Psychology is a science which relies on empirical research to back up its claims. This unit will provide you with the opportunity to plan, design and analyse your own research project within a small group of students. You will be supervised by an academic from the School of Psychology. In this unit, you will continue to understand and explore disciplinary knowledge, while also meeting and collaborating with students through project-based learning. You will identify and solve problems, collect and analyse data and communicate your findings to a diverse audience during a final symposium. All of these skills are highly valued by employers. This unit will foster the ability to work in teams, and this is essential for both professional and research pathways in the future.

Industry and Community Project Units are designed to facilitate a collaborative, inquiry-based approach to complex problem solving. During the course of the unit, you will work in interdisciplinary teams on authentic, complex problem-based projects developed with the University's external Partners and driven by industry and community needs. Your team will identify a unique aspect to address within the broader problem and you will engage in self-directed inquiry-based research with the support of your Project Supervisor. Your Project Partner will provide valuable insights into the industry as well as access to information that will assist with your investigation and provide context for your final recommendations. In addition to deepening your understanding around specific project-related issues, you will develop skills in professional reflection, collaboration and complex problem solving through evidence-based teaching approaches and workshops. By doing this unit you will develop a toolkit that will provide you with adaptivity and agility to successfully navigate diverse and dynamic graduate career paths.

Industry and Community Project Units are designed to facilitate a collaborative, inquiry-based approach to complex problem solving. During the course of the unit, you will work in interdisciplinary teams on authentic, complex problem-based projects developed with the University's external Partners and driven by industry and community needs. Your team will identify a unique aspect to address within the broader problem and you will engage in self-directed inquiry-based research with the support of your Project Supervisor. Your Project Partner will provide valuable insights into the industry as well as access to information that will assist with your investigation and provide context for your final recommendations. In addition to deepening your understanding around specific project-related issues, you will develop skills in professional reflection, collaboration and complex problem solving through evidence-based teaching approaches and workshops. By doing this unit you will develop a toolkit that will provide you with adaptivity and agility to successfully navigate diverse and dynamic graduate career paths.

"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.

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.

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.

Most of us will spend at least one third of our lives in the workplace. Psychology has given us considerable insights into how people think, feel and behave as they do, and this has great implications for the workplace. Workplace psychology, sometimes called business psychology, refers to the practice of applying psychological principles and practices to a work environment. The goal is to identify and solve problems, increase employee satisfaction and well-being, improve workplace dynamics and to generally make the workplace a better place in which to spend one third of your life. In this unit of study there will be a particular focus on using positive psychology in the workplace. You will be equipped to use psychological principles in the workplace to make the workplace a more productive, fairer and a more need-satisfying experience. Drawing on Self-determination Theory you will explore the concept of the Positive Built Workplace Environment and how the interface between leadership, building design and workplace culture can produce sustainable, flourishing workplaces. You will also explore issues like overcoming procrastination and increasing productivity; positively influencing and leading people in organisations; the formation of effective teams; the psychology of negotiation and conflict resolution; facilitating wellness; preventing stress and burnout; psychopaths in the workplace and the creation of positive workplace experiences. You will also cover issues such as the evaluation of positive workplace interventions with data collection methods including questionnaires, surveys, focus groups, interviews and case studies. This theoretically-grounded but very practical unit of study gives you the tools to enhance the work experiences of yourself and others.

Coaching skills are now an essential part of the contemporary workplace. Research shows that the ability to coach self and others is one of the most important skills for employees, managers and leaders. However, to date, opportunities for learning evidence-based coaching skills at universities has been very limited. By completing this unit you will develop a solid understanding of the psychology of coaching in the workplace, in organisations and in relation to personal life matters and the ability to apply such theories in real life situations. We will explore the theoretical foundations of the psychology of coaching including self-regulation theory, goal theory, change theory and solution-focused approaches to coaching and show how to apply these to real-life issues and goals. Students will leave with a portfolio of applied coaching skills, the ability to conduct both formal and informal coaching conversations, the ability to evaluate and create conceptually coherent coaching processes and having experienced a personal coaching program. Active learning in the form of peer coaching is central to this program and will guide students to integrate their developing knowledge, skills and values about coaching in ways that question and build understanding. Students need to be prepared and willing to engage in peer coaching conversations.

The search for happiness and well-being is ubiquitous. Humans crave a sense of well-being, and resilience in the face of hardships has been a prized attribute since time immemorial. However, it is only relatively recently that psychology has turned its attention to the scientific exploration of well-being, resilience, happiness and the life well-lived. This unit of study teaches skills and pathways for cultivating wellbeing, resilience and happiness in individuals, groups, organisations and communities as a whole. The teaching in this unit advocates scientific methods and promotes critical thinking and analysis of key facets of positive psychology. We will explore the theoretical perspectives and conceptual frameworks that underpin positive psychology. The related empirical research will be examined and critiqued in order to identify best practice interventions and to facilitate the utilisation of this knowledge into effective real world methods. Active learning is a central feature of this unit. Students will be expected to apply positive psychology principles in their own lives and to reflect on these experiences. A wide range of learning approaches will be used including: debates, role plays, case studies, and reflective journal entries. These will form part of the learning and assessment activities.

This unit is designed to provide students with training in the professional skills required to practice agronomy. The unit principally builds on theoretical and applied knowledge gained in third year agronomy (AGRO3004). In this unit students will integrate their knowledge of plant physiology, soil science, experimental design, and biometry to address applied problems in agronomy, namely the issue of sustainability. Students will develop their ability to establish conclusions towards making recommendations for long term sustainability of crop and pasture systems. By implementing and managing a major field and/or glasshouse experiment(s) students will develop their research and inquiry skills. Team work is strongly encouraged in this unit and the integration and reporting of research findings will facilitate critical thinking and development of written communication skills. After completing this unit, students should be able to confidently design and manage a glasshouse/field experiment, and interpret and communicate their findings, by integrating knowledge from across disciplinary boundaries.

Progress in our ability to cure or delay cancer is rapidly accelerating. The prospects of patients with cancer have been revolutionised by the genetic, molecular and cellular study of cancers in individual patients. This research is also enabling a more sophisticated understanding of how sub-cellular processes and the many different types of cells in the body interact to support human life. Together we will look at the molecular and cellular origins of the behaviours of cancers: what drives accelerated cancer cell replication, their resistance to cell death and their ability to induce angiogenesis, local invasion and metastasis. We will examine genomic instability, dysregulated cellular metabolism and the role of inflammation and the immune system. We will analyse these behaviours and their relevance to cancer therapy. You will work independently and in groups in face-to-face and online learning activities. You will deepen your knowledge of molecular and cellular biology and hone the intellectual and practical skills required to equip you to participate in the latest developments to improve survival and health for all cancer patients. Upon completion, you will have developed the skills required to launch your career in cancer research, clinical and diagnostic cancer services and/or the corporate system that supports the prevention, diagnosis and treatment of cancer.

We face major health challenges in today's society that require new insights and approaches from bright minds. Tackling the big questions in medical sciences and health requires the research skills that will inform tomorrow's health outcomes for individuals and populations. Immersed in a multidisciplinary medical science and health research environment, you will develop the core skills required to undertake laboratory, clinical and population health research. You will learn to design, execute and evaluate studies, and to scrutinise data and research outcomes. You will work individually and collaboratively in small teams of students from different areas of specialisation to learn theoretical and practical aspects of specific research techniques, as well as the ethical and regulatory frameworks relevant to medical and health research. This unit of study will equip you with knowledge and skills that will enable you to play an active role in finding meaningful solutions to difficult problems in a technical or research setting.

This unit of study focuses on the role and animal and veterinary biosciences in the field of wildlife management management and diseases using project-based, open learning space and research-led teaching approaches. The unit encourages an approach that spans management, wildlife biology and laboratory sciences. In recognition of the power of genetics as a tool in wildlife management and research, a large component of this course reviews fundamental genetic, genomic and immunogenetic principals and their application to understanding, managing and conserving wildlife. This unit also covers themes in Indigenous knowledges related to animal management and conservation as well as cultural competence. At the end of this unit of study, students will demonstrate an understanding of: important issues in wildlife management in Australia and the Asia-pacific region; project management as it applies to multifaceted wildlife research and management issues; application of a range of genetic and physiological methods to the study of ecological issues; the use of appropriate analytical methods and molecular markers in wildlife conservation and management; the underlying genetic structural design of the natural world and how this reflects and influences evolutionary processes in healthy and diseased populations; the use of molecular information to test hypotheses about evolutionary, ecological and social structure of species; how to critically review the ways in which genetic principals are applied to the management and conservation of species; the use of appropriate analytical methods and molecular markers in wildlife conservation and management; how to conduct an investigation into a management problem in wildlife including project design and management recommendations. Students are expected to immerse themselves into the field of conservation, evolutionary genetics and wildlife to develop the ability to critically evaluate the subject. There will be a substantial amount of reading required for the course. There is no formal text; students will be directed to a recommended reading list of both primary and secondary literature.

This unit introduces the concepts of sheep (wool and meat) and beef cattle production in the Australian environment within the context of world food and fibre consumption and production. The key products as well as domestic and export markets for these are presented. The course provides a historical perspective of the basis for each of these industries and describes each of the production systems designed to meet the demand for these products.
Production in both the tropical and temperate regions of Australia will be covered and include the key elements of extensive grazing and intensive feedlot systems. Major issues will include breeds and breeding systems, basic nutrition and production practices and animal welfare issues as they affect the quality and quantity of product marketed.
The concepts of first stage processing of both meat and fibre products in abattoirs and top-making plants respectively will be presented. The major factors that influence the quality of product and therefore grading and market demand will be presented.
Lecture material will be supported with appropriate practical classes, a 2 day trip to the University's 'Arthursleigh' farm and a 5 day study tour to the Riverina to evaluate different commercial production systems. Students will also have an opportunity to compete in the annual Inter Collegiate Meat Judging (ICMJ) competition as a member of the University of Sydney team. This competition involves teams from numerous universities throughout Australia as well as Japan and the USA.

This unit of study is aimed at advanced techniques in Remote Sensing (RS), linked with Geographical Information Systems (GIS), as applied to land management problems. We will review the basic principles of GIS and then focus on advanced RS principles and techniques used for land resource assessment and management. This will be followed by practical training in RS techniques, augmented by land management project development and implementation based on integration of GIS and RS tools. The unit thus consists of three separate but overlapping parts: 1) a short theoretical part which focuses on the concepts of RS; 2) a practical part which aims at developing hands-on skills in using RS tools, and 3) an application-focused module in which students will learn the skills of how to design a land management project and actualize it using integrated GIS and RS techniques.

Lectures and practical work are devoted to the theory, philosophy and practice of plant breeding. The unit addresses screening techniques, conservation of genetic variability, breeding for disease resistance and integration of molecular technology in applied plant breeding, with examples from both field and horticultural crops. The unit is taught in the context [of] climate change, food security and the evolving global intellectual property environment. Classes and practicals are held at the Plant Breeding Institute at Camden and at the Biomedical Building, Australian Technology Park.

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.

This unit explores major episodes in the history of science from the 18th century until the present as well as introducing students to historiographic methods. Special attention is paid to developing practical skills in the history and philosophy of science.

This seminar discusses a range of approaches to the social theory of modern science. We will read key texts on questions such as: what makes science part of Western modernity? What is the role of science in the social transformations of the industrial era? In what sense, if at all, can science be said to offer privileged access to reality? What is the relationship between scientific knowledge and social reality?

An examination of one area of the contemporary literature in the history and philosophy of science. Special attention will be paid to development of research skills in the history and philosophy of science.

Adopting a seminar style, this unit provides students with an advanced knowledge of the skills necessary to conduct their own original research in the sociology, history and philosophy of science. Participants will be given a weekly set of core readings, and specialists both from within the Unit and from outside will present their views on the topic in question. This presentation will form the basis for a discussion involving the students, the academic members of the Unit, and invited speakers. Topics will include: the use of case studies in the philosophy of science, how to conduct oral history projects, institutional history, and sociological methodology.

An intensive reading course, supported by discussion seminars, into core topics in HPS.

In this unit of study you delve deeper into the world of methods used in health research, including both quantitative or qualitative design and analysis. You will attend lectures and interactive workshops, and complete online modules. The unit will build on your prior knowledge (see the prerequisites) and help with your specific Honours project. As part of quantitative methods, we cover experimental and observational (case-control, cohort) study designs, and linear and logistic regression for data analysis. Qualitative approaches include ethnography, grounded theory, phenomenology and narrative. Methods include interview, focus group and text based. The unit will be partly split in streams so that each student will learn either quantitative or qualitative data analysis in depth (not both), depending on their prior learning.

Living organisms are impacted by processes that occur across a very wide range of scales. These range from rapid processes at the molecular and cellular scale to multi-year processes at environmental and evolutionary scales. One of the great challenges for modern systems biology is integrating measurements across these scales to understand gene x environment interactions. This unit will develop your skills in this area through critical analysis of a series of recent research papers on a themed topic in small group discussions. For each paper we will explore principles behind the key methods and the methods' practicality. We will look at how those methods were incorporated into an experimental design to address a biological question. We will critically assess the support for conclusions in their paper and their scientific significance. By doing this unit you will develop skills in reading and interpreting primary scientific literature and an advanced understanding of modern topic in systems biology. You will gain a high level of understanding of the theory of key biochemical and statistical methods for analysis of genes, proteins, and cells in biological systems. You will gain the confidence to apply these insights to planning, conducting and reporting your own research findings.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Research in neuroscience has made tremendous advances in our understanding of the nervous system and its function in health and disease, however we are still far from fully understanding the form and function of the billions of neurons and the trillions of synapses that make up the brain and spinal cord. This unit is designed to introduce you to cutting edge issues in neuroscience Topics will include imaging pain, emotions, cortical development and plasticity, colour vision, addiction and stress, memory and cognitive processing, neuropsychiatric conditions and neurodegenerative disorders. This unit of study will use small group lectures, seminar groups and short research-based projects to engage students in authentic enquiry. You will be encouraged to explore several specific areas of neuroscience research and develop analytic skills and thinking about the processes and methods of doing neuroscience and engage you in debate and discussion, rather than learn facts. You will shape opinion by listening to the ideas of others and improve your skills and insights into problem solving. You will present your views and ideas, listen to those of others and through this appreciate divergent thinking.

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.

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.

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.

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.

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.

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 .

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.

Foundations of Professional Psychology is designed to equip you with the knowledge, critical thinking and practical skills that provide the foundation for professional practice in psychology. It will build upon the background in psychological science established in the undergraduate Psychology program to develop your understanding and capacity for critical evaluation of the theoretical and empirical bases underpinning the construction, implementation and interpretation of major cognitive and personality assessment instruments, and the development and implementation of evidence-based psychological interventions. Through the lectures, practical activities and assessments, you will also develop an understanding of current regulatory and legal contexts, including the National Health Practitioner Regulation Act 2009, NSW, Co-Regulatory Jurisdiction Standards, and mandatory reporting requirements. You will also be introduced to the Australian National Practice Standards for the Mental Health Workforce. Lectures on ethical practice will cover key issues in the psychology profession's Code of Conduct including Professional Relationships and the importance of confidentiality, informed consent and record keeping. The implications of cultural diversity and the factors that need to be considered in culturally informed practice will also be illustrated and evaluated. This unit will meet the accreditation criteria for Honours programs in Psychology and provide students with the essential foundations for psychological practice in a range of contexts.

Health psychology is a rapidly developing sub-discipline of psychology, underlined by a growing number of people affected by, or caring for someone with, a physical and/or mental health condition. Health psychologists examine how biological, psychological, and social/cultural factors affect health, illness, and recovery, and recognise health is more than the absence of disease. Health Psychology is an interdisciplinary field devoted to the study of the promotion and maintenance of health; causes and detection of illness; prevention and treatment of illness; and, improvement of healthcare systems and policy. You will develop advanced understanding of the: impact of acute/chronic/terminal illness on patients and their families; application of psychological theories and methods to assess patient/caregiver outcomes; importance of health literacy and effective clinician-patient-family communication; and, gain skills in the development, evaluation and implementation of interventions to improve patient and caregiver wellbeing. This unit will challenge you to critically evaluate social, cultural, and political aspects of health disparities in Australia, and explore ways to address the needs of the most vulnerable groups in society. At the completion of the unit you will be able to contribute to the development and delivery of evidence-based healthcare programs aimed at addressing current healthcare needs and challenges, particularly in vulnerable populations.

In a world increasingly inundated with technology, data and pseudoscience, you, as a medical science graduate, have a very special responsibility to society. You are one of the few that can help to inform and explain difficult concepts to the broader community. The most important quality you need to develop is that of a confident communicator of and advocate for biomedical science. This cutting-edge contemporary unit will equip you with the critical thinking skills and tools to be an effective communicator of your biomedical knowledge and experience to non-experts. You will build on the many skills you have already developed in your university study and learn how to explain your 4th year project work in ways that are simple, engaging and effective. You will explore how responsible research and innovation and critical thinking underpins modern biomedical science and how modern social media techniques can facilitate information exchange. You will learn from other biomedical scientists who have successfully created media profiles. You will also learn from subject matter experts and use resources to guide your learning and practice. Your growth in this unit will be determined by the completion of assessments through which you will unpack complex ideas using contemporary communication tools. The skill to explain sophisticated concepts in simple and effective ways is key to success in every area of biomedicine. This unit will equip you with the tools to be effective communicators of biomedicine as you move into careers and post-graduate pathways.

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.

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.

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.

This unit will study basic concepts and methods of time series analysis applicable in many real world problems in numerous fields, including economics, finance, insurance, physics, ecology, chemistry, computer science and engineering. This unit will investigate the basic methods of modelling and analyzing of time series data (ie. data containing serially dependence structure). This can be achieved through learning standard time series procedures on identification of components, autocorrelations, partial autocorrelations and their sampling properties. After setting up these basics, students will learn the theory of stationary univariate time series models including ARMA, ARIMA and SARIMA and their properties. Then the identification, estimation, diagnostic model checking, decision making and forecasting methods based on these models will be developed with applications. The spectral theory of time series, estimation of spectra using periodogram and consistent estimation of spectra using lag-windows will be studied in detail. Further, the methods of analyzing long memory and time series and heteroscedastic time series models including ARCH, GARCH, ACD, SCD and SV models from financial econometrics and the analysis of vector ARIMA models will be developed with applications. By completing this unit, students will develop the essential basis for further studies, such as financial econometrics and financial time series. The skills gained through this unit of study will form a strong foundation to work in a financial industry or in a related research organization.

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.

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.