There were 20 postgraduate seminars in 2011. The attendance was as follows:

Congratulations to **Laila El Ghandour** and **Dirk Basson** who attended the most postgraduate seminars (18).

- Monday 21 February

Speaker: Dirk Basson

Title: Sheaves (write-up)

Abstract: Sheaves are a tool with which one can compare local and global information. Given global information like a function defined on a topological space one obtains local information, in this case a function defined on some small open subset. Likewise, if one is give functions defined on many open sets (small), one may sometimes define a new function defined on the whole space (global).

In this talk I will define sheaves, give examples, and state some of their elementary properties. The aim is to give the listener an idea of what sheaves are about and about why they are important. To this end I will speak about the applications of sheaves in various different areas.

- Monday 28 February

Speaker: Hatson John Njagarah

Title: Modeling substance abuse (slides)

Abstract: Substance abuse has been a global menace for some time now. To analyze the variation of the population of substance users, I will present a mathematical model in which the heterogeneous population is subdivided into homogeneous sub-groups. I will use the threshold number of the presented mathematical model, to analyze the stability of steady states and show some of the important parameters in the control of the epidemic. Lastly, I will present some numerical results with respect to the “important parameters”.

- Monday 7 March

Speaker: Alex Bamunoba

Title: Some aspects of Mahler measures (write-up)

Abstract: Given a polynomial f(z) with complex (or for us usually integer) coefficients, the one dimensional logarithmic Mahler measure is defined to be the exponential of the integral of the logarithm of f(z) over the unit circle. I will present the various properties of this measure, discuss Lehmer’s problem and how it generalisesto polynomials in several variables. As an application, we estimate heights and length of polynomials.

- Monday 14 March

Speaker: Rejoyce Gavhi

Title: Interpolatory refinement pairs with properties of symmetry and polynomial filling (slides)

Abstract: Subdivision schemes are important and efficient tools for generating smooth curves and surfaces used in computer aided geometric design (CAGD) and in wavelet decomposition. Out of many subdivision schemes, we consider in this talk particularly interpolatory subdivision schemes (ISS), which have the property that the limit curve interpolates the original control points. We first present the issues of interpolation, symmetry and polynomial filling with respect to a subdivision scheme, eventually leading to a definition of a class of symmetric interpolatory subdivision schemes with the polynomial filling property up to a given odd degree in which all of the above desired properties are combined.

Finally, we derive an explicit characterization formula for the above class. For the case of one degree of freedom in the form of a shape parameter t, we establish an interval for t in which refinable function existence, and therefore also subdivision convergence, are guaranteed.

- Monday 28 March

Speaker : Diego Mirandola

Title: Some basic ideas about Groebner bases and their application to euclidean geometry (write-up)

Abstract: Groebner bases are, basically, “good” basis for the finitely generated ideals of a polynomial ring with several variable. Our guidelines are briefness and analogy with the one variable case: we follow the path monomial ordering – division algorithm – Hilbert basis theorem, which leads to the definition of the Groebner bases. Some examples are used, especially where the theoretical details are omitted. Finally, one of the several applications is shown with an example: here we prove, using the Groebner bases theory, the circle theorem of Apollonius, which is a classical result of euclidean geometry.

Most of the details are omitted: one of our purposes is to invite the audience to an individual deeper study.

- Monday 4 April

Speaker: Thomas Masiwa

Title: On the Frattini-like subgroup (write-up)

Abstract: We introduce maximal subgroups and investigate the influence of the intersection of all maximal subgroups on the structure of a group. Unfortunately some groups, like the Prufer group do not have maximal subgroups. Following Tomkinson we define “major subgroups” and prove that every proper subgroup is contained in a major subgroup. Lastly we introduce a class of groups in which the Frattini-like factor is a T-group, we shall denote this class by T_1 and prove that it is intermediate between T-groups (introduced by Robinson) and T_0-groups (introduced by Fransman).

- Monday 11 April

Speaker : Eric Andriantana (write-up)

Title: Spectra of graphs

Abstract: By attaching an n x n-matrix to each n-vertex graph, we will use results and methods of algebra to deduce theorems about graphs. This will allow us to introduce the spectra and the energy of graphs. The energy of graphs is one of the most popular graph invariants in chemical graph theory. The notions of isospectral graphs and hyperenergetic graphs will be illustrated by examples. Some results on the characterisation of graphs with smallest or largest energy in some classes of graphs such as trees or unicyclic graphs will also be discussed if time allows.

- Monday 18 April

Speaker: Jeanne Onana Eloundou

Title: The Mathematical Modelling of the Stages of Tumour Growth (slides, write-up)

Abstract: For solid tumours to grow and metastasise, they need to pass through two distinct stages: the avascular growth phase in which the tumour remains in a limited diffusion size and the vascular growth phase where the invasion may take place. In order to accomplish the transition from the former to the latter growth phase, solid tumours may secrete a substance known as tumour angiogenesis factor (TAF) into the surrounding tissues. We present some mathematical models dealing with various stages of development of solid tumours and the resulting reaction diffusion equations are solved using the Crank-Nicholson finite differences scheme.

- Monday 28 April

Speaker: Laila El Ghandour

Title: Levy Processes and Connection to Finance (write-up)

Abstract: Levy processes are class of stochastic processes with stationary and independent increments. they are the generalization of very important processes as Brownian Motion, Poisson Processes and Compound Poisson Processes. We present some of general properties of Levy processes, and then we show how the theory of stochastic processes have been used in financial modeling.

- Monday 9 May

Speaker: Tolu Fadina

Title: Pricing and hedging of options using Numerical methods (slides, write-up)

Abstract: An option contract has become a crucial tool in finance both for financial institutions and investors. It is attractive to investors because of its great effect in speculating and reducing risk in investment. This brought up the idea to determine their worth and how much to be traded on in a financial market. We investigate the efficiency of Monte Carlo methods and Fourier cosine series expansion method for pricing and hedging financial derivatives in financial models.

- Monday 16 May

Speaker: Sarudzai Showa

Title: A stage structured HIV model incorporating the life cycle aspects of the virus (write-up)

Abstract: Currently HIV infection is treated by drug combinations that targed the virus at the different stages of the virus’ life cycle. Determining the stages which are most sensitive to treatment is of vital importance on the design of these drug combinations. In this talk I will illustrate how matrix population models can be used to determine the most sensitive stages of the HIV life cycle.

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- Monday 18 July

Speaker: Kelvin Muzundu

Title: Ordered Banach algebras (write-up)

Abstract: Let A be a Banach algebra with unit 1. A subset C of A is called an algebra cone if C contains the unit and is closed under addition, multiplication and positive scalar multiplication. We will discuss how a Banach algbera can be ordered by an algebra cone.

- Monday 25 July

Speaker: Mohamed Omari

Title: Positive Semigroups with Applications (slides, write-up)

Abstract: It has been observed that semigroups can be used to solve a large class of problems commonly known as evolution equations. In this seminar we aim to give a basic introduction to semigroups of linear operators. Furthermore, we present an application of this theory to a model coming from concrete applications, that is, fragmentation process.

- Monday 1 August

Speaker: Andry Rabenantoandro

Title: Introduction to number theory over F_q[T]

Abstract: Elementary number theory deals with arithmetic properties of the integers Z. It turns out that Z has many properties in common with F_q[T]]: both are euclidean (therefore PID, UFD), both have finitely

many units, both have finite residue class ring modulo a non-zero element. One can therefore expect analogous results in Z to hold for F_q [T ]. This is indeed the case. We will give an overview of some

basic results to illustrate these analogies, namely analogues of: Euler and Fermat’s little theorem, Wilson’s theorem, the prime number theorem and Dirichlet’s theorem on primes in arithmetic progression.

- Monday 8 August

Speaker: Rinske Van der Bijl

Title: Bivariate wavelet construction based on solutions of algebraic polynomial identities

Abstract: Over the last two decades, wavelets have been used as filters in numerous applications of multiresolution analysis. In their recent book, Chui and De Villiers showed how, for a given scaling function, the solutions of a certain system of algebraic polynomial identities can be applied to construct a class of (not necessarily orthogonal) synthesis wavelets with vanishing moments of arbitrary order. In our work, we establish an analogous result for bivariate scaling functions with dilation factor 2I, with explicitly calculated examples.

- Monday 22 August

Speaker: Mmile Seitlheko

Title: The Birthday Paradox (write-up)

Abstract: The birthday problem can be stated as follows: Within a group of k people, what is the probability that two or more will share a birthday? It turns out that if k = 23 then two people will share a birthday with a probability just above 50%. In this talk, we will discuss the mathematics behind the birthday paradox and outline some of its applications in cryptology.

- Monday 29 August

Speaker: Frances Odumodu

Title: Elliptic Curves with Complex Multiplication

Abstract: One way of constructing a Galois extension of Q is to adjoin roots of unity. That is, n- torsion points on a unit circle in the complex plane. Here we adjoin special values of the complex exponential function to get an abelian extension of Q (a Galois extension with an abelian Galois group). An analogous way of obtaining a Galois extension of Q is by adjoining the coordinates of the n-torsion points on an elliptic curve. We shall consider two elliptic curves y^2 = x ^3 + x and y^2 = x^3 + 1 which have complex multiplication and use them to construct abelian extensions of the imaginary quadratic fields Q(i) and Q(sqrt(3)) respectively. Here we adjoin special values of the Weierstrass function and its derivative. It is known that every abelian extension of Q(i) is contained in one of these and that this theory works for any quadratic imaginary number field. We shall follow the approach given in Silverman and Tate’s Rational Points on Elliptic Curves.

- Monday 12 September

Speaker: Abey Sherif Kelil

Title: Gauss Quadrature rules for refinable functions (write-up)

Abstract: In this talk, we study Gauss quadrature rules by giving special attention to the refinable functions, especially the cardinal B-splines considered as weight functions. First, we discuss by reviewing some elementary facts about orthogonal polynomials and then, the corresponding weight functions, especially the cardinal splines and the shifted cardinal B-splines with some of their properties will be given. Thereafter, the computational techniques for determining the moments of cardinal splines and the shifted cardinal B-splines are studied and their relationship with the Gaussian quadrature rules will also be discussed. Also, the notion of Lagrange interpolation plays a key role in the determination of the quadrature formula. Furthermore, the role of orthogonal polynomials is of great value since their roots can be used as the nodal values of the Gaussian quadrature.

- Monday 19 September

Speaker: Tovohery Randrianarisoa

Title: Valuations, valuation rings and places (write-up).

Abstract: Places, valuations and valuation rings are widely used in algebra especially in algebraic geometry. The equivalence between them allow us to use their theories together in algebra. The purpose of this talk is to introduce them. We will give some example, and then we will show that valuations, valuation rings and places actually define the same notions.

- Monday 3 October

Speaker: Thibault Monfort

Title: Solvable groups

Abstract: The concept of solvable group is quite important in algebra, particularly in Galois theory, where it is closely related to the solvability of polynomial equations. We’ll see the definition, some characterisations and basic properties of solvable groups.

- Monday, 1 March

Speaker: Leon van Wyk

Title: Centralizers of matrices

- Monday, 8 March

Speaker: Eric Andriantana

Title: Independent subsets and energy of degree-restricted trees (write-up, slides)

Abstract: The Merrifeld-Simmons index (defined as the number of independent vertex subsets of a graph G), the Hosoya index ( defined analogously as the number of matchings, i.e. independent edge subsets), and the graph energy, certainly belong to the most studied parameters in extremal graph theory. They are also objects of interest in mathematical chemistry.

With respect to each of these three parameters, we will discuss the ordering in the set of all trees and the ordering of the trees whose vertex degrees are restricted to 1 or d (for some d >= 3), a natural restriction in the chemical context.

- Monday, 29 March

Speaker: Zurab Janelidze

Title: Map of mathematics at Stellenbosch!

Abstract: This seminar will be devoted to our honors students. I will attempt to sketch a kind of “map” of the mathematics represented in our division, to give to the honors students a view of mathematical opportunities in the division. I hope to get some help from the MSc and PhD students. Hopefully, this will serve as a good introduction to the meeting on Tuesday, organized by Sonja Mouton, where the honors students will be able to speak to the professors directly.

- Monday, 19 April

Speaker: Zurab Janelidze

Title: The category of matrices

Abstract: We will have a look at the mathematical structure that is built using matrices (whose entries are real numbers, or, more generally, elements of an arbitrary ring) and matrix multiplication.

- Monday, 26 April

Speaker: Jean-Claude Dushimimana

Title: Dependence concepts for Levy processes

Abstract: The law of a random vector X on R^d can be characterized by its cumulative distribution function F(x_1, . . . , x_d). The margins of X are the laws of X_i taken separately. It has long been known in statistics that the dependence structure of a random vector on R^d can be disentangled from its margins via the notion of copula. Analogously, in this talk we show that the dependence structure of an R^d-valued Levy process can be completely characterized by a Levy copula. Levy copulas allow to construct multivariate Levy processes and to model their dependence. They can be used to separate the dependence structure from the behavior of the marginal laws. As for random vectors, a version of Sklar’s theorem states that the law of a d-dimensional Levy process is obtained by combining arbitrary univariate Levy processes with an arbitrary Lvy copula. We give examples of Levy copulas and show a method for constructing parametric Levy copulas which turn out to be useful in application.

- Monday, 10 May

Speaker: Ando Razafindrakato

Title: Introduction to subobjects in categories

Abstract: The talk will introduce a general axiomatic setting where it is possible to define a “substructure” of a mathematical structure, which is a common generalization of the notions of a subspace of a vector space, subspace of a topological space, subgroup of a group, subring of a ring, etc. The talk should be accessible to all students who are familiar with the above mentioned structures.

- Thursday, 5 March

Speaker: Bruce Bartlett

Title: Random matrices and the Riemann zeros (write-up)

Abstract: Random matrices and the Riemann zeta function came together during a chance meeting between Hugh Montgomery and Freeman Dyson over tea at Princeton in 1972. Dyson observed that Montgomery’s conjecture for the correlation between pairs of zeros of the zeta function was precisely the same as the known correlation betwen eigenvalues of large random hermitian matrices. I will try to explain how these formulas are derived, at least on the random matrix side of things.

- Thursday, 19 March

Speaker: Magdaleen Marais

Title: Centralizers of matrices (write-up)

Abstract: In this talk we will discuss the centralizer of a 2×2 matrix over a factor ring R/<k> of a unique factorisation domain R, where <k> denotes the principal ideal generated by k in R. We define a class of matrices called k-matrices and find a concrete description of the centralizers of matrices in this class. It turns out that if R is a principal ideal domain, then every 2×2 matrix with entries in R is a k-matrix, so that our concrete description applies. However, this is not the case for all unique factorisation domains.