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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 Review courses 2015-16 1 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. Review courses 2015-16 2 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). __________________________________________________________________________________ 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 Review courses 2015-16 3 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 Review courses 2015-16 4 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. Review courses 2015-16 5 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 Review courses 2015-16 6 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. Review courses 2015-16 7