equations

Abstract: Evolution equations have been used to model a number of physical phenomenon and biological as well as ecological processes. The analysis involved depends on whether the modeller is interested in the application, or the behaviour of the system. In this talk, I will give a general highlight of what evolution equations are, some of their important properties and spaces over which the analysis is done. Using specific examples, I will show an explicit solution of the non-autonomous case on R^{n} and reaction-diffusion dynamics of a single species ecological problem.

- Monday 15 October

Speaker: Antoine Bahizi

Title: Assessing the impact of climate change on regional and global biodiversity: a meta analysis (write-up)

Abstract: Peer-review publication has become a standard vessel for sharing the knowledge gained in scientific research, with its number increasing exponentially in the recent decades. To this end, robust statistical methods that allow us to be able to synthesize and generalize these knowledge points in published literature, and to develop and test overarching hypotheses, are urgently needed by the scientific community. We here use the methodology of meta-analysis and test a few ecological hypotheses on the influence of global climate change on biodiversity maintenance in regional ecosystems. Based on works by Hans van Houwelingen and colleagues, we develop a maximum likelihood estimate of parameters in fixed and random effect models. This approach was then applied to a real dataset collected from literature on the impact of the ambient temperature increase on species performance. Finally, we conclude by highlighting the advantage and limitation of this approach.

From time to time, students may volunteer to write a “What is…” article, a two-page introduction to a mathematical topic in the style of the “What is…” column of the Notices of the AMS.

- What is… Modal Logic? by Walter Cloete, March 2011. Modal logic attempts to describe ‘additional modes of truth’ that are not captured by propositional logic, such as ‘known to be true’, ‘possibly true’ and ‘true at some future time’.
- What is…Bourbaki? by Ando Razafindrakoto, February 2013. A perspective on the French mathematical group “Bourbaki”.