# SOUTH AFRICAN THEORY SCHOOL (SATS) COURSES PRESENTED IN 2021

The SATS pilot programme ran successfully in 2021 with 36 students from 12 South African universities with courses in Nonlinear Dynamics, General Relativity, Cosmology, Machine Learning, Quantum Field Theory, and Advanced Mathematical Methods.

Here are the details of the courses that were presented:

### Advanced Methods for Mathematical Physics

**Outline:**

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

**Skills outcome:**

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

- complex analysis
- spectral analysis
- 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

### Advanced Methods for Mathematical Physics

**Outline:**

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

**Skills outcome:**

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

- complex analysis
- spectral analysis
- 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

### Nonlinear Hamiltonian Dynamics and Chaos

**Outline:**

- 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 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

### Introduction to Machine Learning for Theoretical Physicists

**Outline:**

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

- Introduction to Machine Learning
- Naïve Bayes and Probability
- Decision Trees
- Linear Regression
- Logistic Regression
- A brief overview of Neural Networks
- K-means Clustering
- Practical Application of ML Methods
- Principal Components Analysis
- A brief overview of Reinforcement Learning

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

- An introduction to Pytorch
- A fully connected feed-forward NN
- Convolutional NN’s
- Recurrent NN’s

**PART 3:** *Energy Based Models* (around 5 lectures)

- Restricted Boltzmann Machines
- Hopfield Networks
- 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

### Introduction to Cosmology

**Outline:**

- The Cosmological Principle
- Cosmological Models
- Inflationary Cosmology
- Cosmic Acceleration
- Large-scale Structure.

**Skills outcome: **

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 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

### Introduction to General Relativity

**Outline:**

- Lie groups and symmetries
- Review of special relativity
- Tensor calculus
- Differential forms
- Manifolds
- Geodesics
- Curvature and the Riemann tensor
- Killing vectors
- Maximally symmetric spaces
- Einstein’s equations
- Spherically symmetric solutions
- Basics of black hole physics.

**Skills outcome:**

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

- 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.
- be able to study the motion of free particles on curved spaces by use of the geodesic equation.
- understand the mathematical description of symmetry via the concept of Killing vectors
- reflect on the difficulty of solving Einstein’s equations and provide examples of techniques for doing so.
- 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

### Quantum Field Theory I

**Outline:**

- Postulates of QM and SR
- Quantizing the free scalar field
- Interpreting the results
- Connecting to experiments; in and out states; LSZ reduction
- Lehman-Kallen representation; Gell-Mann–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.

**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

### Quantum Field Theory II

**Outline:**

- Brief introduction to group theory and representations and their importance in quantum state space and constraining potential Lagrangians
- Non-relativistic 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 spin-1 gauge fields. BRST gauge fixing. Non-abelian gauge theory
- 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