# Physics

Unit outlines will be available though Find a unit outline two weeks before the first day of teaching for 1000-level and 5000-level units, or one week before the first day of teaching for all other units.

## PHYSICS (HONOURS)

The Bachelor of Advanced Studies (Honours) (Physics) requires 48 credit points from this table including:

(i) 18 credit points of 4000-level Honours coursework selective units from List 1, and

(ii) 6 credit points of 4000-level Honours coursework selective units from List 1, or List 2, or List 3, and

(iii) 24 credit points of 4000-level Honours research project units

#### Honours Coursework Selective

##### List 1

**PHYS4121 Advanced Electrodynamics and Photonics**

Credit points: 6 Teacher/Coordinator: Bruce Yabsley Session: Semester 1 Classes: lectorial (integrated lecture and tutorial)3 hr/week for 12 weeks Prerequisites: An average of at least 65 in 144 cp of units including (PHYS3x35 or PHYS3x40 or PHYS3941) Assumed knowledge: A major in physics including third-year electromagnetism and third-year optics Assessment: 4 x quizzes (10%), 3 x written assignments (30%), final exam (60%) Campus: Camperdown/Darlington, Sydney Mode of delivery: Normal (lecture/lab/tutorial) day

The electromagnetic force is the only one of the four fundamental forces of nature that is compatible with the four realms of mechanics (classical, quantum, relativistic and relativistic quantum mechanics) and therefore, the study of electrodynamics is fundamental to understanding how the laws of physics may be unified, but also to identify gaps in our knowledge. Drawing upon the foundations of classical electromagnetism and optics laid in the undergraduate physics major, this unit provides an advanced-level treatment of topics in electrodynamics and photonics underlying cutting-edge modern research. Starting with the mathematically elegant covariant formalism of the Maxwell equations, from which special relativity derives, the unit covers topics such as the origin of radiation from relativistic particles and from atoms, which are important in astrophysics and particle physics as well as optical and quantum physics. This then introduces the theme of light-matter interactions, which reveals how light can be manipulated and controlled, leading to fascinating phenomena such as optical tweezers, topological insulators and metamaterials. The unique properties and applications of confined electromagnetic waves and their nonlinear interactions are studied in depth, followed by the physics of laser light. The unit is completed with the contemporary research topic of quantum optics. In studying these topics, you will learn advanced theoretical concepts and associated mathematical methods in physics, including tensor calculus, Greens function method, multipole expansion in field theory, and coupled mode theory. By doing this unit, you will be able to synthesise your knowledge of physics and gain new insights into how to identify and apply relevant aspects of physics-based concepts and techniques to solve modern research problems.

**PHYS4122 Astrophysics and Space Science**

Credit points: 6 Teacher/Coordinator: Bruce Yabsley Session: Semester 1 Classes: 3 hrs/weekintegrated lecture/computer lab for 10 weeks Prerequisites: An average of at least 65 in 144 cp of units Assumed knowledge: A major in physics Assessment: 3 x quizzes (10%), 2 x written assignments (20%), computer lab quiz (10%), final exam (60%) Campus: Camperdown/Darlington, Sydney Mode of delivery: Normal (lecture/lab/tutorial) day

Modern astrophysics covers a vast range of scales, from processes within the Solar System which allow for direct testing, to processes that take place in distant places and times, such as the formation of galaxies. Nonetheless, the same physics underpins all of these situations: the plasma of the solar system meets the interstellar medium, which provides the building blocks for galaxies. This unit provides an advanced-level treatment of three major topics in astrophysics: the formation and evolution of galaxies, the structure and morphology of galaxies, and the physics of plasma in our Solar System. You will learn about the behaviour of gas and plasma throughout the Universe, and their effect on phenomena from galaxy structure to space weather. By doing this unit, you will learn how to synthesise your knowledge of physical concepts and processes, and how these concepts and techniques are used to solve modern research problems.

**PHYS4123 General Relativity and Cosmology**

Credit points: 6 Teacher/Coordinator: Bruce Yabsley Session: Semester 2 Classes: lectorial (integrated lecture and tutorial) 4 hrs/week for 8 weeks (total 30 hours) Prerequisites: An average of at least 65 in 144 cp of units Assumed knowledge: A major in physics and knowledge of special relativity Assessment: 2 x quizzes (10%), 3 x written assignments (30%), final exam (60%) Campus: Camperdown/Darlington, Sydney Mode of delivery: Normal (lecture/lab/tutorial) day

Einstein's General Theory of Relativity represents a pinnacle of modern physics, providing the most accurate description of the action of gravity across the cosmos. To Newton, gravity was simply a force between masses, but Einstein's mathematical language describes gravity in terms of the bending and stretching of space-time. In this course, students will review Einstein's principle of relativity, and the mathematical form of special relativity, and the flat space-time this implies. This will be expanded and generalised to consider Einstein's principle of equivalence and the implications for particle and photon motion with curved space-time. Students will explore the observational consequences of general relativity in several space-time metrics, in particular the Schwarzschild black hole, the Morris-Thorne wormhole, and the Alcubierre warp drive, elucidating the nature of the observer in determining physical quantities. Building on this knowledge, students will understand Einstein's motivation in determining the field equations, relating the distribution of mass and energy to the properties of space-time. Students will apply the field equations, including deriving the cosmological Friedmann-Robertson-Walker metric from the assumption of constant curvature, and using this to determine the universal expansion history and key observables. Students will obtain a complete picture of our modern cosmological model, understanding the constituents of the universe, the need for inflation in the earliest epochs, and the ultimate fate of the cosmos.

**PHYS4124 Physics of the Standard Model**

Credit points: 6 Teacher/Coordinator: Bruce Yabsley Session: Semester 2 Classes: lectures 3 hrs/week for 10 weeks Prerequisites: An average of at least 65 in 144 cp of units including (PHYS3X34 or PHYS3X42 or PHYS3X43 or PHYS3X44) Assumed knowledge: A major in physics including third-year quantum physics and third-year particle physics Assessment: 3 x written assignments (45%), final exam (55%) Campus: Camperdown/Darlington, Sydney Mode of delivery: Normal (lecture/lab/tutorial) day

Our current understanding of the basic building blocks of matter and interactions between them is called the Standard Model of Particle Physics (SM). This most fundamental description of Nature incorporates three of the four basic interactions which govern how the Universe works, the electromagnetic, weak and strong interactions. This unit investigates the mathematical underpinnings of the SM, a quantum field theory constructed upon fundamental notions of symmetry including Lorentz and local gauge invariance. It also explores the notion of spontaneous symmetry breaking, the Higgs field and the way that fundamental particles acquire mass. The interplay between theory and experiment, which has driven the SM's development, is highlighted. Finally, limitations of the model and possible extensions which could overcome them are discussed. You will learn how the SM is constructed based on symmetry principles, quantum fields and their space-time derivatives; how to derive equations of motion for the fields using the Action Principle; and how predictions for physical observables such as cross sections and decay rates can be calculated starting from the SM Lagrangian density. By studying examples of both recent and historically significant measurements confirming or challenging the SM, you will gain experience in reading and interpreting the scientific literature. Through this unit you will develop an appreciation of humankind's most contemporary and successful attempt to describe Nature in terms of fundamental laws.

**PHYS4125 Quantum Field Theory**

Credit points: 6 Teacher/Coordinator: Bruce Yabsley Session: Semester 1 Classes: lectures and tutorial/discussion sessions 3 hrs/week for 12 weeks Prerequisites: An average of at least 65 in 144 cp of units including (PHYS3x34 or PHYS3x42 or PHYS3x43 or PHYS3x44 or PHYS3x35 or PHYS3x40 or PHYS3941 or PHYS3x36 or PHYS3x68 or MATH3x63 or MATH4063 or MATH3x78 or MATH4078) Assumed knowledge: A major in physics including third-year quantum physics Assessment: 4 x written assignments (50% total), final exam (50%) Campus: Camperdown/Darlington, Sydney Mode of delivery: Normal (lecture/lab/tutorial) day

Quantum Field Theory (QFT) is the basic mathematical framework that is used for a consistent quantum-mechanical description of relativistic systems, such as fundamental subatomic particles in particle physics. The tools of QFT are also used for description of quasi-particles and critical phenomena in condensed matter physics and other related fields. This course introduces major concepts and technical tools of QFT. The course is largely self-contained and covers alsoLagrangian and Hamiltonian formalisms for classical fields, elements of group theory and path integral formulation of quantum mechanics. The main topics include second quantization of various fields and description of their interactions, with the main focus on the most accurate fundamental theory of quantum electromagnetism. The last part of the course deals the concept of the renormalisation group, and its applications to critical phenomena in condensed matter systems. By completing this course, you will obtain knowledge of major concepts and tools of contemporary fundamental physics, that can be employed in a wide range of physics and physics-based research, starting from the description of profound effects in condensed matter physics and ending by the understanding of basic building blocks of the Universe .

Textbooks

L.H. Ryder, Quantum Field Theory, Cambridge University Press, (1996), F. Mandl and G. Shaw, Quantum FieldTheory, Wiley-Blackwell, (2010), M.E. Peskin and D.V. Schroeder: An Introduction to quantum field theory, Adison-Wesley (1995), T. Lancaster and S. J. Blundell Quantum Field Theory for the Gifted Amateur, Oxford University Press, (2014)

**PHYS4126 Quantum Nanoscience**

Credit points: 6 Teacher/Coordinator: Bruce Yabsley Session: Semester 2 Classes: lecture 3 hrs/week for 10 weeks Prerequisites: An average of at least 65 in 144 cp of units Assumed knowledge: A major in physics including third-year quantum physics and third-year condensed matter physics Assessment: 4 x quizzes (10%), essay (40%), final exam (50%) Campus: Camperdown/Darlington, Sydney Mode of delivery: Normal (lecture/lab/tutorial) day

Modern nanofabrication and characterisation techniques now allow us to build devices that exhibit controllable quantum features and phenomena. We can now demonstrate the thought experiments posed by the founders of quantum mechanics a century ago, as well as explore the newest breakthroughs in quantum theory. We can also develop new quantum technologies, such as quantum computers. This unit will investigate the latest research results in quantum nanoscience across a variety of platforms. You will be introduced to the latest research papers in the field, published in high-impact journals, and gain an appreciation and understanding of the diverse scientific elements that come together in this research area, including materials, nanofabrication, characterisation, and fundamental theory. You will learn to assess an experiment's demonstration of phenomena in quantum nanoscience, such as quantum coherence and entanglement, mesoscopic transport, exotic topological properties, etc. You will acquire the ability to approach a modern research paper in physics, and to critically analyse the results in the context of the wider scientific community. By doing this unit you will develop the capacity to undertake research, experimental and/or theoretical, in quantum nanoscience.

##### List 2

**PHYS4015 Neural Dynamics and Computation**

Credit points: 6 Session: Semester 2 Classes: lecture 3 hrs/week Prerequisites: 144cp of units including (MATH1x01 or MATH1x21 or MATH1906 or MATH1931) and MATH1x02 Assumed knowledge: First- and second-year physics Assessment: homework assignment (30%), group presentation (20%), final exam (50%) Campus: Camperdown/Darlington, Sydney Mode of delivery: Normal (lecture/lab/tutorial) day

What is the neural code? How do neural circuits communicate information? What happens in our brain when we make a decision? Computational modelling and theoretical analysis are important tools for addressing these fundamental questions and for determining the functioning mechanisms of the brain. This interdisciplinary unit will provide a thorough and up-to-date introduction to the fields of computational neuroscience and neurophysics. You will learn to develop basic models of how neurons process information and perform quantitative analyses of real neural circuits in action. These models include neural activity dynamics at many different scales, including the biophysical, the circuit and the system levels. Basic data analytics of neural recordings at these levels will also be explored. In addition, you will become familiar with the computational principles underlying perception and cognition, and algorithms of neural adaptation and learning, which will provide knowledge for building-inspired artificial intelligence. Your theoretical learning will be complemented by inquiry-led practical classes that reinforce the above concepts. By doing this unit, you will develop essential modelling and quantitative analysis skills for studying how the brain works.

**SCIE4001 Science Communication**

Credit points: 6 Teacher/Coordinator: Dr Alice E Motion Session: Semester 1 Classes: lecture 2-3 hrs/week, workshops 1-2hrs/week Prerequisites: 144 credit points of units of study and including a minimum of 24 credit points at the 3000- or 4000-level and 18 credit points of 3000- or 4000-level units from Science Table A. Assumed knowledge: Completion of a major in a science discipline. Basic knowledge of other sciences is beneficial. Experience in communication such as delivering oral presentations and producing written reports. An awareness of science in a societal context, e.g., of disciplinary applications. Assessment: seminar/workshop attendance and completion of 'course notebook' (10%; individual), written article communicating science topic to specific audience (25%; individual), illustrating science (sound/figure/animation/diagram etc; 15%), 3 minute presentation of science topic to specific audience (25%; individual), group report (25%) Campus: Camperdown/Darlington, Sydney Mode of delivery: Normal (lecture/lab/tutorial) day

Note: Mid-year honours students would take this unit of study in S1 (their second semester of study).

"If you can't explain it simply, you don't understand it well enough". This quote is widely attributed to Albert Einstein, but regardless of its provenance, it suggests that one measure of an expert's knowledge can be found in their ability to translate complex ideas so that they are accessible to anyone. The communication of science to the public is essential for science and society. In order to increase public understanding and appreciation of science, researchers must be able to explain their results, and the wider context of their research, to non-experts. This unit will explore some theoretical foundations of science communications, identify outstanding practitioners and empower students to produce effective science communication in different media. In this unit you will learn the necessary skills and techniques to tell engaging and informative science stories in order to bring complex ideas to life, for non-expert audiences. By undertaking this unit you will develop a greater understanding of the wider context of your honours unit, advance your communication skills and be able to explain your honours research to non-expert audiences such as friends, family or future employers. These transferable skills will equip you for future research - where emphasis is increasingly placed on public communication and/or outreach - or professional pathways - where effective communication of complex ideas is highly valued.

**SCIE4002 Experimental Design and Data Analysis**

Credit points: 6 Session: Intensive March Classes: 4 x 1 hr lectures/week, for six weeks, either online or face-to-face and 1 x 2 hour workshop/week for six weeks Prerequisites: 144 credit points of units of study and including a minimum of 24 credit points at the 3000- or 4000-level and 18 credit points of 3000- or 4000-level units from Science Table A. Prohibitions: ENVX3002 or STAT3X22 or STAT4022 or STAT3X12 Assumed knowledge: Completion of units in quantitative research methods, mathematics or statistical analysis at least at 1000-level. Assessment: design critique (20%), research plan (30%), analysis critique (20%), 2 x analysis quizzes (15% each) Campus: Camperdown/Darlington, Sydney Mode of delivery: Block mode

An indispensable attribute of an effective scientific researcher is the ability to collect, analyse and interpret data. Central to this process is the ability to create hypotheses and test these by using rigorous experimental designs. This modular unit of study will introduce the key concepts of experimental design and data analysis. Specifically, you will learn to formulate experimental aims to test a specific hypothesis. You will develop the skills and understanding required to design a rigorous scientific experiment, including an understanding of concepts such as controls, replicates, sample size, dependent and independent variables and good research practice (e. g. blinding, randomisation). By completing this unit you will develop the knowledge and skills required to appropriately analyse and interpret data in order to draw conclusions in the context of an advanced research project. From this unit of study, you will emerge with a comprehensive understanding of how to optimise the design and analysis of an experiment to most effectively answer scientific questions.

**SCIE4003 Ethics in Science**

Credit points: 6 Teacher/Coordinator: A/Prof Hans Pols Session: Intensive August,Intensive March Classes: part a: lecture/seminars 4hr/week for 3 weeks, in which all students participate, followed by two modules, part b (human ethics) and part c (animal ethics), from which students select one; each module comprises 8 hours of workshops over 1-2 weeks Prerequisites: 144 credit points of units of study and including a minimum of 24 credit points at the 3000- or 4000-level and 18 credit points of 3000- or 4000-level units from Science Table A Prohibitions: HSBH3004 or HPSC3107 Assumed knowledge: Successful completion of a Science major. Assessment: essay (40%), presentation (20%), final exam (40%) Campus: Camperdown/Darlington, Sydney Mode of delivery: Block mode

In the contemporary world, a wide variety of ethical concerns impinge upon the practice of scientific research. In this unit you will learn how to identify potential ethical issues within science, acquire the tools necessary to analyse them, and develop the ability to articulate ethically sound insights about how to resolve them. In the first portion of the unit, you will be familiarised with how significant developments in post-World War II science motivated sustained ethical debate among scientists and in society. In the second portion of the unit, you will select from either a Human Ethics module or an Animal Ethics module and learn the requirements of how to ensure your research complies with appropriate national legislation and codes of conduct. By undertaking this unit you will develop the ability to conduct scientific research in an ethically justifiable way, place scientific developments and their application in a broader social context, and analyse the social implications and ethical issues that may potentially arise in the course of developing scientific knowledge.

##### List 3

**MATH4311 Algebraic Topology**

Credit points: 6 Session: Semester 2 Classes: 3 x 1hr lecture/week, 1 x 1hr tutorial/week Assumed knowledge: Familiarity with abstract algebra and basic topology, e.g., (MATH2922 or MATH2961 or equivalent) and (MATH2923 or equivalent). Assessment: tutorial participation (10%), 2 x homework assignments (40%), final exam (50%) Campus: Camperdown/Darlington, Sydney Mode of delivery: Normal (lecture/lab/tutorial) day

One of the most important aims of algebraic topology is to distinguish or classify topological spaces and maps between them up to homeomorphism. Invariants and obstructions are key to achieve this aim. A familiar invariant is the Euler characteristic of a topological space, which was initially discovered via combinatorial methods and has been rediscovered in many different guises. Modern algebraic topology allows the solution of complicated geometric problems with algebraic methods. Imagine a closed loop of string that looks knotted in space. How would you tell if you can wiggle it about to form an unknotted loop without cutting the string? The space of all deformations of the loop is an intractable set. The key idea is to associate algebraic structures, such as groups or vector spaces, with topological objects such as knots, in such a way that complicated topological questions can be phrased as simpler questions about the algebraic structures. In particular, this turns questions about an intractable set into a conceptual or finite, computational framework that allows us to answer these questions with certainty. In this unit you will learn about fundamental group and covering spaces, homology and cohomology theory. These form the basis for applications in other domains within mathematics and other disciplines, such as physics or biology. At the end of this unit you will have a broad and coherent knowledge of Algebraic Topology, and you will have developed the skills to determine whether seemingly intractable problems can be solved with topological methods.

**MATH4312 Commutative Algebra**

Credit points: 6 Session: Semester 1 Classes: lecture 3 hrs/week and tutorial 1 hr/week Assumed knowledge: Familiarity with abstract algebra, e.g., MATH2922 or equivalent. Assessment: 2 x submitted assignments (20% each), final exam (60%) Campus: Camperdown/Darlington, Sydney Mode of delivery: Normal (lecture/lab/tutorial) day

Commutative Algebra provides the foundation to study modern uses of Algebra in a wide array of settings, from within Mathematics and beyond. The techniques of Commutative Algebra underpin some of the most important advances of mathematics in the last century, most notably in Algebraic Geometry and Algebraic Topology. This unit will teach students the core ideas, theorems, and techniques from Commutative Algebra, and provide examples of their basic applications. Topics covered include affine varieties, Noetherian rings, Hilbert basis theorem, localisation, the Nullstellansatz, ring specta, homological algebra, and dimension theory. Applications may include topics in scheme theory, intersection theory, and algebraic number theory. On completion of this unit students will be thoroughly prepared to undertake further study in algebraic geometry, algebraic number theory, and other areas of mathematics. Students will also gain facility with important examples of abstract ideas with far-reaching consequences.

**MATH4314 Representation Theory**

Credit points: 6 Session: Semester 1 Classes: 3 x 1hr lecture/week, 1 x 1hr tutorial/week Prohibitions: MATH3966 Assumed knowledge: Familiarity with abstract algebra, specifically vector space theory and basic group theory, e.g., MATH2922 or MATH2961 or equivalent. Assessment: tutorial participation (10%), 2 x homework assignments (40%), final exam (50%) Campus: Camperdown/Darlington, Sydney Mode of delivery: Normal (lecture/lab/tutorial) day

Representation theory is the abstract study of the possible types of symmetry in all dimensions. It is a fundamental area of algebra with applications throughout mathematics and physics: the methods of representation theory lead to conceptual and practical simplification of any problem in linear algebra where symmetry is present. This unit will introduce you to the basic notions of modules over associative algebras and representations of groups, and the ways in which these objects can be classified. You will learn the special properties that distinguish the representation theory of finite groups over the complex numbers, and also the unifying principles which are common to the representation theory of a wider range of algebraic structures. By learning the key concepts of representation theory you will also start to appreciate the power of category-theoretic approaches to mathematics. The mental framework you will acquire from this unit of study will enable you both to solve computational problems in linear algebra and to create new mathematical theory.

**MATH4313 Functional Analysis**

Credit points: 6 Session: Semester 1 Classes: lecture 3 hrs/week, tutorials 1 hr/week Assumed knowledge: Real Analysis (e.g., MATH2X23 or equivalent), and, preferably, knowledge of Metric Spaces. Assessment: 3 x homework assignments (total 30%), final exam (70%) Campus: Camperdown/Darlington, Sydney Mode of delivery: Normal (lecture/lab/tutorial) day

Functional analysis is one of the major areas of modern mathematics. It can be thought of as an infinite-dimensional generalisation of linear algebra and involves the study of various properties of linear continuous transformations on normed infinite-dimensional spaces. Functional analysis plays a fundamental role in the theory of differential equations, particularly partial differential equations, representation theory, and probability. In this unit you will cover topics that include normed vector spaces, completions and Banach spaces; linear operators and operator norms; Hilbert spaces and the Stone-Weierstrass theorem; uniform boundedness and the open mapping theorem; dual spaces and the Hahn-Banach theorem; and spectral theory of compact self-adjoint operators. A thorough mechanistic grounding in these topics will lead to the development of your compositional skills in the formulation of solutions to multifaceted problems. By completing this unit you will become proficient in using a set of standard tools that are foundational in modern mathematics and will be equipped to proceed to research projects in PDEs, applied dynamics, representation theory, probability, and ergodic theory.

**MATH4315 Variational Methods**

Credit points: 6 Session: Semester 2 Classes: lectures 3 hrs/week, tutorial 1 hr/week Assumed knowledge: Assumed knowledge of MATH2X23 or equivalent; MATH4061 or MATH3961 or equivalent; MATH3969 or MATH4069 or MATH4313 or equivalent. That is, real analysis, basic functional analysis and some acquaintance with metric spaces or measure theory. Assessment: 2 x homework assignments (20% each), final exam (60%) Campus: Camperdown/Darlington, Sydney Mode of delivery: Normal (lecture/lab/tutorial) day

Variational and spectral methods are foundational in mathematical models that govern the configurations of many physical systems. They have wide-ranging applications in areas such as physics, engineering, economics, differential geometry, optimal control and numerical analysis. In addition they provide the framework for many important questions in modern geometric analysis. This unit will introduce you to a suite of methods and techniques that have been developed to handle these problems. You will learn the important theoretical advances, along with their applications to areas of contemporary research. Special emphasis will be placed on Sobolev spaces and their embedding theorems, which lie at the heart of the modern theory of partial differential equations. Besides engaging with functional analytic methods such as energy methods on Hilbert spaces, you will also develop a broad knowledge of other variational and spectral approaches. These will be selected from areas such as phase space methods, minimax theorems, the Mountain Pass theorem or other tools in the critical point theory. This unit will equip you with a powerful arsenal of methods applicable to many linear and nonlinear problems, setting a strong foundation for understanding the equilibrium or steady state solutions for fundamental models of applied mathematics.

**MATH4061 Metric Spaces**

Credit points: 6 Teacher/Coordinator: Dr Leo Tzou Session: Semester 1 Classes: Three 1 hour lectures and one 1 hour tutorial per week. Prerequisites: An average mark of 65 or above in 12cp from the following units (MATH2X21 or MATH2X22 or MATH2X23 or MATH3061 or MATH3066 or MATH3063 or MATH3076 or MATH3078 or MATH3962 or MATH3963 or MATH3968 or MATH3969 or MATH3971 or MATH3974 or MATH3976 or MATH3977 or MATH3978 or MATH3979) Prohibitions: MATH3961 Assumed knowledge: Real analysis and vector spaces. For example (MATH2922 or MATH2961) and (MATH2923 or MATH2962) Assessment: Quiz (10%), two assignments (2 x 10%) and a final exam (70%) Campus: Camperdown/Darlington, Sydney Mode of delivery: Normal (lecture/lab/tutorial) day

At the end of this unit you will have received a broad introduction and gained a variety of tools to apply them within your further mathematical studies and/or in other disciplines.

**MATH4062 Rings, Fields and Galois Theory**

Credit points: 6 Teacher/Coordinator: Dr Leo Tzou Session: Semester 1 Classes: 3 lectures 3 hrs/week; 1 tutorial 1 hr/week Prerequisites: (MATH2922 or MATH2961) or a mark of 65 or greater in (MATH2022 or MATH2061) or 12cp from (MATH3061 or MATH3066 or MATH3063 or MATH3076 or MATH3078 or MATH3962 or MATH3963 or MATH3968 or MATH3969 or MATH3971 or MATH3974 or MATH3976 or MATH3977 or MATH3978 or MATH3979) Prohibitions: MATH3062 or MATH3962 Assessment: 4 x homework assignments (4 x 5%), tutorial participation (10%), final exam (70%) Campus: Camperdown/Darlington, Sydney Mode of delivery: Normal (lecture/lab/tutorial) day

This unit of study lies at the heart of modern algebra. In the unit we investigate the mathematical theory that was originally developed for the purpose of studying polynomial equations. In a nutshell, the philosophy is that it should be possible to completely factorise any polynomial into a product of linear factors by working over a large enough field (such as the field of all complex numbers). Viewed like this, the problem of solving polynomial equations leads naturally to the problem of understanding extensions of fields. This in turn leads into the area of mathematics known as Galois theory. The basic theoretical tool needed for this program is the concept of a ring, which generalises the concept of a field. The course begins with examples of rings, and associated concepts such as subrings, ring homomorphisms, ideals and quotient rings. These tools are then applied to study quotient rings of polynomial rings. The final part of the course deals with the basics of Galois theory, which gives a way of understanding field extensions. Along the way you will see some beautiful gems of mathematics, including Fermat's Theorem on primes expressible as a sum of two squares, solutions to the ancient Greek problems of trisecting the angle, squaring the circle, and doubling the cube, and the crown of the course: Galois' proof that there is no analogue of the quadratic formula for the general quintic equation. On completing this unit of study you will have obtained a deep understanding of modern abstract algebra.

**MATH4068 Differential Geometry**

Credit points: 6 Teacher/Coordinator: Dr Florica Cirstea Session: Semester 2 Classes: Three 1 hour lectures and one 1 hour tutorial per week. Prerequisites: (A mark of 65 or greater in 12cp of MATH2XXX units of study) or [12cp from (MATH3061 or MATH3066 or MATH3063 or MATH3076 or MATH3078 or MATH3961 or MATH3962 or MATH3963 or MATH3969 or MATH3971 or MATH3974 or MATH3976 or MATH3977 or MATH3978 or MATH3979)] Prohibitions: MATH3968 Assumed knowledge: Vector calculus, differential equations and real analysis, for example MATH2X21 and MATH2X23 Assessment: The grade is determined by student works throughout the semester, including Quiz 1 (10%), Assignment 1 (15%), Assignment 2 (15%), and Exam (60%). Moreover, to provide flexibility, the final grade is taken as the maximum between the above calculated score and the score of the exam out of 100. Campus: Camperdown/Darlington, Sydney Mode of delivery: Normal (lecture/lab/tutorial) day

This unit is an introduction to Differential Geometry, one of the core pillars of modern mathematics. Using ideas from calculus of several variables, we develop the mathematical theory of geometrical objects such as curves, surfaces and their higher-dimensional analogues. For students, this provides the first taste of the investigation on the deep relation between geometry and topology of mathematical objects, highlighted in the classic Gauss-Bonnet Theorem. Differential geometry also plays an important part in both classical and modern theoretical physics. The unit aims to develop geometrical ideas such as curvature in the context of curves and surfaces in space, leading to the famous Gauss-Bonnet formula relating the curvature and topology of a surface. A second aim is to remind the students about all the content covered in the mathematical units for previous years, most importantly the key ideas in vector calculus, along with some applications. It also helps to prepare the students for honours courses like Riemannian Geometry. By doing this unit you will further appreciate the beauty of mathematics which originated from the need to solve practical problems, develop skills in understanding the geometry of the surrounding environment, prepare yourself for future study or the workplace by developing advanced critical thinking skills and gain a deep understanding of the underlying rules of the Universe.

**MATH4069 Measure Theory and Fourier Analysis**

Credit points: 6 Teacher/Coordinator: Dr Leo Tzou Session: Semester 2 Classes: Three 1 hour lectures and one 1 hour tutorial per week. Prerequisites: (A mark of 65 or greater in 12cp of MATH2XXX units of study) or [12cp from the following units (MATH3061 or MATH3066 or MATH3063 or MATH3076 or MATH3078 or MATH3961 or MATH3962 or MATH3963 or MATH3969 or MATH3971 or MATH3974 or MATH3976 or MATH3977 or MATH3978 or MATH3979)] Prohibitions: MATH3969 Assumed knowledge: (MATH2921 and MATH2922) or MATH2961 Assessment: 2 x quiz (20%), 2 x written assignment (20%), final exam (60%) Campus: Camperdown/Darlington, Sydney Mode of delivery: Normal (lecture/lab/tutorial) day

Measure theory is the study of fundamental ideas as length, area, volume, arc length and surface area. It is the basis for Lebesgue integration theory used in advanced mathematics ever since its development in about 1900. Measure theory is also a key foundation for modern probability theory. The course starts by establishing the basics of measure theory and the theory of Lebesgue integration, including important results such as Fubini's Theorem and the Dominated Convergence Theorem which allow us to manipulate integrals. These ideas are applied to Fourier Analysis which leads to results such as the Inversion Formula and Plancherel's Theorem. The Radon-Nikodyn Theorem provides a representation of measures in terms of a density. Key ideas of this theory are applied in detail to probability theory to provide a rigorous framework for probability which takes in and generalizes familiar ideas such as distributions and conditional expectation. When you complete this unit you will have acquired a new generalized way of thinking about key mathematical concepts such as length, area, integration and probability. This will give you a powerful set of intellectual tools and equip you for further study in mathematics and probability.

**MATH4411 Applied Computational Mathematics**

Credit points: 6 Session: Semester 1 Classes: lecture 3 hrs/week, computer lab/tutorial 1 hr/week Assumed knowledge: A thorough knowledge of vector calculus (e.g., MATH2X21) and of linear algebra (e.g., MATH2X22). Some familiarity with partial differential equations (e.g., MATH3X78) and mathematical computing (e.g., MATH3X76) would be useful. Assessment: 3 x homework assignments (total 60%), final exam (40%) Campus: Camperdown/Darlington, Sydney Mode of delivery: Normal (lecture/lab/tutorial) day

Computational mathematics fulfils two distinct purposes within Mathematics. On the one hand the computer is a mathematician's laboratory in which to model problems too hard for analytical treatment and to test existing theories; on the other hand, computational needs both require and inspire the development of new mathematics. Computational methods are an essential part of the tool box of any mathematician. This unit will introduce you to a suite of computational methods and highlight the fruitful interplay between analytical understanding and computational practice. In particular, you will learn both the theory and use of numerical methods to simulate partial differential equations, how numerical schemes determine the stability of your method and how to assure stability when simulating Hamiltonian systems, how to simulate stochastic differential equations, as well as modern approaches to distilling relevant information from data using machine learning. By doing this unit you will develop a broad knowledge of advanced methods and techniques in computational applied mathematics and know how to use these in practice. This will provide a strong foundation for research or further study.

**MATH4412 Advanced Methods in Applied Mathematics**

Credit points: 6 Session: Semester 2 Classes: lecture 3 hrs/week, computer lab/tutorial 1 hr/week Assumed knowledge: A thorough knowledge of vector calculus (e.g., MATH2X21) and of linear algebra (e.g., MATH2X22). Some familiarity with partial differential equations (e.g., MATH3X78) and mathematical computing (e.g., MATH3X76) would be useful. Assessment: 2 x homework assignments (total 40%), final exam (60%) Campus: Camperdown/Darlington, Sydney Mode of delivery: Normal (lecture/lab/tutorial) day

Mathematical approaches to many real-world problems are underpinned by powerful and wide ranging mathematical methods and techniques that have become standard in the field and should be in the toolbag of all applied mathematicians. This unit will introduce you to a suite of those methods and give you the opportunity to engage with applications of these methods to well-known problems. In particular, you will learn both the theory and use of asymptotic methods which are ubiquitous in applications requiring differential equations or other continuous models. You will also engage with methods for probabilistic models including information theory and stochastic models. By doing this unit you will develop a broad knowledge of advanced methods and techniques in applied mathematics and know how to use these in practice. This will provide a strong foundation for using mathematics in a broad sweep of practical applications in research, in industry or in further study.

**MATH4413 Applied Mathematical Modelling**

Credit points: 6 Session: Semester 1 Classes: 2 x 1hr lectures per week, 2 x 1hr tutorials/workshops per week (indicative program) Assumed knowledge: MATH2X21 and MATH3X63 or equivalent. That is, a knowledge of linear and simple nonlinear ordinary differential equations and of linear, second order partial differential equations. Assessment: tutorial participation (10%), homework assignments (20%), presentation assignment (20%), final exam (50%) Campus: Camperdown/Darlington, Sydney Mode of delivery: Normal (lecture/lab/tutorial) day

Applied Mathematics harnesses the power of mathematics to give insight into phenomena in the wider world and to solve practical problems. Modelling is the key process that translates a scientific or other phenomenon into a mathematical framework through applying suitable assumptions, identifying important variables and deriving a well-defined mathematical problem. Mathematicians then use this model to explore the real-world phenomenon, including making predictions. Good mathematical modelling is something of an art and is best learnt by example and by writing, refining and analysing your own models. This unit will introduce you to some classic mathematical models and give you the opportunity to analyse, explore and extend these models to make predictions and gain insights into the underlying phenomena. You will also engage with modelling in depth in at least one area of application. By doing this unit you will develop a broad knowledge of advanced mathematical modelling methods and techniques and know how to use these in practice. This will provide a strong foundation for applying mathematics and modelling to many diverse applications and for research or further study.

**MATH4414 Advanced Dynamical Systems**

Credit points: 6 Session: Semester 2 Classes: lecture 3 hrs/week, computer lab/tutorial 1 hr/week Assumed knowledge: Assumed knowledge is vector calculus (e.g., MATH2X21), linear algebra (e.g., MATH2X22), dynamical systems and applications (e.g., MATH4063 or MATH3X63) or equivalent. Some familiarity with partial differential equations (e.g., MATH3978) and mathematical computing (e.g., MATH3976) is also assumed. Assessment: 2 x homework assignments (40%), final exam (60%) Campus: Camperdown/Darlington, Sydney Mode of delivery: Normal (lecture/lab/tutorial) day

In applied mathematics, dynamical systems are systems whose state is changing with time. Examples include the motion of a pendulum, the change in the population of insects in a field or fluid flow in a river. These systems are typically represented mathematically by differential equations or difference equations. Dynamical systems theory reveals universal mechanisms behind disparate natural phenomena. This area of mathematics brings together sophisticated theory from many areas of pure and applied mathematics to create powerful methods that are used to understand and control the dynamical building blocks which make up physical, biological, chemical, engineered and even sociological systems. By doing this unit you will develop a broad knowledge of methods and techniques in dynamical systems, and know how to use these to analyse systems in nature and in technology. This will provide a strong foundation for using mathematics in a broad sweep of applications and for research or further study.

**MATH4063 Dynamical Systems and Applications**

Credit points: 6 Teacher/Coordinator: Prof Georg Gottwald Session: Semester 1 Classes: Three lectures, one tutorial per week Prerequisites: (A mark of 65 or greater in 12cp of MATH2XXX units of study) or [12cp from (MATH3061 or MATH3066 or MATH3076 or MATH3078 or MATH3961 or MATH3962 or MATH3968 or MATH3969 or MATH3971 or MATH3974 or MATH3976 or MATH3977 or MATH3978 or MATH3979)] Assumed knowledge: Linear ODEs (for example, MATH2921), eigenvalues and eigenvectors of a matrix, determinant and inverse of a matrix and linear coordinate transformations (for example, MATH2922), Cauchy sequence, completeness and uniform convergence (for example, MATH2923) Assessment: Midterm exam (25%), two assignments (20% in total), final exam (55%). Campus: Camperdown/Darlington, Sydney Mode of delivery: Normal (lecture/lab/tutorial) day

The theory of ordinary differential equations is a classical topic going back to Newton and Leibniz. It comprises a vast number of ideas and methods. The theory has many applications and stimulates new developments in almost all areas of mathematics. The emphasis is on qualitative analysis including phase-plane methods, bifurcation theory and the study of limit cycles. The more theoretical part includes existence and uniqueness theorems, linearisation, and analysis of asymptotic behaviour. The applications in this unit will be drawn from predator-prey systems, population models, chemical reactions, and other equations and systems from mathematical biology. You will learn how to use ordinary differential equations to model biological, chemical, physical and/or economic systems and how to use different methods from dynamical systems theory and the theory of nonlinear ordinary differential equations to find the qualitative outcome of the models. By doing this unit you will develop skills in using and analyzing nonlinear differential equations which will prepare you for further studies in mathematics, systems biology or physics or for careers in mathematical modelling.

**MATH4074 Fluid Dynamics**

Credit points: 6 Teacher/Coordinator: Prof Georg Gottwald Session: Semester 1 Classes: Three 1 hour lectures and one 1 hour tutorial per week. Prerequisites: (A mark of 65 or above in 12cp of MATH2XXX ) or (12cp of MATH3XXX ) Prohibitions: MATH3974 Assumed knowledge: (MATH2961 and MATH2965) or (MATH2921 and MATH2922) Assessment: Assignment 1 (10%), Assignment 2 (10%), Assignment 3 (10%), Exam (70%) Campus: Camperdown/Darlington, Sydney Mode of delivery: Normal (lecture/lab/tutorial) day

Fluid Dynamics is the study of systems which allow for a macroscopic description in some continuum limit. It is not limited to the study of liquids such as water but includes our atmosphere and even car traffic. Whether a system can be treated as a fluid, depends on the spatial scales involved. Fluid dynamics presents a cornerstone of applied mathematics and comprises a whole gamut of different mathematical techniques, depending on the question we ask of the system under consideration. The course will discuss applications from engineering, physics and mathematics: How and in what situations a system which is not necessarily liquid can be described as a fluid? The link between an Eulerian description of a fluid and a Lagrangian description of a fluid, the basic variables used to describe flows, the need for continuity, momentum and energy equations, simple forms of these equations, geometric and physical simplifying assumptions, streamlines and stream functions, incompressibility and irrotationality and simple examples of irrotational flows. By the end of this unit, students will have received a basic understanding into fluid mechanics and have acquired general methodology which they can apply in their further studies in mathematics and/or in their chosen discipline.

**MATH4076 Computational Mathematics**

Credit points: 6 Teacher/Coordinator: Prof Georg Gottwald Session: Semester 1 Classes: Three 1 hour lectures and one 1 hour laboratory per week. Prerequisites: [A mark of 65 or above in (12cp of MATH2XXX) or (6cp of MATH2XXX and 6cp of STAT2XXX or DATA2X02)] or (12cp of MATH3XXX) Assumed knowledge: (MATH2X21 and MATH2X22) or (MATH2X61 and MATH2X65) Assessment: Quiz (15%), Assignment (15%), Assignment (15%), Final Exam (55%) Campus: Camperdown/Darlington, Sydney Mode of delivery: Normal (lecture/lab/tutorial) day

Sophisticated mathematics and numerical programming underlie many computer applications, including weather forecasting, computer security, video games, and computer aided design. This unit of study provides a strong foundational introduction to modern interactive programming, computational algorithms, and numerical analysis. Topics covered include: (I) basics ingredients of programming languages such as syntax, data structures, control structures, memory management and visualisation; (II) basic algorithmic concepts including binary and decimal representations, iteration, linear operations, sources of error, divide-and-concur, algorithmic complexity; and (III) basic numerical schemes for rootfinding, integration/differentiation, differential equations, fast Fourier transforms, Monte Carlo methods, data fitting, discrete and continuous optimisation. You will also learn about the philosophical underpinning of computational mathematics including the emergence of complex behaviour from simple rules, undecidability, modelling the physical world, and the joys of experimental mathematics. When you complete this unit you will have a clear and comprehensive understanding of the building blocks of modern computational methods and the ability to start combining them together in different ways. Mathematics and computing are like cooking. Fundamentally, all you have is sugar, fat, salt, heat, stirring, chopping. But becoming a good chef requires knowing just how to put things together in creative ways that work. In previous study, you should have learned to cook. Now you're going to learn how to make something someone else might want to pay for more than one time.

**MATH4077 Lagrangian and Hamiltonian Dynamics**

Credit points: 6 Teacher/Coordinator: Prof Georg Gottwald Session: Semester 2 Classes: Three 1 hour lectures and one 1 hour tutorial per week. Prerequisites: (A mark of 65 or greater in 12cp of MATH2XXX units of study) or [12cp from (MATH3061 orMATH3066 or MATH3063 or MATH3076 or MATH3078 or MATH3961 or MATH3962 or MATH3963 or MATH3968 or MATH3969 or MATH3971 or MATH3974 or MATH3976 or MATH3978 or MATH3979)] Prohibitions: MATH3977 Assumed knowledge: 6cp of 1000 level calculus units and 3cp of 1000 level linear algebra and (MATH2X21 or MATH2X61) Assessment: One 2 hour exam (70%), two mid-term quizzes (10% each) and one assignment (10%). Campus: Camperdown/Darlington, Sydney Mode of delivery: Normal (lecture/lab/tutorial) day

Lagrangian and Hamiltonian dynamics are a reformulation of classical Newtonian mechanics into a mathematically sophisticated framework that can be applied in many different coordinate systems. This formulation generalises elegantly to modern theories of relativity and quantum mechanics. The unit develops dynamics from the Principle of Least Action using the calculus of variations. Emphasis is placed on the relation between the symmetry and invariance properties of the Lagrangian and Hamiltonian functions and conservation laws. Coordinate and canonical transformations are introduced to make apparently complicated dynamical problems appear simpler. In this unit you will also explore connections between geometry and different physical theories beyond classical mechanics. You will be expected to solve fully dynamical systems of some complexity including planetary motion and to investigate stability using perturbation analysis. You will use Hamilton-Jacobi theory to solve problems ranging from geodesic motion (shortest path between two points) on curved surfaces to relativistic motion in the vicinity of black holes. This unit is a useful preparation for units in dynamical systems and chaos, and complements units in differential equations, quantum theory and general relativity.

**MATH4078 PDEs and Applications**

Credit points: 6 Teacher/Coordinator: Prof Georg Gottwald Session: Semester 2 Classes: 3 lectures 1 hr/week; tutorial 1 hr/week Prerequisites: (A mark of 65 or greater in 12cp of 2000 level units) or [12cp from (MATH3061 or MATH3066 or MATH3063 or MATH3076 or MATH3961 or MATH3962 or MATH3963 or MATH3968 or MATH3969 or MATH3971 or MATH3974 or MATH3976 or MATH3977 or MATH3979)] Prohibitions: MATH3078 or MATH3978 Assumed knowledge: (MATH2X61 and MATH2X65) or (MATH2X21 and MATH2X22) Assessment: Final exam (70%), 2 assignments (15%+15%). To pass the course, students must achieve at least 50% on the final exam. Campus: Camperdown/Darlington, Sydney Mode of delivery: Normal (lecture/lab/tutorial) day

The aim of this unit is to introduce some fundamental concepts of the theory of partial differential equations (PDEs) arising in Physics, Chemistry, Biology and Mathematical Finance. The focus is mainly on linear equations but some important examples of nonlinear equations and related phenomena re introduced as well. After an introductory lecture, we proceed with first-order PDEs and the method of characteristics. Here, we also nonlinear transport equations and shock waves are discussed. Then the theory of the elliptic equations is presented with an emphasis on eigenvalue problems and their application to solve parabolic and hyperbolic initial boundary-value problems. The Maximum principle and Harnack's inequality will be discussed and the theory of Green's functions.

**MATH4079 Complex Analysis**

Credit points: 6 Teacher/Coordinator: Prof Georg Gottwald Session: Semester 1 Classes: Lecture 3 hrs/week; tutorial 1 hr/week Prerequisites: (A mark of 65 or above in 12cp of MATH2XXX) or (12cp of MATH3XXX) Prohibitions: MATH3979 or MATH3964 Assumed knowledge: Good knowledge of analysis of functions of one real variable, working knowledge of complex numbers, including their topology, for example MATH2X23 or MATH2962 or MATH3068 Assessment: 2 x assessment (30%), final exam worth (70%) (requires pass mark of 50% or more) Campus: Camperdown/Darlington, Sydney Mode of delivery: Normal (lecture/lab/tutorial) day

The unit will begin with a revision of properties of complex numbers and complex functions. This will be followed by material on conformal mappings, Riemann surfaces, complex integration, entire and analytic functions, the Riemann mapping theorem, analytic continuation, and Gamma and Zeta functions. Finally, special topics chosen by the lecturer will be presented, which may include elliptic functions, normal families, Julia sets, functions of several complex variables, or complex manifolds.

**STAT4528 Probability and Martingale Theory**

Credit points: 6 Session: Semester 1 Classes: 3 x 1hr lectures and 1 x 1hr tutorial per week Prohibitions: STAT4028 Assumed knowledge: STAT2X11 or equivalent and STAT3X21 or equivalent; that is, a good foundational knowledge of probability and some acquaintance with stochastic processes. Assessment: 12 x weekly homework (40%), final exam (60%) Campus: Camperdown/Darlington, Sydney Mode of delivery: Normal (lecture/lab/tutorial) day

Probability Theory lays the theoretical foundations that underpin the models we use when analysing phenomena that involve chance. This unit introduces the students to modern probability theory (based on measure theory) that was developed by Andrey Kolmogorov. You will be introduced to the fundamental concept of a measure as a generalisation of the notion of length and Lebesgue integration which is a generalisation of the Riemann integral. This theory provides a powerful unifying structure that brings together both the theory of discrete random variables and the theory of continuous random variables that were introduced earlier in your studies. You will see how measure theory is used to put other important probabilistic ideas into a rigorous mathematical framework. These include various notions of convergence of random variables, 0-1 laws, conditional expectation, and the characteristic function. You will then synthesise all these concepts to establish the Central Limit Theorem and to thoroughly study discrete-time martingales. Originally used to model betting strategies, martingales are a powerful generalisation of random walks that allow us to prove fundamental results such as the Strong Law of Large Numbers or analyse problems such as the gambler's ruin. By doing this unit you will become familiar with many of the theoretical building blocks that are required for any in-depth study in probability, stochastic systems or financial mathematics.

**MATH4511 Arbitrage Pricing in Continuous Time**

Credit points: 6 Session: Semester 1 Classes: 3 x 1hr lectures and 1 x 1hr tutorials per week Assumed knowledge: Familiarity with basic probability (eg STAT2X11), with differential equations (eg MATH3X63, MATH3X78) and with basic numerical analysis and coding (eg MATH3X76), achievement at credit level or above in MATH3XXX or STAT3XXX units or equivalent. Assessment: 2 x homework assignments (40%), final exam (60%) Campus: Camperdown/Darlington, Sydney Mode of delivery: Normal (lecture/lab/tutorial) day

The aim of Financial Mathematics is to establish a theoretical background for building models of securities markets and provides computational techniques for pricing financial derivatives and risk assessment and mitigation. Specialists in Financial Mathematics are widely sought after by major investment banks, hedge funds and other, government and private, financial institutions worldwide. This course is foundational for honours and masters programs in Financial Mathematics. Its aim is to introduce the basic concepts and problems of securities markets and to develop theoretical frameworks and computational tools for pricing financial products and hedging the risk associated with them. This unit will focus on two ideas that are fundamental for Financial Mathematics. You will learn how the concept of arbitrage and the concept of martingale measure provide a unified approach to a large variety of seemingly unrelated problems arising in practice. You will also learn how to use the wide range of tools required by Financial Mathematics, including stochastic calculus, partial differential equations, optimisation and statistics. By doing this unit, you will learn how to formulate problems that arise in finance as mathematical problems and how to solve them using the concepts of arbitrage and martingale measure. You will also learn how to choose an appropriate computational method and devise explicit numerical algorithms useful for a practitioner.

**MATH4512 Stochastic Analysis**

Credit points: 6 Session: Semester 2 Classes: lecture 3 hrs/week, tutorial 1 hr/week for 13 weeks Assumed knowledge: Students should have a sound knowledge of probability theory and stochastic processes from, for example, STAT2X11 and STAT3021 or equivalent. Assessment: 2 x homework assignment (20% each), final exam (60%) Campus: Camperdown/Darlington, Sydney Mode of delivery: Normal (lecture/lab/tutorial) day

Capturing random phenomena is a challenging problem in many disciplines from biology, chemistry and physics through engineering to economics and finance. There is a wide spectrum of problems in these fields, which are described using random processes that evolve with time. Hence it is of crucial importance that applied mathematicians are equipped with tools used to analyse and quantify random phenomena. This unit will introduce an important class of stochastic processes, using the theory of martingales. You will study concepts such as the Ito stochastic integral with respect to a continuous martingale and related stochastic differential equations. Special attention will be given to the classical notion of the Brownian motion, which is the most celebrated and widely used example of a continuous martingale. By completing this unit, you will learn how to rigorously describe and tackle the evolution of random phenomena using continuous time stochastic processes. You will also gain a deep knowledge about stochastic integration, which is an indispensable tool to study problems arising, for example, in Financial Mathematics.

**MATH4513 Topics in Financial Mathematics**

Credit points: 6 Session: Semester 2 Classes: lecture 3 hrs/week, tutorial 1 hr/week for 13 weeks Assumed knowledge: Students are expected to have working knowledge of Stochastic Processes, Stochastic Calculus and mathematical methods used to price options and other financial derivatives, for example as in MATH4511 or equivalent Assessment: 2 x homework assignments (20% each), final exam (60%) Campus: Camperdown/Darlington, Sydney Mode of delivery: Normal (lecture/lab/tutorial) day

Securities and derivatives are the foundation of modern financial markets. The fixed-income market, for example, is the dominant sector of the global financial market where various interest-rate linked securities are traded, such as zero-coupon and coupon bonds, interest rate swaps and swaptions. This unit will investigate short-term interest rate models, the Heath-Jarrow-Morton approach to instantaneous forward rates and recently developed models of forward London Interbank Offered Rates (LIBORs) and forward swap rates. You will learn about pricing and hedging of credit derivatives, another challenging and practically important problem and become familiar with stochastic models for credit events, dependent default times and credit ratings. You will learn how to value and hedge single-name and multi-name credit derivatives such as vulnerable options, corporate bonds, credit default swaps and collateralized debt obligations. You will also learn about the most recent developments in Financial Mathematics, such as robust pricing and nonlinear evaluations. By doing this unit, you will get a solid grasp of mathematical tools used in valuation and hedging of fixed income securities, develop a broad knowledge of advanced quantitative methods related to interest rates and credit risk and you will learn to use powerful mathematical tools to address important real-world quantitative problems in the finance industry.

Textbooks

1. M. Musiela and M. Rutkowski, "Martingale Methods in Financial Modelling." Springer, Berlin, 2nd Edition, 2005. 2. T. R. Bielecki, M. Jeanblanc and M. Rutkowski, "Credit Risk Modeling." Osaka University Press, Osaka, 2009.

**MATH4071 Convex Analysis and Optimal Control**

Credit points: 6 Teacher/Coordinator: Prof Georg Gottwald Session: Semester 1 Classes: Lecture 3hours/week, tutorial 1hr/week Prerequisites: [A mark of 65 or above in 12cp of (MATH2XXX or STAT2XXX or DATA2X02)] or [12cp of (MATH3XXX or STAT3XXX)] Prohibitions: MATH3971 Assumed knowledge: MATH2X21 and MATH2X23 and STAT2X11 Assessment: Assignment (15%), assignment (15%), exam (70%) Campus: Camperdown/Darlington, Sydney Mode of delivery: Normal (lecture/lab/tutorial) day

The questions how to maximise your gain (or to minimise the cost) and how to determine the optimal strategy/policy are fundamental for an engineer, an economist, a doctor designing a cancer therapy, or a government planning some social policies. Many problems in mechanics, physics, neuroscience and biology can be formulated as optimisation problems. Therefore, optimisation theory is an indispensable tool for an applied mathematician. Optimisation theory has many diverse applications and requires a wide range of tools but there are only a few ideas underpinning all this diversity of methods and applications. This course will focus on two of them. We will learn how the concept of convexity and the concept of dynamic programming provide a unified approach to a large number of seemingly unrelated problems. By completing this unit you will learn how to formulate optimisation problems that arise in science, economics and engineering and to use the concepts of convexity and the dynamic programming principle to solve straightforward examples of such problems. You will also learn about important classes of optimisation problems arising in finance, economics, engineering and insurance.

**STAT4028 Probability and Mathematical Statistics**

Credit points: 6 Session: Semester 1 Classes: 3 x 1hr lectures/week, 1 x 1hr tutorial or laboratory class/week Prohibitions: STAT4528 Assumed knowledge: STAT3X23 or equivalent: that is, a sound working and theoretical knowledge of statistical inference. Assessment: 12 x weekly homework (40%), final exam (60%) Campus: Camperdown/Darlington, Sydney Mode of delivery: Normal (lecture/lab/tutorial) day

Probability Theory lays the theoretical foundations that underpin the models we use when analysing phenomena that involve chance. This unit introduces the students to modern probability theory and applies it to problems in mathematical statistics. You will be introduced to the fundamental concept of a measure as a generalisation of the notion of length and Lebesgue integration which is a generalisation of the Riemann integral. This theory provides a powerful unifying structure that bring together both the theory of discrete random variables and the theory of continuous random variables that were introduce to earlier in your studies. You will see how measure theory is used to put other important probabilistic ideas into a rigorous mathematical framework. These include various notions of convergence of random variables, 0-1 laws, and the characteristic function. You will then synthesise all these concepts to establish the Central Limit Theorem and also verify important results in Mathematical Statistics. These involve exponential families, efficient estimation, large-sample testing and Bayesian methods. Finally you will verify important convergence properties of the expectation-maximisation (EM) algorithm. By doing this unit you will become familiar with many of the theoretical building blocks that are required for any in-depth study in probability or mathematical statistics.

**STAT4026 Statistical Consulting**

Credit points: 6 Teacher/Coordinator: Dr John Ormerod Session: Semester 1 Classes: lecture 1 hr/week; workshop 2hrs/week Prerequisites: At least 12cp from STAT2X11 or STAT2X12 or DATA2X02 or STAT3XXX Prohibitions: STAT3926 Assessment: 4 x reports (40%), take-home exam report (40%), oral presentation (20%) Practical field work: Face to face client consultation: approximately 1 - 1.5 hrs/week Campus: Camperdown/Darlington, Sydney Mode of delivery: Normal (lecture/lab/tutorial) day

In our ever-changing world, we are facing a new data-driven era where the capability to efficiently combine and analyse large data collections is essential for informed decision making in business and government, and for scientific research. Statistics and data analytics consulting provide an important framework for many individuals to seek assistance with statistics and data-driven problems. This unit of study will provide students with an opportunity to gain real-life experience in statistical consulting or work with collaborative (interdisciplinary) research. In this unit, you will have an opportunity to have practical experience in a consultation setting with real clients. You will also apply your statistical knowledge in a diverse collection of consulting projects while learning project and time management skills. In this unit you will need to identify and place the client's problem into an analytical framework, provide a solution within a given time frame and communicate your findings back to the client. All such skills are highly valued by employers. This unit will foster the expertise needed to work in a statistical consulting firm or data analytical team which will be essential for data-driven professional and research pathways in the future.

**STAT4027 Advanced Statistical Modelling**

Credit points: 6 Session: Semester 2 Classes: 2 x 1 hr lecture/week, 1 x 1 hour tutorial/lab/week Prerequisites: STAT3X12 and STAT3X13 Assumed knowledge: A three year major in statistics or equivalent including familiarity with material in DATA2X02 and STAT3X22 (applied statistics and linear models) or equivalent Assessment: 3 x homework assignments (30%), 2 x report and presentation (30%), final exam (40%) Campus: Camperdown/Darlington, Sydney Mode of delivery: Normal (lecture/lab/tutorial) day

Applied Statistics fundamentally brings statistical learning to the wider world. Some data sets are complex due to the nature of their responses or predictors or have high dimensionality. These types of data pose theoretical, methodological and computational challenges that require knowledge of advanced modelling techniques, estimation methodologies and model selection skills. In this unit you will investigate contemporary model building, estimation and selection approaches for linear and generalised linear regression models. You will learn about two scenarios in model building: when an extensive search of the model space is possible; and when the dimension is large and either stepwise algorithms or regularisation techniques have to be employed to identify good models. These particular data analysis skills have been foundational in developing modern ideas about science, medicine, economics and society and in the development of new technology and should be in the toolkit of all applied statisticians. This unit will provide you with a strong foundation of critical thinking about statistical modelling and technology and give you the opportunity to engage with applications of these methods across a wide scope of applications and for research or further study.

**STAT4021 Stochastic Processes and Applications**

Credit points: 6 Session: Semester 1 Classes: lecture 3 hrs/week, workshop 1 hr/week Prohibitions: STAT3011 or STAT3911 or STAT3021 or STAT3003 or STAT3903 or STAT3005 or STAT3905 or STAT3921. Assumed knowledge: STAT2011 or STAT2911, and MATH1003 or MATH1903 or MATH1907 or MATH1023 or MATH1923 or MATH1933 or equivalent. That is, students are expected to have a thorough knowledge of basic probability and integral calculus and to have achieved at credit level or above in their studies in these topics. Assessment: 2 x homework assignments (10%), 2 x in-class quizzes (30%), final exam (60%) Campus: Camperdown/Darlington, Sydney Mode of delivery: Normal (lecture/lab/tutorial) day

A stochastic process is a mathematical model of time-dependent random phenomena and is employed in numerous fields of application, including economics, finance, insurance, physics, biology, chemistry and computer science. In this unit you will rigorously establish the basic properties and limit theory of discrete-time Markov chains and branching processes and then, building on this foundation, derive key results for the Poisson process and continuous-time Markov chains, stopping times and martingales. You will learn about various illustrative examples throughout the unit to demonstrate how stochastic processes can be applied in modeling and analysing problems of practical interest, such as queuing, inventory, population, financial asset price dynamics and image processing. By completing this unit, you will develop a solid mathematical foundation in stochastic processes which will become the platform for further studies in advanced areas such as stochastic analysis, stochastic differential equations, stochastic control and financial mathematics.

**STAT4022 Linear and Mixed Models**

Credit points: 6 Session: Semester 1 Classes: 3 x 1hr lectures, 1 x 1 hr tutorial and 1 x 1 hr computer workshop/week Prohibitions: STAT3012 or STAT3912 or STAT3022 or STAT3922 or STAT3004 or STAT3904. Assumed knowledge: Material in DATA2X02 or equivalent and MATH1X02 or equivalent; that is, a knowledge of applied statistics and an introductory knowledge to linear algebra, including eigenvalues and eigenvectors. Assessment: 2 x homework assignment (10%), 3 x tutorial quiz (35%), final exam (55%) Campus: Camperdown/Darlington, Sydney Mode of delivery: Normal (lecture/lab/tutorial) day

Classical linear models are widely used in science, business, economics and technology. This unit will introduce the fundamental concepts of analysis of data from both observational studies and experimental designs using linear methods, together with concepts of collection of data and design of experiments. You will first consider linear models and regression methods with diagnostics for checking appropriateness of models, looking briefly at robust regression methods. Then you will consider the design and analysis of experiments considering notions of replication, randomization and ideas of factorial designs. Throughout the course you will use the R statistical package to give analyses and graphical displays. This unit includes material in STAT3022, but has an additional component on the mathematical techniques underlying applied linear models together with proofs of distribution theory based on vector space methods.

**STAT4023 Theory and Methods of Statistical Inference**

Credit points: 6 Session: Semester 2 Classes: 3 x 1hr lecture/week, 1 x 2hr workshop/week Prohibitions: STAT3013 or STAT3913 or STAT3023 or STAT3923 Assumed knowledge: STAT2X11 and (DATA2X02 or STAT2X12) or equivalent. That is, a grounding in probability theory and a good knowledge of the foundations of applied statistics. Assessment: weekly homework assignments (5%), 2 x in-class quizzes (20%), 5 x computer lab reports (10%), computer exam (10%), final exam (55%) Campus: Camperdown/Darlington, Sydney Mode of delivery: Normal (lecture/lab/tutorial) day

In today's data-rich world, more and more people from diverse fields need to perform statistical analyses, and indeed there are more and more tools to do this becoming available. It is relatively easy to "point and click" and obtain some statistical analysis of your data. But how do you know if any particular analysis is indeed appropriate? Is there another procedure or workflow which would be more suitable? Is there such a thing as a "best possible" approach in a given situation? All of these questions (and more) are addressed in this unit. You will study the foundational core of modern statistical inference, including classical and cutting-edge theory and methods of mathematical statistics with a particular focus on various notions of optimality. The first part of the unit covers aspects of distribution theory which are applied in the second part which deals with optimal procedures in estimation and testing. The framework of statistical decision theory is used to unify many of the concepts that are introduced in this unit. You will rigorously prove key results and apply these to real-world problems in laboratory sessions. By completing this unit, you will develop the necessary skills to confidently choose the best statistical analysis to use in many situations.

**HPSC4101 Philosophy of Science**

Credit points: 6 Teacher/Coordinator: Professor Dean Rickles Session: Semester 1 Classes: One 2 hour seminar per week, individual consultation. Prerequisites: 12 credit points of HPSC3XXX or PHIL3XXX or HSTY3XXX Assessment: 5000 wd essay (50%) Seminar presentation (50%) Campus: Camperdown/Darlington, Sydney Mode of delivery: Normal (lecture/lab/tutorial) day

In this course we explore a range of issues from within the philosophy of physics. We focus on the interpretation of the theories physics provides, examining how these theories might describe our world. The course will assume some basic mathematical literacy, but most technical matters will be introduced in class.

Textbooks

Weekly readings

#### Honours Core Research Project

**PHYS4103 Physics Honours Project A**

Credit points: 6 Teacher/Coordinator: Bruce Yabsley Session: Semester 1,Semester 2 Classes: individual work supported by supervisor(s) Assessment: oral presentation (10%), thesis (90%) Practical field work: Honours in Physics may involve fieldwork in some cases, e.g. at offsite research facilities. Arrangements differ depending on the specific research topic. Campus: Camperdown/Darlington, Sydney Mode of delivery: Supervision

Research is the lifeblood of the physical sciences, and participation in research is both a privilege and an opportunity for learning and growth. In this unit you will complete a research project in physics. Together with your supervisor, you will identify a novel research question and a method to address it. Typical activities during the project include review and synthesis of the literature, specialised study in a sub-field of physics, design and conduct of experiments, data mining, theoretical work, modelling, calculation, simulation, coding, and the use of specialised software. Your project will be assessed through an oral presentation to your peers and colleagues, and a forty-page research report. Successful completion of your honours research project will demonstrate that you have mastered significant research and professional skills for undertaking postgraduate studies, or a range of careers in industry.

**PHYS4104 Physics Honours Project B**

Credit points: 6 Teacher/Coordinator: Bruce Yabsley Session: Semester 1,Semester 2 Classes: individual work supported by supervisor(s) Corequisites: PHYS4103 Assessment: oral presentation (10%), thesis (90%) Practical field work: Honours in Physics may involve fieldwork in some cases, e.g. at offsite research facilities. Arrangements differ depending on the specific research topic. Campus: Camperdown/Darlington, Sydney Mode of delivery: Supervision

Research is the lifeblood of the physical sciences, and participation in research is both a privilege and an opportunity for learning and growth. In this unit you will complete a research project in physics. Together with your supervisor, you will identify a novel research question and a method to address it. Typical activities during the project include review and synthesis of the literature, specialised study in a sub-field of physics, design and conduct of experiments, data mining, theoretical work, modelling, calculation, simulation, coding, and the use of specialised software. Your project will be assessed through an oral presentation to your peers and colleagues, and a forty-page research report. Successful completion of your honours research project will demonstrate that you have mastered significant research and professional skills for undertaking postgraduate studies, or a range of careers in industry.

**PHYS4105 Physics Honours Project C**

Credit points: 6 Teacher/Coordinator: Bruce Yabsley Session: Semester 1,Semester 2 Classes: individual work supported by supervisor(s) Corequisites: PHYS4104 Assessment: oral presentation (10%), thesis (90%) Practical field work: Honours in Physics may involve fieldwork in some cases, e.g. at offsite research facilities. Arrangements differ depending on the specific research topic. Campus: Camperdown/Darlington, Sydney Mode of delivery: Supervision

Research is the lifeblood of the physical sciences, and participation in research is both a privilege and an opportunity for learning and growth. In this unit you will complete a research project in physics. Together with your supervisor, you will identify a novel research question and a method to address it. Typical activities during the project include review and synthesis of the literature, specialised study in a sub-field of physics, design and conduct of experiments, data mining, theoretical work, modelling, calculation, simulation, coding, and the use of specialised software. Your project will be assessed through an oral presentation to your peers and colleagues, and a forty-page research report. Successful completion of your honours research project will demonstrate that you have mastered significant research and professional skills for undertaking postgraduate studies, or a range of careers in industry.

**PHYS4106 Physics Honours Project D**

Credit points: 6 Teacher/Coordinator: Bruce Yabsley Session: Semester 1,Semester 2 Classes: individual work supported by supervisor(s) Corequisites: PHYS4105 Assessment: oral presentation (10%), thesis (90%) Practical field work: Honours in Physics may involve fieldwork in some cases, e.g. at offsite research facilities. Arrangements differ depending on the specific research topic. Campus: Camperdown/Darlington, Sydney Mode of delivery: Supervision