Download The course will have an Algebra and a Linear

Structured Master's in Mathematical Sciences - August 2015 intake
Review Course Abstracts
9-27 November
Algebra
Karin-Therese Howell (Stellenbosch)
The course will have an Algebra and a Linear Algebra component. Under Algebra we will discuss
functions, relations, partitions, groups, morphisms, quotients, the Isomorphism Theorems for
groups and direct products. As part of the Linear Algebra component we will discuss linear
spaces, linear functionals and operators, matrices, change of basis, eigenvalues and eigenvectors
and the normal form.
Differential Equations
Precious Sibanda and Joseph Malinzi (KwaZulu Natal)
30 November - 18 December
Model Theory
Gareth Boxall (Stellenbosch)
Model theory is a branch of mathematical logic which has applications in several other areas of
mathematics including algebraic geometry, number theory and combinatorics. This course will
introduce model theory from the beginning, including some preliminaries on ordinals and cardinals.
The focus will then be on reviewing several interesting applications. In particular, we shall devote
significant time to non-standard analysis.
Des Johnston (Heriot-Watt)
Finite-dimensional Quantum Mechanics and Quantum Computing
The course comprises an introduction to quantum mechanics using elementary methods in a finitedimensional context with applications to quantum computing. A version of this course has been
delivered in previous years at AIMS-South Africa by Bernd Schroers. The course starts with a review of
some basic linear algebra, and using the language of matrices and vectors introduces quantum
mechanics for finite-dimensional systems in the style of Heisenberg. The framework is developed to
consider multi-spin systems and density matrices. Quantum entanglement (Einstein's "Spooky action at
a distance") and Bell's inequalities, which demonstrate the quantum nature of reality, are then studied.
The course ends with applications of the formalism to some quantum circuits and simple quantum
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algorithms in quantum computing such as the Deutsch algorithm.
The Numerical Solution of Differential Equations
Gabriel Lord and Lyonell Boulton (Heriot-Watt)
The aim of this course will be to provide an overview of the basic techniques, tools and theory for
solving both ordinary differential equations (ODE) and partial differential equations (PDE) numerically.
For the ODE part of the course, we will focus our attention on the theory of numerical solution of
evolution problems, and its many applications. Simple examples will be discussed, including diffusion of
heat in a bar. More complex example, such as the transport and diffusion of chemicals (e.g oil) through
an underground reservoir, will also be considered. For the PDE part of the course, we will focus on the
finite element method in two and three dimensions. We will include applications from spectral theory,
the theory of sound and elementary models from quantum mechanics. During this course, the students
will be presented with various numerical algorithms. All these algorithms will be made available in Sage,
Octave and FreeFem++.
4-22 January 2016
Flat and Hyperbolic Geometry of Surfaces
Jayadev Athreya and Steve Bradlow
We will study the geometry of surfaces through a series of concrete examples and computational
challenges. We will introduce the concepts of a surface, of flat geometry, and of basic hyperbolic
geometry. We will use as a guide the book entitled "Mostly Surfaces" by Rich Schwartz and will cover a
selection of topics chosen from the following (with associated computational challenges):
-Topology of Surfaces (associated computational project: computing Euler characteristic)
-Euclidean Geometry review (associated computational project: constructing quadratic irrationals)
-Hyperbolic Geometry and Surfaces (associated computational project: computing fundamental
domains)
-Flat Geometry and Veech Groups (associated computational project: computing sets of saddle
connections)
Rafael Nepomechie (Miami)
Quantum Mechanics and Quantum Spin Chains
The first week will be devoted to reviewing elements of classical mechanics (Lagrangian, Hamiltonian,
and Poisson brackets) and linear algebra using Dirac bra-ket notation. The eigenvalue problem and
simultaneous diagonalization will be emphasized. In the second week, the principles of quantum
mechanics will be introduced, using the free particle, the harmonic oscillator and the two-state (spin1/2) system as the working examples. Students will then learn about higher (s > 1/2) spin, which
provides an ideal introduction to Lie algebras and their irreducible representations. Using the notion of
tensor product to describe a composite system of two spins, students will be introduced to the ClebschGordan theorem. Students will gain valuable "hands on" experience by using Sage to explicitly obtain
both irreducible and reducible representations of the su(2) generators. The capstone of this course,
making non-trivial use of all of these concepts, will be an introduction to the Heisenberg quantum spin
chain (a ring of N spin-1/2 spins with nearest-neighbor isotropic interactions), which is of fundamental
importance in theoretical physics. Students will first study this model numerically for small values of N.
They will classify the states according to energy, momentum and spin, and understand degeneracies
with the help of the Clebsch-Gordan theorem. Finally, they will reproduce these results from the Bethe
ansatz, and go further by computing the ground-state properties in the infinite-N limit. Additional
mathematical and computer techniques (including the Newton-Raphson method, recursive
programming, and Fourier transforms) will also be reviewed and used.
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Financial Mathematics
(To be confirmed)
6-15 January 2016
Mathematics in Industry Study Group
The Mathematics in Industry Study Group is a five-day workshop at which academic researchers and
graduate students work collaboratively with representatives from industry on research problems
submitted by local industry. (See https://www.aims.ac.za/en/research-centre/workshopsconferences/past/mathematics-in-industry-study-group-misgsa-2015 for details of the 2015 workshop).
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25 January to 12 February 2016
Industrial mathematics
Neville Fowkes (Western Australia)
One of the interesting features of industrial and scientific modelling is that the same phenomena occur
across disciplines in slightly different guises. This is why the study of `archetypal problems' is so
important. An archetypal problem should `strip away' the inessentials leaving the basic issues exposed.
A list of archetypal problems is of course impossible because no agreement even between experienced
`modellers' is possible, but the list would not vary by too much and largely differ because of familiarity
with the context. In this course I shall present a range of archetypal problems. The appropriate
mathematical procedure to use to extract results depends not only on the problem type but also on the
question of interest. Also very often an approximate solution is better than an exact solution. I shall
illustrate this by examining a broad range of useful techniques in the context of archetypal problems
and problems arising out of continuum mechanics/industrial contexts. Techniques may include: scaling,
asymptotics, singular perturbation techniques, variational methods, analytic vs numerical methods,
classification of partial differential equations, Fourier methods. Phenomena may include diffusion,
nonlinear vibrations, waves, shock dynamics, boundary layers, buckling, enzyme kinetics.
Analytical Techniques in Mathematical Biology
Wilson Lamb (Strathclyde)
Mathematical models arising in the natural sciences often involve equations which describe how the
phenomena under investigation evolve in time. When time is regarded as a continuous variable such
evolution equations usually take the form of differential equations. In this course a number of
mathematical techniques will be presented for analysing a range of evolution equations that arise in
Biomathematics, particularly in population dynamics. The emphasis will be placed on determining
qualitative features of solutions, such as the long-term behaviour. Different types of equations will be
examined, but a unifying theme will be provided by developing methods from a dynamical systems
point of view and using some results from functional analysis. To fix ideas, the course will begin with
some simple one-dimensional models from population dynamics, such as the Malthusian and Verhulst
equations. Structured population models arising in epidemiology, such as the SIS and SIR models, and
multispecies models, such as the Lotka –Volterra predator-prey equations, will be considered next. The
latter models result in non-linear systems of ordinary differential equations and their analysis involves
higher (but still finite) dimensional dynamical systems theory. To give an indication of the need, in some
problems, to work within an infinite-dimensional setting, we shall conclude by examining the notion of
diffusion-driven (or Turing) instability in reaction-diffusion type partial differential equations and discuss
mathematical models of pattern formation (e.g. in animal coats) that involve such equations.
Fluid Dynamics
Grae Worster
Fluids are all around us, from the air we breath to the oceans that determine our climate and from oil
that powers our industries to metals that are cast into machinery. The study of fluid dynamics requires
sophisticated applications of mathematics and the ability to translate physical problems into
mathematical language and back again. The course begins by building a fundamental understanding of
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viscous fluid flows in the context of unidirectional flows. In more general, higher dimensional flows,
pressure gradients are generated within a fluid to deflect the flow around obstacles rather than the fluid
being compressed in front of them, and an understanding of the coupling between momentum and
mass conservation through the pressure field is key to the understanding and analysis of fluid motions.
We will use simple experiments to illustrate and motivate our mathematical understanding of fluid flow.
Prerequisite for the course is fluency with differential equations and vector calculus. No previous
knowledge of fluid dynamics will be assumed.
Introduction to Random Systems, Information Theory and Related Topics
Stephane Ouvry ( Paris-Sud University)
This course is an introduction to various random systems, probability theory, Shannon information
entropy and some related topics, with a special emphasis on their mathematical aspects. In
particular I will present selected lectures on
• Probability calculus and the central limit theorem
• Application to Monte Carlo sampling
• Application to random walks on a line
• Random walks on a square lattice and their algebraic area
• Random permutations and their application to
– the statistical “curse” problem in sailing boat regattas
– the Diaconis shuffling cards trick
• Shannon statistical entropy and information theory
• LZW compression algorithm
• Application to the identification of languages
Several topics covered in the lectures will be illustrated by numerical simulations and algorithms. A
few examples :
• Generate closed and opened random walks of a given length on a square lattice and their
algebraic area
• Enumerate and simulate score performances in boat regattas with ranks given by random
permutations etc
22-26 February
Entrepreneurship
Graham Richards (Oxford)
Students tend to think of their futures as PhD perhaps followed by a Postdoc and then an academic
career. This is realistic for only a minority; the rest must join major corporations, government or
increasingly start their own businesses.
This course aims to show how the latter can be done, providing the essential know-how about
intellectual property, funding and business plans. Case studies and an opportunity for groups of
students to suggest their own plans will be provided.
References:
Spin-outs: creating businesses from university intellectual property.
Graham Richards, Harriman-House 2009
University intellectual property: a source of finance and impact.
Graham Richards, Harriman-House 2012
22 February to 11 March
Computational Algebra
Wolfram Decker and Gerhard Pfister, Kaiserslautern
Groebner bases and Buchberger's algorithm for ideals and modules will be studied. Applications to
commutative algebra, selected problems in singularity theory and algebraic geometry will be given as
well as applications to electronics and engineering. The course includes an introduction to the computer
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algebra system SINGULAR and its programming language.
Bioinformatics
Paolo Zuliani (Newcastle)
We present some of the most used mathematical and computational techniques for modelling
biological systems, with a focus on genetic and biochemical networks. We introduce metabolic
modelling using flux balance analysis, and signalling pathway modelling using ODEs (Ordinary
Differential Equations) and stochastic discrete models (Gillespie's algorithm). For biological
sequence analysis we introduce hidden Markov models and related algorithms (e.g., expectationmaximisation). Finally, we present very recent work on parameter synthesis for ODE models, using
SMT (Satisfiability Modulo Theory) techniques. ***This course does NOT assume any Biology
background.***
Knot Theory
Vaughan Jones (California, Berkely)
We begin with the definition of knots and links and their elementary theory-fundamental group of
the complement and Reidemeister moves. We define the Alexander polynomial from the
fundamental group using some algebra.
We then turn to Kauffman’s bracket and how it gives the Jones polynomial with a discussion of
tangle decomposition and the diagrammatic Temperley-Lieb algebra. Alternating knots and links.
The HOMFLYPT two-variable polynomial and the Kauffman two-variable polynomial.
We then take up the study of braids and representations of the braid group and the various
algebras related to the polynomials-Hecke algebra, Temperley-Lieb algebra and BMW algebra and
other appearances of those algebras in mathematics and physics.
14 March to 1 April
Introduction to Quantum Field Theory
Robert de Mello Koch (Witwatersrand)
Quantum Field Theory provides the successful unification of quantum mechanics and special relativity.
The course begins with a description of classical field theory, using classical electrodynamics as a
concrete example. In particular, we discuss both the Lagrangian and Hamiltonian versions of the
dynamics and we derive Noether's theorem. The need for a description employing fields in
any theory consistent with Special Relativity is developed.
The real and complex scalar fields are dealt with in the homework problems.
The quantum field theory of the real scalar field is then developed, using canonical quantization. The
spectrum of the theory is obtained and time ordered correlation functions are introduced. Using the
interactionpicture, the Feynman rules for interacting quantum field obtained. If time permits we will
recover these results using a path integral approach.
The course aims to provide a solid conceptual framework. For example, we explicitly describe how a
simple two point function computation leads to the Yukawa force as an exchange of mesons. We also
explain why much of the structure of the theory follows directly from the structure
of quantum mechanics and special relativity.
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Ecological and Evolutionary Dynamics
Cang Hui, Andriamihaja Ramanatoanina, P Landi (AIMS/Stellenbosch)
Part I – Ecology.
1. The history of ecology and the concept of niche.
2. Biotic interactions – Introducing concepts and models on density dependence (e.g. from
Malthusian to Logistic equations; overcrowding and Allee effect), matrix population model; LotkaVolterra-type model (competition, predation, mutualism; functional response; limiting similarity;
optimal harvest; competitive exclusion; paradox of enrichment, etc.).
3. Spatial distributions – Introducing aggregation (spatial autocorrelation) and its measurements;
occupancy-scaling and occupancy-abundance relationship; niche theory and species distribution
models; analytic and simulation models (reaction-diffusion; cellular automaton; individual based
model; metapopulation), covariance and spatial synchrony.
4. Biodiversity patterns – Introducing species-area curves, species accumulation curves
(rarefaction curves); null model test; neutral theory; niche-based theory; island biogeography.
5. Ecological networks –Introducing network structures (nestedness, modularity, connectance,
node-degree, etc.) and network function (e.g. stability and biodiversity maintenance, productivity);
food webs; nutrient cycles; adaptive networks; regime shift.
Part II – Evolution.
6. The history of evolution and the concept of fitness.
7. Population genetics – Introducing principles of Mendelian genetics (e.g. concepts of genotype
and phenotype, drift, mutation, natural selection, migration/gene flow; Hardy-Weinberg principle).
8. Evolutionary games – Introducing game theory, Prisoner’s dilemma; replicator equation;
evolutionarily stable strategy; behavioural ecology (optimal foraging, the evolution of sex, optimal
clutch size; altruistic behaviours, mimicry, etc.); adaptive immune response and HIV.
9. Trait evolution – Introducing invasion fitness, fitness landscape, adaptive dynamics; Darwin’s
race; evolutionary invasion analysis; co-evolution; evolutionary suicide; evolutionary trap; red
queen dynamics; red king? Green beard?
10. Phylogeny – Introducing tree/clustering analysis and basic modern phylogenetic modelling.
Phylogenetic analysis and phylogeographic analysis. Speciation modes. Coalescent model.
Approximation Theory
Walter van Assche (KU Leuven)
Most of the course will deal with approximation of (real or complex) functions on a set X. We start
by explaining the notion of linear approximation in a normed space. The existence and the
uniqueness of the best approximant are important theoretical issues, the characterization and
computation are more practical aspects, but most of the course consists in understanding how
good the best approximant is: do the approximants converge to the given function, what kind of
convergence, how fast do they converge?
Least squares approximation is often used when the functions belong to a Hilbert space. The best
approximant is then an orthogonal projection onto the approximation space and this is most easily
done by using an orthogonal basis in the approximation space. The basis of trigonometric functions
gives Fourier analysis, but we also introduce bases of orthogonal polynomials and extend Fourier
analysis to this setting.
If the given function is known at a number of points only, then interpolation is used where one looks
for a function in the approximation space which is equal to the given data at the interpolation
points. We deal with polynomial interpolation (Lagrange interpolation) and extend this to Hermite
interpolation and Hermite-Fejér interpolation. Contrary to popular belief, interpolation is not such a
good way to approximate functions, unless the interpolation points are well chosen and the
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function is reasonably smooth. Interpolation is used primarily in numerical quadrature, i.e., in
approximating the integral of a given function. Here orthogonal polynomials are again quite useful
since the zeros of orthogonal polynomials are optimal interpolation points for quadrature (Gauss
quadrature).
Instead of approximating functions by trigonometric functions or polynomials, one can also
approximate by rational functions. This is particularly useful if one wants to get information about
singularities of the function. We explain the notion of Padé approximation and show that the
denominators of Padé approximation at infinity are orthogonal polynomials. The asymptotic
distribution of the zeros of orthogonal polynomials gives information about the singularities of the
function. This asymptotic distribution can be obtained by solving an extremal problem in logarithmic
potential theory, which is explained in the last part of the course.
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