SOUTH AFRICAN THEORY AND COMPUTATIONAL SCHOOL: PAST COURSES

Outline:

Part I: Structures on manifolds

1. Vectors
2. Tensors
3. Tensor products (with applications to quantum mechanics)
4. Symmetric and skew-symmetric tensors (with applications to bosons and fermions)
5. Cartesian tensors (with applications to Maxwell electrodynamics)

Part II: Differential calculus on manifolds

1. Vector and convector fields
2. Differentiating tensors – the Lie bracket and Lie derivative
3. Exterior calculus
4. Applications to Maxwell electrodynamics and Hamiltonian mechanics
5. Covariant derivatives

Part III: Integration on manifolds

1. Manifolds – charts, atlases, coverings etc.
2. P-form integration
3. Stokes’ theorem
4. Spin textures
5. Homotopy and the Hopf map.
6. 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 semi-rigorous 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 in-person and uploaded to YouTube.

Method of evaluation:
Weekly problem sets and a final take-home 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 2004-2006 and was a member at the Institute for Advanced Study in Princeton in 2016-2017. His research interests lie primarily in understanding emergent phenomena, from condensed matter to neurophysics. His recent focus has been on low-dimensional quantum field theory, topological quantum matter and quantum chaos in disordered systems.

E: jeffmurugan @ gmail.com | W: Lecturer’s personal website  |  Publications

Outline:

1. Introduction to Machine Learning (ML)
2. Introduction to trustworthy Machine Learning
3. Privacy Enhancing Technologies (PETs)
4. Differential Privacy
5. Federated Learning
6. Explainability in Machine Learning
7. Machine Learning robustness
8. 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 KwaZulu-Natal, 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

Outline:

1. Context
   1. the role & responsibilities of a software architect
   2. what is software architecture, including
      1. software architecture vs application design
2. Software Architecture Requirements
   1. specifying, verifying and quantifying software architecture requirements including:
      1. quantified quality requirements
         1. making appropriate quality requirement trade-off decisions
      2. integration and access requirements and
      3. architectural constraints
3. Elements of Software Architecture Design
   1. architectural patterns
   2. architectural tactics
   3. integration patterns
   4. reference architecture and frameworks
4. Systematic Method for Software Architecture Design (SyMAD)
5. Software architecture description
   1. ISO/IEEE standard for a software architecture description
   2. Overview of Software Architecture description frameworks
   3. SyMAD Model based description
   4. Generating text based descriptions from models
6. Software Architecture Recovery
   1. Why?
   2. Overview of architecture recovery methods
   3. Manually recovering a software architecture description from code
7. Software Architecture Analysis
   1. Why?
   2. Overview of analysis methods
   3. The Attribute Trade-Off 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 60-minute lecture per week on Tuesdays from 18h00-19h00.

Method of evaluation:
Group projects and 1 final exam.

Lecturer biography:
After completing a PhD and a post-doc 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 S-PLANE Automation. Here he developed an application server for real-time safety-critical systems, as well as infrastructure for model-based 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

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 in-person at the University of Cape Town. All lectures and tutorials will be broadcast synchronously; the lectures and tutorials will also be recorded in-person and made available online. There are 2 one-hour lectures per week.

Method of evaluation:
Two assignments and a take-home exam.

Lecturer biography:
Tommie Meyer is a professor in Computer Science at UCT and co-director 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 A-rating 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

Outline:

1. Some elementary group theory – definitions, finite groups, products of groups
2. Continuous groups – orthogonal and rotation groups, SO(3), The Lorentz group, SU(2) and SL(2,C)
3. Representation theory – finite dimensional reps., infinite-dimensional reps, SO(2), irreducible representations of SO(3), characters
4. 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 post-graduate 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 2016-2020 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: INSPIRE-HEP profile  |  Publications

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 10:00-12:00). 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

Outline:

1. The Cosmological Principle
2. Cosmological Models
3. Inflationary Cosmology
4. Cosmological Perturbation Theory
5. Large-scale Structure Formation

Skills outcome:
The course will offer hot topics in modern cosmology.

At the end of the course, students are expected to:

1. understand the assumptions in cosmology that led to the formulation of the standard cosmological model
2. derive the cosmological field equations and analyse their solutions
3. demonstrate length and time scales of the universe
4. 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
5. understand the physical processes and mechanisms that lead to large-scale structure formation
6. 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 in-person discussions.

Method of evaluation:
Weekly assignments, 2-3 projects that test both theoretical and computational skills, and an exit-assessment 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 North-West 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

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

Outline:

1. Postulates of QM and SR
2. Quantizing the free scalar field
3. Interpreting the results
4. Connecting to experiments; in and out states; LSZ reduction
5. Lehman-Kallen representation; Gell-Mann–Low theorem; cross sections
6. Feynman rules for scalar fields
7. 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 high-energy quantum chromodynamics applications.

E: wa.horowitz @ gmail.com | W: Lecturer’s personal website | Publications

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 sub-fields 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 KwaZulu-Natal, 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

Outline:

1. 1. Intro to machine learning
   2. Statistical theory and naive bases
   3. Regression tests and training ML models
   4. Gradient free optimisation methods
   5. Classification tasks and model evaluation
   6. Ensemble modeling
   7. Unsupervised learning: Clustering
   8. Unsupervised learning: Dimensionality Reduction
   9. Reinforcement Learning
   10. Intro to Neural Networks
   11. Pytorch
   12. Convolutional Neural Networks
   13. 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

Outline:

1. Introduction
2. First order differential equations
3. Second order differential equations; Preliminaries
4. Integral and differential operators
5. Generalized Green’s identity
6. Green’s identity and adjoint boundary conditions
7. Second order self adjoint operators
8. Green’s functions
9. Properties and construction of Green’s function
10. Generalized Green’s function
11. Second order differential equations with inhomogeneous boundary conditions, initial value problem
12. The Sturm Liouville problem
13. Series representation of the Green’s function
14. Preliminaries to special functions
15. The hypergeometric functions
16. 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: Geo-Net School, Benin, 2019; CIMPA School, Burundi, 2021; East African Institute for Fundamental Research (EAIFR), Rwanda, 2019, 2021; NITheCS Mini-School, 2022.

Her current research interests include new trends in quantization procedures, PT-symmetry 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 Burkina-Faso, where she was born, and currently a dual Italian citizen. She studied Mathematics at the University of Ouagadougou (now Université Joseph Ki-Zerbo), 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), Porto-Novo, 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.

E: laure.gouba @ gmail.com | Publications

Outline:

1. Brief introduction to group theory and representations and their importance in quantum state space and constraining potential Lagrangians
2. Non-relativistic quantum rotations and spin
3. Irreducible representations of the Lorentz group SO(3,1)
4. Free 2D Weyl spinor fields
5. Interacting 2D Weyl spinor fields
6. 4D Majorana and Dirac fields
7. Free spin-1 gauge fields. BRST gauge fixing. Non-abelian gauge theory
8. Spinor helicity techniques. BCFW recursion.

Skills outcome:
Students should have a thorough understanding of quantum field theories for particles up to spin-1.

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 high-energy quantum chromodynamics applications.

E: wa.horowitz @ gmail.com | W: Lecturer’s personal website | Publications

Outline:

1. Winding number in complex analysis and the generalised Cauchy theorem
2. Integral representations and differential equations
3. Advanced method for Green’s functions – resolvents and operator methods
4. Spectral analysis in quantum mechanics
5. Integral equations.

Skills outcome:
At the end of the course, the students are expected to have a working knowledge of:

1. complex analysis
2. spectral analysis
3. some more mathematical aspects of quantum mechanics.

Prerequisites:
Basic knowledge of complex analysis (contour integrals, Cauchy’s theorem etc), differential equations (mostly ordinary); some quantum mechanics is useful but not essential.

Dates:
1 semester, 2 August — 1 November, 2021; meeting twice a week for 2 hours per lecture.

Method of evaluation:
Weekly problem sets.

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 2004 to 2006 and was a member at the Institute for Advanced Study in Princeton in 2016 – 2017. His research interests lie primarily in understanding emergent phenomena, from condensed matter to neurophysics. His recent focus has been on low-dimensional quantum field theory, topological quantum matter and quantum chaos in disordered systems.

E: jeffmurugan @ gmail.com | W: Lecturer’s personal website | Publications

Outline:

1. Winding number in complex analysis and the generalised Cauchy theorem
2. Integral representations and differential equations
3. Advanced method for Green’s functions – resolvents and operator methods
4. Spectral analysis in quantum mechanics
5. Integral equations.

Skills outcome:
At the end of the course, the students are expected to have a working knowledge of:

1. complex analysis
2. spectral analysis
3. some more mathematical aspects of quantum mechanics.

Prerequisites:
Basic knowledge of complex analysis (contour integrals, Cauchy’s theorem etc), differential equations (mostly ordinary); some quantum mechanics is useful but not essential.

Dates:
1 semester, 2 August — 1 November, 2021; meeting twice a week for 2 hours per lecture.

Method of evaluation:
Weekly problem sets.

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 2004 to 2006 and was a member at the Institute for Advanced Study in Princeton in 2016 – 2017. His research interests lie primarily in understanding emergent phenomena, from condensed matter to neurophysics. His recent focus has been on low-dimensional quantum field theory, topological quantum matter and quantum chaos in disordered systems.

E: jeffmurugan @ gmail.com | W: Lecturer’s personal website | Publications

1. Chaos
2. Autonomous Hamiltonian systems and symplectic mappings
3. Numerical integration of Hamilton equations of motion
4. Poincaré surface of section
5. Integrals of motion
6. Symplectic integrators
7. Variational equations
8. Tangent Map Method
9. Maximum Lyapunov exponent
10. Spectrum of Lyapunov exponents
11. Chaos indicators.

Skills outcome:
Students will learn basic concepts of nonlinear dynamics, particularly in the framework of Hamiltonian systems. At the end of the course, they will be able to implement by themselves several modern numerical techniques to investigate and quantify the chaotic behaviour of systems coming from a wide variety of scientific fields, like for example classical mechanics, astronomy, optics, electromagnetism, solid state physics, as well as quantum and statistical mechanics.

Prerequisites:
A good background on Applied Mathematics, Mathematical Modelling, Mathematical Physics and Applied Computing (e.g. numerical analysis, numerical methods, ordinary differential equations, dynamical systems, Newtonian, Lagrangian and Hamiltoninan mechanics) is requited.

Good computational and programming skills in some computational environment and/or computer language (e.g. C, Python, Matlab, Mathematica) is essential.

Dates:
1 semester, 16 August — 12 November 2021; synchronous lecture videos.

Method of evaluation:
During the run of the course: Four (4) assignments (each one followed by a written report) related to the implementation of some numerical methods to particular dynamical systems.

At the end of the course: a final project (in the form of a written report) concerning the theory and applications of a modern chaos detection technique. Students will be working on their final project during the whole second half of the course’s duration.

Lecturer biography:
Associate Professor Haris Skokos acquired a PhD in Physics-Nonlinear Dynamics from the University of Athens (Greece). Over the years, he worked at several research institutes in Europe: the University of Athens (Greece), the Academy of Athens (Greece), the University of Patras (Greece), the Observatory of Paris (France), the Max Planck Institute for the Physics of Complex Systems in Dresden (Germany) and the Aristotle University of Thessaloniki (Greece), and then joined the University of Cape Town in 2013. His research activity belongs to the field of applied mathematics and computational physics, in particular, to nonlinear dynamical systems and chaotic dynamics.

E: haris.skokos @ gmail.com | W: Lecturer’s personal website | Publications

Outline:

PART 1: A broad sweep of ML (around 10 lectures)

1. Introduction to Machine Learning
2. Naïve Bayes and Probability
3. Decision Trees
4. Linear Regression
5. Logistic Regression
6. A brief overview of Neural Networks
7. K-means Clustering
8. Practical Application of ML Methods
9. Principal Components Analysis
10. A brief overview of Reinforcement Learning

PART 2: A deeper dive into deep learning (around 15 lectures)

1. An introduction to Pytorch
2. A fully connected feed-forward NN
3. Convolutional NN’s
4. Recurrent NN’s

PART 3: Energy Based Models (around 5 lectures)

1. Restricted Boltzmann Machines
2. Hopfield Networks
3. Self-supervised learning and modern energy based models

Skills outcome:
Students will come away with a broad overview of machine learning ideas and techniques and will be able to implement them for themselves using Python, and Pytorch in particular. They will understand and know where to apply the three main branches of ML to different datasets and use cases.

Prerequisites:
Python, including familiarity with object oriented coding + calculus and linear algebra.

Dates:
1 semester, 16 August — 12 November 2021; asynchronous video lecture delivery.

Method of evaluation:
Weekly coding challenges and an essay at the end on a subject in ML chosen by the student (and guided by the lecturer/tutor).

Lecturer biography:
Jonathan Shock is a senior lecturer at the University of Cape Town. He has a PhD in theoretical physics from the University of Southampton, focusing on string theory, and continues to work in this field along with researching in machine learning, neuroscience and medical data analysis.

E: jon.shock @ gmail.com | W: Lecturer’s personal website | Publications

Outline:

1. The Cosmological Principle
2. Cosmological Models
3. Inflationary Cosmology
4. Cosmic Acceleration
5. Large-scale Structure.

Skills outcome:
At the end of the course, students are expected to:

1. Understand the assumptions in cosmology that led to the formulation of the standard cosmological model
2. Derive the cosmological field equations and analyse their solutions
3. Demonstrate length and time scales of the universe
4. 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
5. Understand the physical processes and mechanisms that lead to large-scale structure formation.

Prerequisites:
Introduction to General Relativity.

Dates:
4th quarter, 20 September — 12 November; synchronous lecture videos.

Method of evaluation:
Weekly assignments and an 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 North-West University from 2014 to 2015 after which he joined the faculty at this same institution. He is currently an Associate Professor of Physics and his research interests lie in gravitation and cosmology.

E:  amare.abbebe @ gmail.com | W: Lecturer’s personal website | Publications

Outline:

1. Lie groups and symmetries
2. Review of special relativity
3. Tensor calculus
4. Differential forms
5. Manifolds
6. Geodesics
7. Curvature and the Riemann tensor
8. Killing vectors
9. Maximally symmetric spaces
10. Einstein’s equations
11. Spherically symmetric solutions
12. Basics of black hole physics.

Skills outcome:
At the end of the course the student is expected to:

1. show familiarity with the basic tools of GR such as the use of tensors, the metric, the meaning of curvature and the various curvature tensors.
2. be able to study the motion of free particles on curved spaces by use of the geodesic equation.
3. understand the mathematical description of symmetry via the concept of Killing vectors
4. reflect on the difficulty of solving Einstein’s equations and provide examples of techniques for doing so.
5. discuss the main features of static black holes such as event horizons, and (qualitatively) the notion of temperature and entropy for black holes and the paradoxes it leads to.

Dates:
3rd quarter, 2 August – 17 September; asynchronous recorded lectures with a live weekly tutorial/discussion session.

Method of evaluation:
Weekly assignments and a final project.

Prerequisites:
Special relativity, familiarity with coordinate systems for Euclidean space (Cartesian, spherical etc). Some of the exercises will require symbolic manipulation software, preferably SageMath but the choice is up to the student.

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

Outline:

1. Postulates of QM and SR
2. Quantizing the free scalar field
3. Interpreting the results
4. Connecting to experiments; in and out states; LSZ reduction
5. Lehman-Kallen representation; Gell-Mann–Low theorem; cross sections
6. Feynman rules for scalar fields
7. 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.

Dates:
One quarter: 2 August – 17 September 2021.
Synchronous lecture videos twice per week, one synchronous tutorial per week.

Method of evaluation:
Weekly problem sets and an exam.

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, and then joined 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 high-energy quantum chromodynamics applications.

E: wa.horowitz @ gmail.com | W: Lecturer’s personal website | Publications

Outline:

1. Brief introduction to group theory and representations and their importance in quantum state space and constraining potential Lagrangians
2. Non-relativistic quantum rotations and spin
3. Irreducible representations of the Lorentz group SO(3,1)
4. Free 2D Weyl spinor fields
5. Interacting 2D Weyl spinor fields
6. 4D Majorana and Dirac fields
7. Free spin-1 gauge fields. BRST gauge fixing. Non-abelian gauge theory
8. Spinor helicity techniques. BCFW recursion.

Skills outcome:
At the end of the course, the students are expected to have a thorough understanding of quantum field theories for particles up to spin-1.

Prerequisites:
Quantum Field Theory I.

Dates:
One quarter: 27 September — 1 November 2021;
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, and then joined 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 high-energy quantum chromodynamics applications.

E: wa.horowitz @ gmail.com | W: Lecturer’s personal website | Publications