WELCOME TO THE SOUTH AFRICAN THEORY AND COMPUTATIONAL SCHOOL (SATACS)
SATACS is a decentralised, semivirtual, national teaching programme in theoretical and computational sciences. Our goal is to provide teaching of the highest quality and at a level similar to that found in elite postgraduate programmes around the world.
A critically important aim is to provide a pathway to this worldleading material for students who are registered at a university, with a particular emphasis on transformation. Crucially, we want to build and promote a South African community of students who will work together, learn together, and support each other through inperson and, especially, virtual platforms.
We intend for the knowledge presented in these courses to broaden and deepen the base of knowledge for students to then use to perform research of the highest quality and have a background that allows engagement with the elite research and researchers of the world. Therefore, courses are pitched at an Honours / Masters level. Click to view the courses presented in 2021 and 2022.
Click to download a summary of the courses that start in semester 2, 2023. Please note that individual courses will only be run if there is sufficient student interest.
COURSES FOR 2023:
First semester
Differential Geometry  Jeff Murugan
Outline:
Part I: Structures on manifolds
 Vectors
 Tensors
 Tensor products (with applications to quantum mechanics)
 Symmetric and skewsymmetric tensors (with applications to bosons and fermions)
 Cartesian tensors (with applications to Maxwell electrodynamics)
Part II: Differential calculus on manifolds
 Vector and convector fields
 Differentiating tensors – the Lie bracket and Lie derivative
 Exterior calculus
 Applications to Maxwell electrodynamics and Hamiltonian mechanics
 Covariant derivatives
Part III: Integration on manifolds
 Manifolds – charts, atlases, coverings etc.
 Pform integration
 Stokes’ theorem
 Spin textures
 Homotopy and the Hopf map.
 An application to molecular folding – twists and writhes.
Skills outcome:
Geometry is the language of physics, from quantum mechanics, to modern gauge theory to general relativity. This course will give a semirigorous introduction and overview of some of the basics of differential and integral calculus on generally curved manifolds. By the end of the course, students will have a working understanding of notions such as Lie derivatives, exterior calculus, integration on manifolds and covariant derivatives. This should provide a good platform for subsequent courses such as general relativity, cosmology and Lie group theory.
Prerequisites:
Vector calculus is essential. Some quantum mechanics would be recommended but I will cover anything required in the course.
Dates:
Roughly 13 February – 19 May 2023. More details will be available in the first lecture.
Lecture format:
The course will run over 2 days a week for 1.5 hours per lecture.
As this will be running at UCT, it will be in hybrid format. All lectures will be recorded inperson and uploaded to YouTube.
Method of evaluation:
Weekly problem sets and a final takehome exam.
Lecturer biography:
Jeff Murugan is Professor of Mathematical Physics and head of the Laboratory for Quantum Gravity & Strings at the University of Cape Town. He received a PhD in Noncommutative Geometry in String Theory from UCT and Oxford in 2004. He has held a postdoctoral position at Brown University from 20042006 and was a member at the Institute for Advanced Study in Princeton in 20162017. His research interests lie primarily in understanding emergent phenomena, from condensed matter to neurophysics. His recent focus has been on lowdimensional quantum field theory, topological quantum matter and quantum chaos in disordered systems.
E: jeffmurugan @ gmail.com  W: Lecturer’s personal website  Publications
Trustworthy Machine Learning  Makhamisa Senekane
Outline:
 Introduction to Machine Learning (ML)
 Introduction to trustworthy Machine Learning
 Privacy Enhancing Technologies (PETs)
 Differential Privacy
 Federated Learning
 Explainability in Machine Learning
 Machine Learning robustness
 Fairness and Bias in Machine Learning.
Skills outcome:
At the end of the course, the students are expected to:
 Appreciate the need to build trustworthy ML models
 Have a firm understanding and application of PETs such as Differential Privacy and Federated Learning
 Understand and be able to use the ML explainability frameworks
 Be well equipped to the ways of building robustness in Machine Learning models
 Appreciate the value of fairness and the dangers of bias in ML models.
Prerequisites:
Linear Algebra; Probability Theory; Calculus; Programming basics, especially with Python.
Dates:
13 February – 30 April 2023
Lecture format:
Synchronous virtual weekly lecture sessions (2 hours) and tutorial sessions (2 hours). All the sessions will be recorded.
Method of evaluation:
Four assignments, each worth 25% of the total mark.
Lecturer biography:
Makhamisa Senekane is a Senior Researcher in the Institute for Intelligent Systems at the University of Johannesburg. Prior to that, he lectured in the Department of Physics and Electronics at the National University of Lesotho. He was also a Senior Lecturer in the Faculty of Information and Communication Technology at Limkwokwing University of Creative Technology (Lesotho). Further, he lectured in the Faculty of Computing at Botho University (Maseru Campus). He has a PhD in Physics from the University of KwaZuluNatal, MSc.Eng in Electrical Engineering from the University of Cape Town, and B.Eng in Electronics Engineering from the National University of Lesotho. His research interests include data science, data security, data privacy, artificial intelligence (machine learning and natural language processing), and quantum information processing (quantum cryptography, quantum computing, and quantum machine learning).
E: smakhamisa @ uj.ac.za  W: Lecturer’s personal website  Publications
Software Architecture  Fritz Solms
Outline:
 Context
 the role & responsibilities of a software architect
 what is software architecture, including
 software architecture vs application design
 Software Architecture Requirements
 specifying, verifying and quantifying software architecture requirements including:
 quantified quality requirements
 making appropriate quality requirement tradeoff decisions
 integration and access requirements and
 architectural constraints
 quantified quality requirements
 specifying, verifying and quantifying software architecture requirements including:
 Elements of Software Architecture Design
 architectural patterns
 architectural tactics
 integration patterns
 reference architecture and frameworks
 Systematic Method for Software Architecture Design (SyMAD)
 Software architecture description
 ISO/IEEE standard for a software architecture description
 Overview of Software Architecture description frameworks
 SyMAD Model based description
 Generating text based descriptions from models
 Software Architecture Recovery
 Why?
 Overview of architecture recovery methods
 Manually recovering a software architecture description from code
 Software Architecture Analysis
 Why?
 Overview of analysis methods
 The Attribute TradeOff Analysis Method (ATAM)
Skills outcome:
A solid understanding of software architecture versus application design, the concepts and methods and standards applicable to software architecture design.
Prerequisites:
3rd year Computer Science or Software Engineering.
Dates:
7 March – 1 August 2023.
Lecture format:
20 weeks with 1 synchronous 60minute lecture per week on Tuesdays from 18h0019h00.
Method of evaluation:
Group projects and 1 final exam.
Lecturer biography:
After completing a PhD and a postdoc in Theoretical Physics, Fritz was a senior lecturer at Applied Mathematics at the University of Johannesburg. Here he founded with Prof Steeb the International School of Scientific Computing. After having been three years a quantitative analyst and software architect at Standard Corporate and Merchant Bank, Fritz formed a training and consulting company, Solms Training & Consulting (SolmsTC.com). The company provides training and consulting services around software application and software architecture design, as well as a range of software development technologies.
In 2011 he joined the Computer Science Department of the University of Pretoria where he founded and led the Software Architecture and Software Engineering Research (SESAr) Group. Subsequently he was employed as lead software architect at SPLANE Automation. Here he developed an application server for realtime safetycritical systems, as well as infrastructure for modelbased development.
He has currently shifted his focus back onto SolmsTC.com. Since 2017, Fritz is also a research associate of the School of Data Science and Computational Thinking at the University of Stellenbosch, where he supervises some postgraduate students.
E: fritz @ solmstc.com  W: SolmsTC.com; FritzSolms.me  Publications: see lecturer’s websites
Logics for Artificial Intelligence  Tommie Meyer
Outline:
This course will introduce students to logics used in the area of Knowledge Representation and Reasoning – a subarea of Artificial Intelligence.
Skills outcome:
Logic plays a central role in many areas of Artificial Intelligence. This course will introduce students to Propositional Logic, as well as Description Logics, a family of logics frequently used in the area of Knowledge Representation and Reasoning. Description Logics are frequently used to represent formal ontologies.
Prerequisites:
Familiarity with basic discrete mathematics, a basic understanding of algorithms and the analysis of algorithms.
Dates:
11 April to 2 June 2023.
Lecture format:
The course will be presented inperson at the University of Cape Town. All lectures and tutorials will be broadcast synchronously; the lectures and tutorials will also be recorded inperson and made available online. There are 2 onehour lectures per week.
Method of evaluation:
Two assignments and a takehome exam.
Lecturer biography:
Tommie Meyer is a professor in Computer Science at UCT and codirector of the Centre for Artificial Intelligence Research (CAIR). Prior to this he held positions at the CSIR in Pretoria, National ICT Australia, the University of New South Wales in Australia, the University of Pretoria, and the University of South Africa. He is recognised internationally as an expert in Knowledge Representation and Reasoning. He is one of only three South African Computer Scientists to have obtained an Arating from the NRF. He is a member of SAICSIT, AAAI, ACM, and ASSAf.
E: tommie.meyer @ uct.ac.za  W: https://tommiemeyer.org.za
Publications: https://tommiemeyer.org.za/publications
Second semester
Symmetries in Physics  An Introduction to Group Theory  Jaco van Zyl
Outline:
 Some elementary group theory – definitions, finite groups, products of groups
 Continuous groups – orthogonal and rotation groups, SO(3), The Lorentz group, SU(2) and SL(2,C)
 Representation theory – finite dimensional reps., infinitedimensional reps, SO(2), irreducible representations of SO(3), characters
 Group representations in quantum mechanics – SU(2) and SL(2,C) representations, spinors.
Skills outcome:
Group theory and representations of said groups are foundational to modern mathematical and theoretical physics, from quantum mechanics to cosmology. Students will exit this course with a working knowledge of group and representation theory at the postgraduate level.
Prerequisites:
Dates:
26 July – 31 October 2023.
Lecture format:
Fully online.
Method of evaluation:
Problem sets and a final project.
Lecturer biography:
Jaco van Zyl is a postdoctoral fellow in the Laboratory for Quantum Gravity & Strings at the University of Cape Town. After completing his PhD in 2015 at Stellenbosch University he joined the Mandelstam Institute for Theoretical Physics at the University of the Witwatersrand as a postdoctoral fellow from 20162020 and the Department of Physics as a sessional lecturer in 2021.
His research interests include holography, conformal field theory, quantum chaos and quantum complexity. His recent work is funded by the ‘Quantum Technologies for Sustainable Development‘ grant from NITheCS, with the primary aim of developing means of quantum energy storage.
E: hjrvanzyl @ gmail.com  W: INSPIREHEP profile  Publications
Special Topics in Category Theory  Zurab Janelidze
Outline:
In this course we will explore various topics in category theory. The choice of topics can range from basic to advanced and will depend on the existing knowledge of the subject among the participants. Category theory provides a unifying language for conceptualising phenomena across different disciplines, including subjects within pure mathematics, as well as some aspects of quantum physics, computer science, biology, and others. Visit the course website.
Skills outcome:
This course introduces students to basic concepts of category theory, which are useful when applying category theory as a language of conceptualisation in various disciplines. Upon completing the course, you would have gained the skills of making sense of, working with and applying these concepts.
Prerequisites:
Experience with mathematical thinking and working with a symbolic language (for example, experience with mathematical formalisms). Anyone interested in the course is advised to look through the notes and videos.
Dates:
29 July – 4 November 2023 (Saturdays 10:0012:00). Join via this Zoom link.
Method of evaluation:
Students will be assessed based on assignments and presentations.
Lecturer biography:
Zurab Janelidze is a professor of mathematics at Stellenbosch University. He is an Associate of NITheCS and a principal investigator in one of the research programmes at NITheCS. He serves on the editorial boards of two international journals in his field of expertise, category theory, and recently joined the editorial board of Afrika Matematika, a journal of the African Mathematical Union. Prof Janelidze is currently the president of the South African Mathematical Society. He is passionate about discovering and teaching mathematics, as well as looking for mathematical structures in other art forms.
E: zurab @ sun.ac.za  W: Lecturer’s personal website  Publications
Cosmology  Amare Abebe
Outline:
 The Cosmological Principle
 Cosmological Models
 Inflationary Cosmology
 Cosmological Perturbation Theory
 Largescale Structure Formation
Skills outcome:
The course will offer hot topics in modern cosmology.
At the end of the course, students are expected to:
 understand the assumptions in cosmology that led to the formulation of the standard cosmological model
 derive the cosmological field equations and analyse their solutions
 demonstrate length and time scales of the universe
 apply the specialised and integrated knowledge of general relativity and cosmology to critically analyse the shortcomings of the Big Bang Model, and the need to introduce inflation, dark energy, and dark matter
 understand the physical processes and mechanisms that lead to largescale structure formation
 critically analyse the standard cosmological model and understand the need to look at new paradigms beyond the standard model.
Prerequisites:
Introduction to General Relativity.
Dates:
17 July — 30 October 2023
Lecture format:
Weekly synchronous lecture videos and tutorials, with the possibility of inperson discussions.
Method of evaluation:
Weekly assignments, 23 projects that test both theoretical and computational skills, and an exitassessment exam.
Lecturer biography:
Amare Abebe received his PhD in cosmology from the University of Cape Town in 2013. He held a postdoc position at the NorthWest University from 2014 to 2015 after which he joined the faculty at this same institution. He is currently a Professor of Physics and his broad research interests lie in gravitation and cosmology.
E: amare.abebe @ nithecs.ac.za  W: Lecturer’s personal website  Publications
Integrable Systems  Konstantinos Zoubos
Outline:
This course will introduce classical and quantum integrable systems. We will start with integrability in classical dynamical systems, introducing concepts such as Lax pairs, the inverse scattering method and solitons. Afterwards we will focus on the quantum case, covering integrable spin chains and the Bethe ansatz, a brief introduction to quantum groups, and statistical systems such as the Ising, RSOS and percolation models. Prerequisites for the course include lagrangian mechanics, quantum mechanics at Honours level, and group theory. Some necessary aspects from statistical mechanics will be reviewed.
Skills outcome:
Students will develop a useful background for the understanding of topics such as critical phenomena, 2d CFT and many body scattering. Some technical aspects that will need to be introduced (quantum groups, Jacobi theta functions and many more) play important roles in many areas of physics and mathematics.
Prerequisites:
Classical Mechanics and Quantum Mechanics at Honours level, as well as Lie algebras.
Dates:
24 July – 8 September 2023.
Lecture format:
Online lectures (perhaps 3 hours a week) plus a weekly tutorial.
Method of evaluation:
Weekly homework assignments, for most of which students will need to use Sagemath.
Lecturer biography:
Konstantinos Zoubos is Associate Professor at the Physics Department of the University of Pretoria. His research interests are in supersymmetric Quantum Field Theory and String Theory, with an emphasis on integrable structures and the tools to analyse them, such as quantum groups.
E: konstantinos.zoubos @ up.ac.za  W: Lecturer’s personal website
Quantum Field Theory I  W. A. Horowitz
Outline:
 Postulates of QM and SR
 Quantizing the free scalar field
 Interpreting the results
 Connecting to experiments; in and out states; LSZ reduction
 LehmanKallen representation; GellMann–Low theorem; cross sections
 Feynman rules for scalar fields
 Introduction to QED, QED Feynman rules, and trace technology for cross sections.
Skills outcome:
Students will leave the course with a deep understanding of 1) free scalar quantum field theory and 2) Feynman calculus for computing cross sections involving scalar particles. Students should also have a good facility for computing Feynman diagrams and cross sections related to QED processes.
Prerequisites:
A course in advanced quantum mechanics and a course in which special relativity was treated in some detail.
Dates:
25 July — 2 September 2023.
Lecture format:
Synchronous lecture videos four times per week, one synchronous tutorial per week.
Method of evaluation:
Weekly problem sets and a project.
Lecturer biography:
Associate Professor W. A. Horowitz received his PhD in Physics from Columbia University in 2008. He held a postdoctoral research position at the Ohio State University from 2008 to 2010 before joining the faculty at the University of Cape Town. Prof Horowitz is an expert in the use of perturbative quantum field theory and AdS/CFT methods in phenomenological highenergy quantum chromodynamics applications.
E: wa.horowitz @ gmail.com  W: Lecturer’s personal website  Publications
Adversarial Artificial Intelligence  Makhamisa Senekane
Outline:
 Introduction to Artificial Intelligence (AI)
 Categories of Adversarial attacks
 Adversarial attacks in Machine Learning (ML)
 Adversarial attacks in Natural Language Processing (NLP) – Adversarial attacks in Computer Vision (CV)
 Defense strategies against the adversarial attacks in AI.
Skills outcome:
At the end of the course, the students are expected to:
 Have an understanding of various subfields of Artificial Intelligence
 Be familiar with the adversarial attacks on AI systems
 Have a firm understanding of defense strategies against adversarial attacks on AI systems.
Prerequisites:
Linear Algebra; Probability Theory; Calculus; Basics of cryptography; Programming basics, especially with Python.
Dates:
7 August — 29 October 2023.
Lecture format:
Synchronous virtual weekly lecture sessions (2 hours) and tutorial sessions (2 hours). All the sessions will be recorded.
Method of evaluation:
Four assignments, each worth 25% of the total mark.
Lecturer biography:
Makhamisa Senekane is a Senior Researcher in the Institute for Intelligent Systems at the University of Johannesburg. Prior to that, he lectured in the Department of Physics and Electronics at the National University of Lesotho. He was also a Senior Lecturer in the Faculty of Information and Communication Technology at Limkwokwing University of Creative Technology (Lesotho). Further, he lectured in the Faculty of Computing at Botho University (Maseru Campus). He has a PhD in Physics from the University of KwaZuluNatal, MSc.Eng in Electrical Engineering from the University of Cape Town, and B.Eng in Electronics Engineering from the National University of Lesotho. His research interests include data science, data security, data privacy, artificial intelligence (machine learning and natural language processing), and quantum information processing (quantum cryptography, quantum computing, and quantum machine learning).
E: smakhamisa @ uj.ac.za  W: Lecturer’s personal website  Publications
Machine Learning: from Linear Regression to Deep Learning  Jonathan Shock
Outline:

 Intro to machine learning
 Statistical theory and naive bases
 Regression tests and training ML models
 Gradient free optimisation methods
 Classification tasks and model evaluation
 Ensemble modeling
 Unsupervised learning: Clustering
 Unsupervised learning: Dimensionality Reduction
 Reinforcement Learning
 Intro to Neural Networks
 Pytorch
 Convolutional Neural Networks
 Recurrent Neural Networks
PLUS… a project based on the above topics on a research area of the student’s choice
Skills outcome:
A practical, working knowledge of a wide variety of machine learning techniques including supervised, unsupervised and reinforcement learning.
Prerequisites:
Python, including familiarity with object oriented coding + calculus and linear algebra.
Dates:
 14 August – 10 November 2023 (semester 1)
 8 January – 29 March 2024 (semester 2)
Lecture format:
Weekly discussion sessions based on guided, asynchronous video lectures.
Method of evaluation:
Weekly assignments and a project at the end.
Lecturer biography:
Jonathan Shock is an Associate Professor at the University of Cape Town. He has a PhD in theoretical physics from the University of Southampton, focusing on string theory. He continues to work in this field along with researching in machine learning, neuroscience and medical data analysis.
E: jon.shock @ gmail.com  W: www.shocklab.net  Publications
Ordinary Differential Equations  Laure Gouba
Outline:
 Introduction
 First order differential equations
 Second order differential equations; Preliminaries
 Integral and differential operators
 Generalized Green’s identity
 Green’s identity and adjoint boundary conditions
 Second order self adjoint operators
 Green’s functions
 Properties and construction of Green’s function
 Generalized Green’s function
 Second order differential equations with inhomogeneous boundary conditions, initial value problem
 The Sturm Liouville problem
 Series representation of the Green’s function
 Preliminaries to special functions
 The hypergeometric functions
 The confluent hypergeometric functions.
Prerequisites:
Elementary differential and integral calculus, vector analysis, theory of systems of algebraic equations.
Dates:
21 August – 20 October 2023
Lecture format:
Synchronous virtual lecture videos.
Method of evaluation:
Weekly problem sets and a final exam.
Lecturer biography:
Dr Laure Gouba is a mathematical physicist in visit at the Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, Italy. She is a visiting lecturer at the African Institute for Mathematical Sciences (AIMS) and a member of the UNESCO International Chair of Mathematical Physics and Applications (ICMPA), Université Abomey Calavi (UAC), Cotonou, Benin. Dr Gouba has been involved in various teaching activities in Africa: GeoNet School, Benin, 2019; CIMPA School, Burundi, 2021; East African Institute for Fundamental Research (EAIFR), Rwanda, 2019, 2021; NITheCS MiniSchool, 2022.
Her current research interests include new trends in quantization procedures, PTsymmetry in quantum mechanics, coherent states, noncommutative quantum mechanics, quantum correlations and quantum cosmology. Dr Gouba is a referee for many international journal in mathematical physics. On January 2020, she has obtained the Italian National Scientific Qualification (ASN) for the title of Associate Professor in Mathematical Physics.
Dr Gouba is native from BurkinaFaso, where she was born, and currently a dual Italian citizen. She studied Mathematics at the University of Ouagadougou (now Université Joseph KiZerbo), where she obtained a DEA (Research Master) in Mathematics in 1999. She enrolled in a PhD programme in 2001 at Institut de Mathématiques et de Sciences Physiques (IMSP), PortoNovo, Benin, and obtained a PhD degree in Mathematical Physics in 2005. Dr Gouba worked for AIMS as a senior tutor from 2006 to 2008. She was a postdoctoral fellow at NITheP (now NITheCS) from 2008 to 2010. Dr. Laure Gouba is an alumna and a full member of the Organization for Women in Sciences for the Developing World.
Quantum Field Theory II  W. A. Horowitz
Outline:
 Brief introduction to group theory and representations and their importance in quantum state space and constraining potential Lagrangians
 Nonrelativistic quantum rotations and spin
 Irreducible representations of the Lorentz group SO(3,1)
 Free 2D Weyl spinor fields
 Interacting 2D Weyl spinor fields
 4D Majorana and Dirac fields
 Free spin1 gauge fields. BRST gauge fixing. Nonabelian gauge theory
 Spinor helicity techniques. BCFW recursion.
Skills outcome:
Students should have a thorough understanding of quantum field theories for particles up to spin1.
Prerequisites:
Quantum Field Theory I.
Dates:
12 September — 21 October 2023.
Lecture format:
Synchronous lecture videos twice per week, one synchronous tutorial per week.
Method of evaluation:
Weekly problem sets and a project.
Lecturer biography:
Associate Professor W. A. Horowitz received his PhD in Physics from Columbia University in 2008. He held a postdoctoral research position at the Ohio State University from 2008 to 2010 before joining the faculty at the University of Cape Town. Prof Horowitz is an expert in the use of perturbative quantum field theory and AdS/CFT methods in phenomenological highenergy quantum chromodynamics applications.
E: wa.horowitz @ gmail.com  W: Lecturer’s personal website  Publications
HOW TO APPLY
Please complete the Student Application Form and have your supervisor complete the Supervisor’s Support Form. To apply, you require a Google account. Please click if you need to create a free Google account.
NGACoursework
NGACoursework is an initiative funded by the South African National Graduate Academy for Mathematical and Statistical Sciences (NGA(MaSS)). It provides a centralised platform to pool the resources of all South African universities, to offer a more consistent and wider range of masters level coursework in mathematics and statistical sciences. Such pooledresource models have proven to be successful internationally.
NGACoursework is presented online and geared toward MSc and PhD students who are registered at South African universities. Staff and postdocs may also participate. NGACoursework is free of charge for all participants.
Quality assurance of all courses is the mandate of the NGA (MaSS). Participants who wish to negotiate credit recognition at their home university should discuss this with the steering committee and their home university administration ahead of time.