# Mathematics Descriptions

## MATHEMATICS

## Mathematics major

A major in Mathematics requires 48 credit points from this table including:

(i) 12 credit points of 1000-level units as follows:

(a) 6 credit points of calculus units; 3 credit points of linear algebra units; and 3 credit points of statistics* or discrete mathematics units or

(b) 6 credit points of calculus units; 3 credit points of linear algebra units; and 3 credit points of statistics^ for students in the Mathematical Sciences program

(ii) 12 credit points of 2000-level core units

(iii) 6 credit points of 2000-level selective units

(iv) 6 credit points of 3000-level interdisciplinary project units

(v) 12 credit points of 3000-level or 4000-level selective units

*BSc students may substitute DATA1001 or ENVX1002 and students not enrolled in the BSc may substitute DATA1001, ECMT1010 or BUSS1020

^If elective space allows, students may substitute DATA1001/1901 for the statistics unit

## Mathematics minor

A minor in Mathematics requires 36 credit points from this table including:

(i) 12 credit points of 1000-level units as follows: 6 credit points of calculus units; 3 credit points of linear algebra units; and 3 credit points of statistics or discrete mathematics units

(ii) 12 credit points of 2000-level core units

(iii) 6 credit points of 2000-level selective units

(iv) 6 credit points of 3000-level or 4000-level selective units

### Units of study

The units of study are listed below.

#### 1000-level units of study

###### Calculus

**MATH1021 Calculus Of One Variable**

Credit points: 3 Teacher/Coordinator: A/Prof Sharon Stephen Session: Intensive January,Semester 1,Semester 2 Classes: 2x1-hr lectures; 1x1-hr tutorial/wk Prohibitions: MATH1011 or MATH1901 or MATH1906 or ENVX1001 or MATH1001 or MATH1921 or MATH1931 Assumed knowledge: HSC Mathematics Extension 1 or equivalent. Assessment: 2 x quizzes (30%), 2 x assignments (5%), online quizzes (10%), final exam (55%) Mode of delivery: Normal (lecture/lab/tutorial) day

Calculus is a discipline of mathematics that finds profound applications in science, engineering, and economics. This unit investigates differential calculus and integral calculus of one variable and the diverse applications of this theory. Emphasis is given both to the theoretical and foundational aspects of the subject, as well as developing the valuable skill of applying the mathematical theory to solve practical problems. Topics covered in this unit of study include complex numbers, functions of a single variable, limits and continuity, differentiation, optimisation, Taylor polynomials, Taylor's Theorem, Taylor series, Riemann sums, and Riemann integrals.

Textbooks

Calculus of One Variable (Course Notes for MATH1021)

**MATH1921 Calculus Of One Variable (Advanced)**

Credit points: 3 Teacher/Coordinator: A/Prof Sharon Stephen Session: Semester 1 Classes: 2x1-hr lectures; 1x1-hr tutorial/wk Prohibitions: MATH1001 or MATH1011 or MATH1906 or ENVX1001 or MATH1901 or MATH1021 or MATH1931 Assumed knowledge: (HSC Mathematics Extension 2) OR (Band E4 in HSC Mathematics Extension 1) or equivalent. Assessment: 2 x quizzes (20%); 2 x assignments (10%); final exam (70%) Mode of delivery: Normal (lecture/lab/tutorial) day

Note: Department permission required for enrolment

Calculus is a discipline of mathematics that finds profound applications in science, engineering, and economics. This unit investigates differential calculus and integral calculus of one variable and the diverse applications of this theory. Emphasis is given both to the theoretical and foundational aspects of the subject, as well as developing the valuable skill of applying the mathematical theory to solve practical problems. Topics covered in this unit of study include complex numbers, functions of a single variable, limits and continuity, differentiation, optimisation, Taylor polynomials, Taylor's Theorem, Taylor series, Riemann sums, and Riemann integrals. Additional theoretical topics included in this advanced unit include the Intermediate Value Theorem, Rolle's Theorem, and the Mean Value Theorem.

Textbooks

As set out in the Junior Mathematics Handbook

**MATH1931 Calculus Of One Variable (SSP)**

Credit points: 3 Teacher/Coordinator: A/Prof Sharon Stephen Session: Semester 1 Classes: 2x1-hr lectures; and 1x1-hr tutorial/wk Prohibitions: MATH1001 or MATH1011 or MATH1901 or ENVX1001 or MATH1906 or MATH1021 or MATH1921 Assumed knowledge: (HSC Mathematics Extension 2) OR (Band E4 in HSC Mathematics Extension 1) or equivalent. Assessment: Seminar participation (10%); 3 x special assignments (10%); 2 x quizzes (16%); 2 x assignments (8%); final exam (56%) Mode of delivery: Normal (lecture/lab/tutorial) day

Note: Department permission required for enrolment

Note: Enrolment is by invitation only

The Mathematics Special Studies Program is for students with exceptional mathematical aptitude, and requires outstanding performance in past mathematical studies. Students will cover the material of MATH1921 Calculus of One Variable (Adv), and attend a weekly seminar covering special topics on available elsewhere in the Mathematics and Statistics program.

**MATH1023 Multivariable Calculus and Modelling**

Credit points: 3 Teacher/Coordinator: A/Prof Sharon Stephen Session: Intensive January,Semester 1,Semester 2 Classes: 2x1-hr lectures; 1x1-hr tutorial/wk Prohibitions: MATH1013 or MATH1903 or MATH1907 or MATH1003 or MATH1923 or MATH1933 Assumed knowledge: Knowledge of complex numbers and methods of differential and integral calculus including integration by partial fractions and integration by parts as for example in MATH1021 or MATH1921 or MATH1931 or HSC Mathematics Extension 2 Assessment: 2 x quizzes (30%), 2 x assignments (5%), online quizzes (10%), final exam (55%) Mode of delivery: Normal (lecture/lab/tutorial) day

Calculus is a discipline of mathematics that finds profound applications in science, engineering, and economics. This unit investigates multivariable differential calculus and modelling. Emphasis is given both to the theoretical and foundational aspects of the subject, as well as developing the valuable skill of applying the mathematical theory to solve practical problems. Topics covered in this unit of study include mathematical modelling, first order differential equations, second order differential equations, systems of linear equations, visualisation in 2 and 3 dimensions, partial derivatives, directional derivatives, the gradient vector, and optimisation for functions of more than one variable.

Textbooks

Multivariable Calculus and Modelling (Course Notes for MATH1023)

**MATH1923 Multivariable Calculus and Modelling (Adv)**

Credit points: 3 Teacher/Coordinator: A/Prof Sharon Stephen Session: Semester 2 Classes: 2x1-hr lectures; and 1x1-hr tutorial/wk Prohibitions: MATH1003 or MATH1013 or MATH1907 or MATH1903 or MATH1023 or MATH1933 Assumed knowledge: (HSC Mathematics Extension 2) OR (Band E4 in HSC Mathematics Extension 1) or equivalent. Assessment: 2 x quizzes (20%); 2 x assignments (10%); final exam (70%) Mode of delivery: Normal (lecture/lab/tutorial) day

Note: Department permission required for enrolment

Calculus is a discipline of mathematics that finds profound applications in science, engineering, and economics. This unit investigates multivariable differential calculus and modelling. Emphasis is given both to the theoretical and foundational aspects of the subject, as well as developing the valuable skill of applying the mathematical theory to solve practical problems. Topics covered in this unit of study include mathematical modelling, first order differential equations, second order differential equations, systems of linear equations, visualisation in 2 and 3 dimensions, partial derivatives, directional derivatives, the gradient vector, and optimisation for functions of more than one variable. Additional topics covered in this advanced unit of study include the use of diagonalisation of matrices to study systems of linear equation and optimisation problems, limits of functions of two or more variables, and the derivative of a function of two or more variables.

Textbooks

As set out in the Junior Mathematics Handbook

**MATH1933 Multivariable Calculus and Modelling (SSP)**

Credit points: 3 Teacher/Coordinator: A/Prof Sharon Stephen Session: Semester 2 Classes: 2x1-hr lectures; and 1x1-hr tutorial/wk Prohibitions: MATH1003 or MATH1903 or MATH1013 or MATH1907 or MATH1023 or MATH1923 Assumed knowledge: (HSC Mathematics Extension 2) OR (Band E4 in HSC Mathematics Extension 1) or equivalent. Assessment: Seminar participation (10%); 3 x special assignments (10%); 2 x quizzes (16%); 2 x assignments (8%); final exam (56%) Mode of delivery: Normal (lecture/lab/tutorial) day

Note: Department permission required for enrolment

Note: Enrolment is by invitation only.

The Mathematics Special Studies Program is for students with exceptional mathematical aptitude, and requires outstanding performance in past mathematical studies. Students will cover the material of MATH1923 Multivariable Calculus and Modelling (Adv), and attend a weekly seminar covering special topics on available elsewhere in the Mathematics and Statistics program.

###### Linear algebra

**MATH1002 Linear Algebra**

Credit points: 3 Teacher/Coordinator: A/Prof Sharon Stephen Session: Intensive January,Semester 1 Classes: 2x1-hr lectures; 1x1-hr tutorial/wk Prohibitions: MATH1012 or MATH1014 or MATH1902 Assumed knowledge: HSC Mathematics or MATH1111. Students who have not completed HSC Mathematics (or equivalent) are strongly advised to take the Mathematics Bridging Course (offered in February). Assessment: online quizzes (10%), quiz (15%), assignments (10%), final exam (65%) Mode of delivery: Normal (lecture/lab/tutorial) day

MATH1002 is designed to provide a thorough preparation for further study in mathematics and statistics. It is a core unit of study providing three of the twelve credit points required by the Faculty of Science as well as a Junior level requirement in the Faculty of Engineering.

This unit of study introduces vectors and vector algebra, linear algebra including solutions of linear systems, matrices, determinants, eigenvalues and eigenvectors.

This unit of study introduces vectors and vector algebra, linear algebra including solutions of linear systems, matrices, determinants, eigenvalues and eigenvectors.

Textbooks

Linear Algebra: A Modern Introduction, (4th edition), David Poole

**MATH1902 Linear Algebra (Advanced)**

Credit points: 3 Teacher/Coordinator: A/Prof Sharon Stephen Session: Semester 1 Classes: 2x1-hr lectures; 1x1-hr tutorial/wk Prohibitions: MATH1002 or MATH1014 Assumed knowledge: (HSC Mathematics Extension 2) OR (90 or above in HSC Mathematics Extension 1) or equivalent Assessment: Online quizzes (10%); 4 x assignments (20%); final exam (70%) Mode of delivery: Normal (lecture/lab/tutorial) day

Note: Department permission required for enrolment

This unit is designed to provide a thorough preparation for further study in mathematics and statistics. It is a core unit of study providing three of the twelve credit points required by the Faculty of Science as well as a Junior level requirement in the Faculty of Engineering. It parallels the normal unit MATH1002 but goes more deeply into the subject matter and requires more mathematical sophistication.

Textbooks

As set out in the Junior Mathematics Handbook

###### Discrete mathematics

**MATH1004 Discrete Mathematics**

Credit points: 3 Teacher/Coordinator: A/Prof Sharon Stephen Session: Semester 2 Classes: 2x1-hr lectures; 1x1-hr tutorial/wk Prohibitions: MATH1904 or MATH1064 Assumed knowledge: HSC Mathematics or MATH1111. Students who have not completed HSC Mathematics (or equivalent) are strongly advised to take the Mathematics Bridging Course (offered in February). Assessment: 2 x quizzes (30%); 2 x assignments (5%); final exam (65%) Mode of delivery: Normal (lecture/lab/tutorial) day

This unit provides an introduction to fundamental aspects of discrete mathematics, which deals with 'things that come in chunks that can be counted'. It focuses on the enumeration of a set of numbers, viz. Catalan numbers. Topics include sets and functions, counting principles, discrete probability, Boolean expressions, mathematical induction, linear recurrence relations, graphs and trees.

Textbooks

Introduction to Discrete Mathematics, K G Choo and D E Taylor, Addison Wesley Long-man Australia, (1998)

**MATH1904 Discrete Mathematics (Advanced)**

Credit points: 3 Teacher/Coordinator: A/Prof Sharon Stephen Session: Semester 2 Classes: 2x1-hr lectures; 1x1-hr tutorial/wk Prohibitions: MATH1004 or MATH1064 Assumed knowledge: Strong skills in mathematical problem solving and theory, including coordinate geometry, integral and differential calculus, and solution of polynomial equations equivalent to HSC Mathematics Extension 2 or a Band E4 in HSC Mathematics Extension 1 Assessment: 2 x quizzes (30%); 2 x assignments (5%); final exam (65%) Mode of delivery: Normal (lecture/lab/tutorial) day

Note: Department permission required for enrolment

This unit is designed to provide a thorough preparation for further study in mathematics. It parallels the normal unit MATH1004 but goes more deeply into the subject matter and requires more mathematical sophistication.

Textbooks

As set out in the Junior Mathematics Handbook

###### Statistics

**MATH1005 Statistical Thinking with Data**

Credit points: 3 Teacher/Coordinator: A/Prof Sharon Stephen Session: Intensive January,Semester 1,Semester 2 Classes: 2x1-hr lectures; 1x1-hr lab/wk Prohibitions: MATH1015 or MATH1905 or STAT1021 or ECMT1010 or ENVX1001 or ENVX1002 or BUSS1020 or DATA1001 or DATA1901 Assumed knowledge: HSC Mathematics. Students who have not completed HSC Mathematics (or equivalent) are strongly advised to take the Mathematics Bridging Course (offered in February). Assessment: quizzes (10%), project 1 (10%), project 2 (15%), final exam (65%) Mode of delivery: Normal (lecture/lab/tutorial) day

In a data-rich world, global citizens need to problem solve with data, and evidence based decision-making is essential is every field of research and work.

This unit equips you with the foundational statistical thinking to become a critical consumer of data. You will learn to think analytically about data and to evaluate the validity and accuracy of any conclusions drawn. Focusing on statistical literacy, the unit covers foundational statistical concepts, including the design of experiments, exploratory data analysis, sampling and tests of significance.

This unit equips you with the foundational statistical thinking to become a critical consumer of data. You will learn to think analytically about data and to evaluate the validity and accuracy of any conclusions drawn. Focusing on statistical literacy, the unit covers foundational statistical concepts, including the design of experiments, exploratory data analysis, sampling and tests of significance.

Textbooks

Statistics, (4th Edition), Freedman Pisani Purves (2007)

**MATH1905 Statistical Thinking with Data (Advanced)**

Credit points: 3 Teacher/Coordinator: Prof Qiying Wang Session: Semester 2 Classes: 2x1-hr lectures; 1x1-hr tutorial/wk Prohibitions: MATH1005 or MATH1015 or STAT1021 or ECMT1010 or ENVX1001 or ENVX1002 or BUSS1020 or DATA1001 or DATA1901 Assumed knowledge: (HSC Mathematics Extension 2) OR (90 or above in HSC Mathematics Extension 1) or equivalent Assessment: 2 x quizzes (20%); 2 x assignments (10%); final exam (70%) Mode of delivery: Normal (lecture/lab/tutorial) day

Note: Department permission required for enrolment

This unit is designed to provide a thorough preparation for further study in mathematics and statistics. It is a core unit of study providing three of the twelve credit points required by the Faculty of Science as well as a Junior level requirement in the Faculty of Engineering. This Advanced level unit of study parallels the normal unit MATH1005 but goes more deeply into the subject matter and requires more mathematical sophistication.

Textbooks

A Primer of Statistics (4th edition), M C Phipps and M P Quine, Prentice Hall, Australia (2001)

#### 2000-level units of study

###### Core

**MATH2021 Vector Calculus and Differential Equations**

Credit points: 6 Teacher/Coordinator: Dr Daniel Hauer Session: Semester 1 Classes: 3x1-hr lectures; 1x1-hr tutorial; and 1x1-hr practice class/wk Prerequisites: (MATH1X21 or MATH1931 or MATH1X01 or MATH1906) and (MATH1XX2) and (MATH1X23 or MATH1933 or MATH1X03 or MATH1907) Prohibitions: MATH2921 or MATH2065 or MATH2965 or (MATH2061 and MATH2022) or (MATH2061 and MATH2922) or (MATH2961 and MATH2022) or (MATH2961 and MATH2922) or MATH2067 Assessment: 2 x quizzes (24%); 2 x assignments (16%); final exam (60%) Mode of delivery: Normal (lecture/lab/tutorial) day

This unit opens with topics from vector calculus, including vector-valued functions (parametrised curves and surfaces; vector fields; div, grad and curl; gradient fields and potential functions), line integrals (arc length; work; path-independent integrals and conservative fields; flux across a curve), iterated integrals (double and triple integrals, polar, cylindrical and spherical coordinates; areas, volumes and mass; Green's Theorem), flux integrals (flow through a surface; flux integrals through a surface defined by a function of two variables, through cylinders, spheres and other parametrised surfaces), Gauss' and Stokes' theorems. The unit then moves to topics in solution techniques for ordinary and partial differential equations (ODEs and PDEs) with applications. It provides a basic grounding in these techniques to enable students to build on the concepts in their subsequent courses. The main topics are: second order ODEs (including inhomogeneous equations), higher order ODEs and systems of first order equations, solution methods (variation of parameters, undetermined coefficients) the Laplace and Fourier Transform, an introduction to PDEs, and first methods of solutions (including separation of variables, and Fourier Series).

Textbooks

As set out in the Intermediate Mathematics Handbook

**MATH2921 Vector Calculus and Differential Eqs (Adv)**

Credit points: 6 Teacher/Coordinator: Dr Daniel Hauer Session: Semester 1 Classes: 3x1-hr lectures; 1x1-hr tutorial; and 1x1-hr practice class/wk Prerequisites: [(MATH1921 or MATH1931 or MATH1901 or MATH1906) or (a mark of 65 or above in MATH1021 or MATH1001)] and [MATH1902 or (a mark of 65 or above in MATH1002)] and [(MATH1923 or MATH1933 or MATH1903 or MATH1907) or (a mark of 65 or above in MATH1023 or MATH1003)] Prohibitions: MATH2021 or MATH2065 or MATH2965 or (MATH2061 and MATH2022) or (MATH2061 and MATH2922) or (MATH2961 and MATH2022) or (MATH2961 and MATH2922) or MATH2067 Assessment: Quizzes (10%), assignments (20%); final exam (60%) Mode of delivery: Normal (lecture/lab/tutorial) day

This is the advanced version of MATH2021, with more emphasis on the underlying concepts and mathematical rigour. The vector calculus component of the course will include: parametrised curves and surfaces, vector fields, div, grad and curl, gradient fields and potential functions, lagrange multipliers line integrals, arc length, work, path-independent integrals, and conservative fields, flux across a curve, double and triple integrals, change of variable formulas, polar, cylindrical and spherical coordinates, areas, volumes and mass, flux integrals, and Green's Gauss' and Stokes' theorems. The Differential Equations half of the course will focus on ordinary and partial differential equations (ODEs and PDEs) with applications with more complexity and depth. The main topics are: second order ODEs (including inhomogeneous equations), series solutions near a regular point, higher order ODEs and systems of first order equations, matrix equations and solutions, solution methods (variation of parameters, undetermined coefficients) the Laplace and Fourier Transform, elementary Sturm-Liouville theory, an introduction to PDEs, and first methods of solutions (including separation of variables, and Fourier Series). The unit then moves to topics in solution techniques for ordinary and partial differential equations (ODEs and PDEs) with applications. It provides a more thorough grounding in these techniques to enable students to build on the concepts in their subsequent courses. The main topics are: second order ODEs (including inhomogeneous equations), higher order ODEs and systems of first order equations, solution methods (variation of parameters, undetermined coefficients) the Laplace and Fourier Transform, an introduction to PDEs, and first methods of solutions (including separation of variables, and Fourier Series).

Textbooks

As set out in the Intermediate Mathematics Handbook

**MATH2022 Linear and Abstract Algebra**

Credit points: 6 Teacher/Coordinator: Dr Daniel Hauer Session: Semester 1 Classes: 3x1-hr lectures; 1x1-hr tutorial; and 1x1-hr practice class/wk Prerequisites: MATH1XX2 or (a mark of 65 or above in MATH1014) Prohibitions: MATH2922 or MATH2968 or (MATH2061 and MATH2021) or (MATH2061 and MATH2921) or (MATH2961 and MATH2021) or (MATH2961 and MATH2921) Assessment: 3 x quizzes (30%); 2 x assignments (10%); final exam (60%) Mode of delivery: Normal (lecture/lab/tutorial) day

Linear and abstract algebra is one of the cornerstones of mathematics and it is at the heart of many applications of mathematics and statistics in the sciences and engineering. This unit investigates and explores properties of linear functions, developing general principles relating to the solution sets of homogeneous and inhomogeneous linear equations, including differential equations. Linear independence is introduced as a way of understanding and solving linear systems of arbitrary dimension. Linear operators on real spaces are investigated, paying particular attention to the geometrical significance of eigenvalues and eigenvectors, extending ideas from first year linear algebra. To better understand symmetry, matrix and permutation groups are introduced and used to motivate the study of abstract group theory.

Textbooks

Linear Algebra: A Modern Introduction, (4th edition), David Poole

**MATH2922 Linear and Abstract Algebra (Advanced)**

Credit points: 6 Teacher/Coordinator: Dr Daniel Hauer Session: Semester 1 Classes: 3x1-hr lectures; 1x1-hr tutorial; and 1x1-hr practice class/wk Prerequisites: MATH1902 or (a mark of 65 or above in MATH1002) Prohibitions: MATH2022 or MATH2968 or (MATH2061 and MATH2021) or (MATH2061 and MATH2921) or (MATH2961 and MATH2021) or (MATH2961 and MATH2921) Assessment: Online quizzes (10%); 2 x assignments (20%); final exam (70%) Mode of delivery: Normal (lecture/lab/tutorial) day

Linear and abstract algebra is one of the cornerstones of mathematics and it is at the heart of many applications of mathematics and statistics in the sciences and engineering. This unit is an advanced version of MATH2022, with more emphasis on the underlying concepts and on mathematical rigour. This unit investigates and explores properties of vector spaces, matrices and linear transformations, developing general principles relating to the solution sets of homogeneous and inhomogeneous linear equations, including differential equations. Linear independence is introduced as a way of understanding and solving linear systems of arbitrary dimension. Linear operators on real spaces are investigated, paying particular attention to the geometrical significance of eigenvalues and eigenvectors, extending ideas from first year linear algebra. To better understand symmetry, matrix and permutation groups are introduced and used to motivate the study of abstract groups theory. The unit culminates in studying inner spaces, quadratic forms and normal forms of matrices together with their applications to problems both in mathematics and in the sciences and engineering.

Textbooks

As set out in the Intermediate Mathematics Handbook

###### Selective

**MATH2023 Analysis**

Credit points: 6 Teacher/Coordinator: Dr Daniel Hauer Session: Semester 2 Classes: 3x1-hr lectures; 1x1-hr tutorial; and 1x1-hr practice class/wk Prerequisites: (MATH1X21 or MATH1931 or MATH1X01 or MATH1906) and (MATH1X23 or MATH1933 or MATH1X03 or MATH1907) and (MATH1XX2 or a mark of 65 or above in MATH1014) Prohibitions: MATH2923 or MATH3068 or MATH2962 Assessment: 2 x in-class quizzes (20%); a take-home assignment (10%); and a final exam (70%) Mode of delivery: Normal (lecture/lab/tutorial) day

Analysis grew out of calculus, which leads to the study of limits of functions, sequences and series. It is one of the fundamental topics underlying much of mathematics including differential equations, dynamical systems, differential geometry, topology and Fourier analysis. This unit introduces the field of mathematical analysis both with a careful theoretical framework as well as selected applications. It shows the utility of abstract concepts and teaches an understanding and construction of proofs in mathematics. This unit will be useful to students of mathematics, science and engineering and in particular to future school mathematics teachers, because we shall explain why common practices in the use of calculus are correct, and understanding this is important for correct applications and explanations. The unit starts with the foundations of calculus and the real numbers system. It goes on to study the limiting behaviour of sequences and series of real and complex numbers. This leads naturally to the study of functions defined as limits and to the notion of uniform convergence. Returning to the beginnings of calculus and power series expansions leads to complex variable theory: elementary functions of complex variable, the Cauchy integral theorem, Cauchy integral formula, residues and related topics with applications to real integrals.

Textbooks

As set out in the Intermediate Mathematics Handbook

**MATH2923 Analysis (Advanced)**

Credit points: 6 Teacher/Coordinator: Dr Daniel Hauer Session: Semester 2 Classes: 3x1-hr lectures; 1x1-hr practice class and 1x1-hr tutorial/wk Prerequisites: [(MATH1921 or MATH1931 or MATH1901 or MATH1906) or (a mark of 65 or above in MATH1021 or MATH1001)] and [MATH1902 or (a mark of 65 or above in MATH1002)] and [(MATH1923 or MATH1933 or MATH1903 or MATH1907) or (a mark of 65 or above in MATH1023 or MATH1003)] Prohibitions: MATH2023 or MATH2962 or MATH3068 Assessment: 2 x quizzes (30%); an assignment (10%); and a final exam (60%) Mode of delivery: Normal (lecture/lab/tutorial) day

Analysis grew out of calculus, which leads to the study of limits of functions, sequences and series. It is one of the fundamental topics underlying much of mathematics including differential equations, dynamical systems, differential geometry, topology and Fourier analysis. This advanced unit introduces the field of mathematical analysis both with a careful theoretical frame- work as well as selected applications.This unit will be useful to students with more mathematical maturity who study mathematics, science, or engineering. Starting off with an axiomatic description of the real numbers system, this unit concentrates on the limiting behaviour of sequences and series of real and complex numbers. This leads naturally to the study of functions defined as limits and to the notion of uniform con-vergence. Special attention is given to power series, leading into the theory of analytic functions and complex analysis. Besides a rigorous treatment of many concepts from calculus, you will learn the basic results of complex analysis such as the Cauchy integral theorem, Cauchy integral formula, the residues theorems, leading to useful techniques for evaluating real integrals. By doing this unit, you will develop solid foundations in the more formal aspects of analysis, including knowledge of abstract concepts, how to apply them and the ability to construct proofs in mathematics.

Textbooks

As set out in the Intermediate Mathematics Handbook

**MATH2088 Number Theory and Cryptography**

Credit points: 6 Teacher/Coordinator: Dr Daniel Hauer Session: Semester 2 Classes: Three 1 hour lectures, one 1 hour tutorial and one 1 hour computer laboratory per week. Prerequisites: MATH1002 or MATH1902 or MATH1004 or MATH1904 or MATH1064 or (a mark of 65 or above in MATH1014) Prohibitions: MATH2068 or MATH2988 Assessment: 2 hour exam, assignments, quizzes (100%) Mode of delivery: Normal (lecture/lab/tutorial) day

Cryptography is the branch of mathematics that provides the techniques for confidential exchange of information sent via possibly insecure channels. This unit introduces the tools from elementary number theory that are needed to understand the mathematics underlying the most commonly used modern public key cryptosystems. Topics include the Euclidean Algorithm, Fermat's Little Theorem, the Chinese Remainder Theorem, Mobius Inversion, the RSA Cryptosystem, the Elgamal Cryptosystem and the Diffie-Hellman Protocol. Issues of computational complexity are also discussed.

**MATH2988 Number Theory and Cryptography Adv**

Credit points: 6 Teacher/Coordinator: Prof Martin Wechslberger Session: Semester 2 Classes: 3x1-hr lectures; 1x1-hr tutorial; and 1x1-hr computer lab/wk Prerequisites: MATH1902 or MATH1904 or (a mark of 65 or above in MATH1002 or MATH1004 or MATH1064) Prohibitions: MATH2068 or MATH2088 Assessment: Quiz (10%); 2 x assignments (20%); final 2-hr exam (70%) Mode of delivery: Normal (lecture/lab/tutorial) day

This unit of study is an advanced version of MATH2068, sharing the same lectures but with more advanced topics introduced in the tutorials and computer laboratory sessions.

Textbooks

Number Theory and Cryptography, R. Howlett, School of Mathematics and Statistics, University of Sydney, 2018.

#### 3000-level units of study

###### Interdisciplinary project units

**MATH3888 Projects in Mathematics**

Credit points: 6 Teacher/Coordinator: Prof Mary Myerscough Session: Semester 2 Classes: 2hrs lectures and 3 hrs workshop per week Prerequisites: (MATH2921 or MATH2021 or MATH2065 or MATH2965 or MATH2061 or MATH2961 or MATH2923 or MATH2023) and (MATH2922 or MATH2022 or MATH2061 or MATH2961 or MATH2088 or MATH2988) Assessment: Discipline content assignment (10%), discipline content quiz (20%), Discipline project report (10%), discipline project presentation (10%), reflective task (10%), team work process (10%), interdisciplinary project report (20%), interdisciplinary project presentation (10%) Mode of delivery: Normal (lecture/lab/tutorial) day

Mathematics is ubiquitous in the modern world. Mathematical ideas contribute to philosophy, art, music, economics, business, science, history, medicine and engineering. To really see the power and beauty of mathematics at work, students need to identify and explore interdisciplinary links. Engagement with other disciplines also provides essential foundational skills for using mathematics in the world beyond the lecture room. In this unit you will commence by working on a group project in an area of mathematics that interests you. From this you will acquire skills of teamwork, research, writing and project management as well as disciplinary knowledge. You will then have the opportunity to apply your disciplinary knowledge in an interdisciplinary team to identify and solve problems and communicate your findings to a diverse audience.

**SCPU3001 Science Interdisciplinary Project**

Credit points: 6 Teacher/Coordinator: Prof Pauline Ross Session: Intensive February,Intensive July,Semester 1,Semester 2 Classes: The unit consists of one seminar/workshop per week with accompanying online materials and a project to be determined in consultation with the partner organisation and completed as part of a team with academic supervision. Prerequisites: Completion of 2000-level units required for at least one Science major. Assessment: group plan, group presentation, reflective journal, group project Mode of delivery: Normal (lecture/lab/tutorial) day

This unit is designed for students who are concurrently enrolled in at least one 3000-level Science Table A unit of study to undertake a project that allows them to work with one of the University's industry and community partners. Students will work in teams on a real-world problem provided by the partner. This experience will allow students to apply their academic skills and disciplinary knowledge to a real-world issue in an authentic and meaningful way. Participation in this unit will require students to submit an application to the Faculty of Science.

###### Selective

**MATH3061 Geometry and Topology**

Credit points: 6 Teacher/Coordinator: Dr Leo Tzou Session: Semester 2 Classes: 3x1-hr lectures; 1x1-hr tutorial/wk Prerequisites: 12 credit points of Intermediate Mathematics Prohibitions: MATH3001 or MATH3006 Assumed knowledge: Theory and methods of linear transformations and vector spaces, for example MATH2061, MATH2961 or MATH2022 Assessment: 1 x Geometry assignment (5%); 1 x Topology assignment (5%); 1 x Geometry quiz (12%); 1 x Topology quiz (12%); 2-hr final exam (66%) Mode of delivery: Normal (lecture/lab/tutorial) day

The aim of the unit is to expand visual/geometric ways of thinking. The Geometry section is concerned mainly with transformations of the Euclidean plane (that is, bijections from the plane to itself), with a focus on the study of isometries (proving the classification theorem for transformations which preserve distances between points), symmetries (including the classification of frieze groups) and affine transformations (transformations which map lines to lines). The basic approach is via vectors and matrices, emphasising the interplay between geometry and linear algebra. The study of affine transformations is then extended to the study of collineations in the real projective plane, including collineations which map conics to conics. The Topology section considers graphs, surfaces and knots from a combinatorial point of view. Key ideas such as homeomorphism, subdivision, cutting and pasting and the Euler invariant are introduced first for graphs (1-dimensional objects) and then for triangulated surfaces (2-dimensional objects). Topics include the classification of surfaces, map colouring, decomposition of knots and knot invariants.

**MATH3063 Nonlinear ODEs with Applications**

Credit points: 6 Teacher/Coordinator: Dr Leo Tzou Session: Semester 1 Classes: 3x1-hr lectures; 1x1-hr tutorial/wk Prerequisites: 12 credit points of MATH2XXX units of study Prohibitions: MATH3963 or MATH4063 Assumed knowledge: MATH2061 or MATH2961 or [MATH2X21 and MATH2X22] Assessment: 3 x assignments (20%); 2 x class tests (20%); final exam (60%) Mode of delivery: Normal (lecture/lab/tutorial) day

This unit of study is an introduction to the theory of systems of ordinary differential equations. Such systems model many types of phenomena in engineering, biology and the physical sciences. The emphasis will not be on finding explicit solutions, but instead on the qualitative features of these systems, such as stability, instability and oscillatory behaviour. The aim is to develop a good geometrical intuition into the behaviour of solutions to such systems. Some background in linear algebra, and familiarity with concepts such as limits and continuity, will be assumed. The applications in this unit will be drawn from predator-prey systems, transmission of diseases, chemical reactions and other equations and systems from mathematical biology.

**MATH3066 Algebra and Logic**

Credit points: 6 Teacher/Coordinator: Dr Leo Tzou Session: Semester 1 Classes: 3x1-hr lectures; 1x1-hr tutorial/wk Prerequisites: 6 credit points of Intermediate Mathematics Prohibitions: MATH3062 or MATH3065 Assumed knowledge: Introductory knowledge of group theory. For example as in MATH2X22 Assessment: Quiz (10%); 2 x assignments (30%); cognitive, problem-based final exam (60%) Mode of delivery: Normal (lecture/lab/tutorial) day

This unit of study unifies and extends mathematical ideas and techniques that most participants will have met in their first and second years, and will be of general interest to all students of pure and applied mathematics. It combines algebra and logic to present and answer a number of related questions of fundamental importance in the development of mathematics, from ancient to modern times.

The Propositional and Predicate Calculi are studied as model axiomatic systems in their own right, including proofs of consistency and completeness. The final part of the course introduces precise notions of computability and decidability, through abstract Turing machines, culminating in the unsolvability of the Halting Problem the undecidability of First Order Logic, and a discussion of Godel's Incompleteness Theorem.

Classical and novel arithmetics are introduced, unified and described abstractly using field and ring axioms and the language of field extensions. Quotient rings are introduced, which are used to construct different finite and infinite fields. A construction of the real numbers, by factoring out rings of Cauchy sequences of rationals by the ideal of null sequences, is presented. Axiomatics are placed in the context of reasoning within first order logic and set theory.

The Propositional and Predicate Calculi are studied as model axiomatic systems in their own right, including proofs of consistency and completeness. The final part of the course introduces precise notions of computability and decidability, through abstract Turing machines, culminating in the unsolvability of the Halting Problem the undecidability of First Order Logic, and a discussion of Godel's Incompleteness Theorem.

Classical and novel arithmetics are introduced, unified and described abstractly using field and ring axioms and the language of field extensions. Quotient rings are introduced, which are used to construct different finite and infinite fields. A construction of the real numbers, by factoring out rings of Cauchy sequences of rationals by the ideal of null sequences, is presented. Axiomatics are placed in the context of reasoning within first order logic and set theory.

**MATH3076 Mathematical Computing**

Credit points: 6 Teacher/Coordinator: Prof Georg Gottwald Session: Semester 1 Classes: 3x1-hr lectures; 1x1-hr computer lab/wk Prerequisites: 12cp of MATH2XXX or [6cp of MATH2XXX and (6cp of STAT2XXX or DATA2X02)] Prohibitions: MATH3976 or MATH4076 Assessment: One 3 hour exam (55%), 2 assignments (15%+15%), 1 quiz (15%). To pass the course, students much achieve more than 50% on the final exam. Mode of delivery: Normal (lecture/lab/tutorial) day

This unit of study provides an introduction to programming and numerical methods. Topics covered include computer arithmetic and computational errors, systems of linear equations, interpolation and approximation, solution of nonlinear equations, quadrature, initial value problems for ordinary differential equations and boundary value problems, and optimisation.

**MATH3078 PDEs and Waves**

Credit points: 6 Teacher/Coordinator: Prof Georg Gottwald Session: Semester 2 Classes: 3x 1 hour lectures; 1x1 hour laboratory /wk Prerequisites: 12 credit points of MATH2XXX units of study Prohibitions: MATH3978 or MATH4078 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. 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.

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**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%) 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%) 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.

**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%). 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.

**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. 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%) 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.

**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%) 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%) 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%). 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. 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) 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.