SOUTH AFRICAN THEORY AND COMPUTATIONAL SCHOOL: PAST COURSES

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:
At the end of the course, students will have an ability to choose and implement modern machine learning techniques using Python with appropriate training/validation and testing pipelines, hyperparameter tuning etc.

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

Dates:

• Semester 1: 16 August – 11 November 2022
• Semester 2: January – March 2023 (dates to be confirmed).

Lesson format:
Synchronous and asynchronous video lecture delivery.

Method of evaluation:
Coding problems throughout applied to unseen datasets and a project.

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, and 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

1. Preliminaries

1. Maps
2. Vector spaces and linear algebra
3. Multilinear algebra and tensors
4. Topological spaces
5. Neighbourhoods and Hausdorff spaces

2. Differentiable Manifold

1. Differentiable manifold
2. Calculus on manifolds
3. Flows and Lie Derivatives

3. Affine Manifolds

1. Parallel transport
2. Affine Connection
3. Covariant derivatives
4. Curvature and Torsion

4. Riemannian Geometry

1. Riemann and pseudo-Riemann manifolds
2. Metric connection
3. Levi-Civita connection
4. Applications

5. Symmetries

1. Lie group and Lie Algebra
2. Action of Lie groups on Manifolds
3. Isometries and Conformal Transformations
4. Killing and Conformal Killing Vector Fields

6. Differential Forms and Exterior Calculus: TBD, if time permits

Prerequisites:
Linear Algebra, Multivariable Calculus.

Dates:
14 February – 19 May 2022

Lesson format:
Synchronous lectures with a mixture of face-to-face and synchronous tutorial/discussion sessions.

Method of evaluation:
Problem sets, project and a final exam.

Lecturer biography:
Shajid Haque received his PhD in Theoretical Physics from UW-Madison in 2011. He did postdoctoral research at Wits University from 2012 to 2015 and at UCT from 2015 to 2017.  He is interested in the applications of quantum information theory in quantum many body system, quantum field theory, holography and cosmology.

E: shajid.haque @ uct.ac.za | Publications

Outline:

1. Lie groups and symmetries
2. Review of special relativity
3. Tensor calculus
4. Differential forms
5. Review of differential geometry
6. Geodesics
7. Curvature and the Riemann tensor
8. Killing vectors
9. Maximally symmetric spaces
10. Einstein’s equations
11. The Einstein-Hilbert action
12. Spherically symmetric solutions
13. 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.

Prerequisites:
Students are encouraged to have a background in differential geometry (e.g. from the SATACS Differential Geometry course).

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.

Dates:
2nd quarter, 4 April – 20 May, 2022; asynchronous recorded lectures with two weekly live tutorial/discussion sessions.

Method of evaluation:
Weekly assignments and a final project.

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. The Cosmological Principle
2. Cosmological Models
3. Inflationary Cosmology
4. Cosmic Acceleration
5. Large-scale Structure Formation
6. [Some Advanced Topics] Beyond Standard Cosmology.

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

At the end of the course, the 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 specialized 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:
1 semester, 1 August — 4 November, 2022; weekly synchronous lecture videos and tutorials.

Method of evaluation:
Weekly assignments, projects 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.abbebe @ gmail.com | W: Lecturer’s personal website | Publications

Outline:

1. The Schwarzschild solution
2. Black Holes
3. Penrose Diagrams
4. Gravitational waves

Skills outcome:
These are key modern areas of current research in cosmology/astrophysics and gravity and yet the maths and physics underlying these areas is quite old.  Nonetheless, the maths and physics aren’t taught at most places in the country, if anywhere.  This course will prepare students to enter these fields if they want to – by the end of the course, students will be beautifully trained for the future.  And should they not choose these fields, the work will skill them for other related fields.

Prerequisites:
Introduction to General Relativity.

Dates:
25 July — 4 November 2022; in person and synchronous lecture videos.

Method of evaluation:
Weekly assignments, a project and an exam.

Lecturer biography:
Amanda Weltman received her PhD in Physics from Columbia University in 2007.  She held a postdoctoral researcher position at the University of Cambridge from 2007-2009 before joining the University of Cape Town as a Senior Lecturer.  She is currently a Professor at UCT with research interests in astrophysics, fundamental physics, cosmology, and gravity.

E:  amanda.weltman @ uct.ac.za | W: Lecturer’s personal website | Publications

Outline:
The course will provide an introduction to various privacy tools that can be used in order to enable privacy-preserving Artificial Intelligence (with focus on Machine Learning and Natural Language Processing sub-fields of Artificial Intelligence).

Skills outcome:
The developments in Data Science bring with them the challenge of dealing with sensitive datasets; where the need to preserve the privacy of the participants in the datasets is of utmost importance. This challenge warrants the need to explore privacy-preserving schemes that can be used in order to ensure the privacy of the participants in the datasets. This course is intended to introduce these privacy-preserving schemes to the students, and demonstrate how such schemes can be applied in Machine Learning and Natural Language Processing sub-fields of Artificial Intelligence.

At the end of the course, students would have acquired the skills on implementing Data Privacy tools such as Differential Privacy and Federated Learning in Artificial Intelligence tasks. These acquired skills would then make it possible for students to appropriately deal with sensitive datasets such as healthcare datasets, in order to design privacy-preserving Artificial Intelligence techniques.

Prerequisites:
Python programming basics, Linear Algebra, Calculus, and Probability Theory.

Dates:
1 semester, 4 April — 24 June, 2022; synchronous lecture videos.

Method of evaluation:
Weekly problem sets and an exam.

Lecturer biography:
Makhamisa Senekane is a Senior Researcher in the Institute for Intelligent Systems at the University of Johannesburg. Prior to that, he worked as a Lecturer in the Department of Physics and Electronics at the National University of Lesotho, as a Senior Lecturer in the Faculty of Information and Communication Technology at Limkwokwing University of Creative Technology (Lesotho), and as a Lecturer in the Faculty of Computing at Botho University (Maseru Campus). He has obtained his PhD in Physics from the University of KwaZulu-Natal, his MSc.Eng in Electrical Engineering from the University of Cape Town, and his 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

Outline:

The course will present a survey of and interconnections between various mathematical structures that arise on the overlap of abstract mathematics, physics, and computer science. Just to give an example: Hilbert spaces arise in quantum mechanics, their subspaces lead one to quantum logic, while the structure of subspaces of a Hilbert space is that of a “lattice” from order theory in abstract mathematics, which is a generalization of a Boolean algebra, relevant to computer science. The methodology for comparing mathematical structures in this course will rely on ideas from set theory, which enable description of mathematical structures via “elements” (or “points”), and category theory, which looks at networks of mathematical structured formed by structure preserving functions between them (e.g., linear maps in the case of vector spaces).

Skills outcome:
The goal of the course is for the students to see the bigger picture behind modern abstract mathematics and its links to other sciences, while at the same time acquiring research skills in mathematics (thanks to the problem-solving based nature of the course). The students will leave with an intuition for “how abstract mathematics works”. They will also be able to see how the mathematics that they knew previously fits into a broader realm of mathematical structures. Such course is not offered anywhere in South Africa. It will be an excellent complement to existing specialized courses in universities.

Dates:
1 August — 4 November 2022; synchronous lectures over Zoom.

Method of evaluation:
Group work assignments. Each group work will consist of problems that the students must work together on. Marks will be awarded per group. Some of the group work will be to record a joint video presentation explaining their solutions to given problems.

Lecturers:

This course will be presented by a team of investigators in a 2022 NITheCS research programme in Mathematical Structures. Course convenors are the Principal Investigators (listed alphabetically): Partha Pratim Ghosh (UNISA), Yorick Hardy (WITS), Zurab Janelidze (SU), and Cerene Rathilal (UJ).

E: zurab @ sun.ac.za | 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:
The course is designed as one in advanced mathematical methods, aimed at theoretical/mathematical physics students. Material is drawn from various sources, including Arfken & Weber and Morse & Feshbach. Students will leave the course with a working vocabulary in complex analysis, spectral analysis, and some more mathematical aspects of quantum mechanics. These topics are foundational for a number of topics in theoretical and mathematical physics across the spectrum of the South African theory community.

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:
25 July — 4 November 2022; 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 2004-2006 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

uOutline:
Nonlinear Hamiltonian dynamics is used to study the behavior of systems coming from a wide variety of scientific fields, the most important of them being classical mechanics, astronomy, optics, electromagnetism, solid state physics, quantum mechanics, and statistical mechanics. An important phenomenon appearing in nonlinear systems is chaos, which is attributed to the sensitive dependence of a system’s dynamical evolution on its initial conditions.

In this course we will implement several modern numerical techniques to investigate and quantify the chaotic behavior of low-dimensional Hamiltonian systems and area preserving symplectic maps. In particular, we will discuss the following topics:

• Chaos
• Autonomous Hamiltonian systems and symplectic mappings
• Numerical integration of Hamilton equations of motion
• Poincaré surface of section
• Integrals of motion
• Symplectic integrators
• Variational equations
• Tangent Map Method
• Maximum Lyapunov exponent
• Spectrum of Lyapunov exponents
• Chaos indicators

Skills outcome:
Students will learn basic concepts of nonlinear dynamics (particularly in the framework of Hamiltonian mechanics), will be introduced into various traditional as well as modern techniques of analysing chaotic systems and will also implement these methods by themselves in studying some prototypical dynamical models. Thus, students will acquire and practice some basic tools of nonlinear dynamics, which can be implemented in a wide spectrum of problems coming from various scientific fields

Prerequisites:
Basic knowledge of 2nd year classical mechanics.

Dates:
1 semester, 1 August — 4 November, 2022; synchronous lecture videos.

Method of evaluation:
Students will be asked to do 4 assignments of the form of implementing some numerical methods to a particular dynamical system. Then they will have to submit for evaluation at predefined dates their report for each assignment. Depending on practical constrains they might be requested to orally present their assignment (brief presentation of at most 20 minutes). At the beginning of week 6 each student will get a final project concerning the theory and application of a modern chaos technique which he/she will have to finalize by the end of the course, similarly to the 4 assignments. This assignment will be more demanding as it will also require some bibliographical research from the students along with the more applied aspect of the project (numerical application of the chaos indicators to some dynamical systems).

The course’s final mark (FM) will be defined at a 50% level by your mean mark of your 4 assignments (AS1, AS2, AS3, AS4) and 50% by your final project (FP) mark: FM=(AS1+AS2+AS3+AS4)/4 +FP/2.

Lecturer biography:
Charalampos (Haris) Skokos received his PhD in Nonlinear Dynamical Systems from the University of Athens, Greece.  After a number of prestigious appointments, Prof Skokos joined the faculty of the University of Cape Town in 2013; he is now Associate Professor of Mathematics and Applied Mathematics and Deputy Head of Department.  His research interests are in nonlinear dynamical systems, chaotic dynamics and Hamiltonian systems.

E: haris.skokos @ uct.ac.za | W: Lecturer’s personal website  |  Publications

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 2022; synchronous lecture videos twice a 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

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:
One quarter: 12 September — 21 October 2022; 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

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