# Mathematical Sciences

## MATHEMATICAL SCIENCES

## Mathematical Sciences program

The Mathematical Sciences program is available only to students eligible to enroll in the Dalyell stream.

A program in Mathematical Sciences requires 66 credit points including:

(i) A 48 credit point major in either, Data Science, Financial Mathematics and Statistics, Mathematics or Statistics, and:

(ii) 6 credit points of 1000-level program core units according to the following rules:

(a) For students with a major in Financial Mathematics and Statistics, Mathematics or Statistics, 6 credit points of 1000-level program core (non-Data Science major) units in addition to those counted towards the major.

(b) For students with a major in Data Science, 6 credit points of 1000-level program core (Data Science major) units in addition to those counted towards the major.

(iii) 12 credit points of 2000-level and 3000-level selective units according to the following rules:

(a) For students with a major in Mathematics, 6 credit points of 2000-level program selective (Mathematics major) units and 6 credit points of 3000-level selective units, in addition to those counted towards the major.

(b) For students with a major in Statistics, 6 credit points of 2000- level program selective set one (Statistics major) units, and 6 credit points of 2000-level program selective set two (Statistics major) in addition to those counted towards the major.

(c) For students with a major in Financial Mathematics and Statistics, 6 credit points of 2000-level program selective (Financial Mathematics and Statistics major) units, and 6 credit points of 3000-level selective units, in addition to those counted towards the major.

(d) For students with a major in Data Science, 6 credit points of 2000-level program selective (Data Science major) units in addition to those counted towards the major, and 6 credit points of 3000-level selective units, in addition to those counted towards the major.

### Units of study

The units of study are listed below.

#### 1000-level units of study

###### Program Core (Non-Data Science major)

**DATA1002 Informatics: Data and Computation**

Credit points: 6 Session: Semester 2 Classes: Refer to the unit of study outline https://www.sydney.edu.au/units Prohibitions: INFO1903 OR DATA1902 Assessment: Refer to the assessment table in the unit outline. Mode of delivery: Normal (lecture/lab/tutorial) day

This unit covers computation and data handling, integrating sophisticated use of existing productivity software, e.g. spreadsheets, with the development of custom software using the general-purpose Python language. It will focus on skills directly applicable to data-driven decision-making. Students will see examples from many domains, and be able to write code to automate the common processes of data science, such as data ingestion, format conversion, cleaning, summarization, creation and application of a predictive model.

**DATA1902 Informatics: Data and Computation (Advanced)**

Credit points: 6 Session: Semester 2 Classes: Refer to the unit of study outline https://www.sydney.edu.au/units Prohibitions: INFO1903 OR DATA1002 Assumed knowledge: This unit is intended for students with ATAR at least sufficient for entry to the BSc/BAdvStudies(Advanced) stream, or for those who gained Distinction results or better, in some unit in Data Science, Mathematics, or Computer Science. Students with portfolio of high-quality relevant prior work can also be admitted Assessment: Refer to the assessment table in the unit outline. Mode of delivery: Normal (lecture/lab/tutorial) day

Note: Department permission required for enrolment

This unit covers computation and data handling, integrating sophisticated use of existing productivity software, e. g. spreadsheets, with the development of custom software using the general-purpose Python language. It will focus on skills directly applicable to data-driven decision-making. Students will see examples from many domains, and be able to write code to automate the common processes of data science, such as data ingestion, format conversion, cleaning, summarization, creation and application of a predictive model. This unit includes the content of DATA1002, along with additional topics that are more sophisticated, suited for students with high academic achievement.

###### Program Core (Data Science major)

**MATH1021 Calculus Of One Variable**

Credit points: 3 Teacher/Coordinator: Refer to the unit of study outline https://www.sydney.edu.au/units Session: Intensive January,Semester 1,Semester 2 Classes: Refer to the unit of study outline https://www.sydney.edu.au/units Prohibitions: MATH1901 or MATH1906 or ENVX1001 or MATH1001 or MATH1921 or MATH1931 Assumed knowledge: HSC Mathematics Extension 1 or equivalent Assessment: Refer to the unit of study outline https://www.sydney.edu.au/units 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.

Students are strongly recommended to complete MATH1021 or MATH1921 Calculus Of One Variable (Advanced) before commencing MATH1023 Multivariable Calculus and Modelling or MATH1923 Multivariable Calculus and Modelling (Adv).

Students are strongly recommended to complete MATH1021 or MATH1921 Calculus Of One Variable (Advanced) before commencing MATH1023 Multivariable Calculus and Modelling or MATH1923 Multivariable Calculus and Modelling (Adv).

Textbooks

Calculus of One Variable (Course Notes for MATH1021)

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

Credit points: 3 Teacher/Coordinator: Refer to the unit of study outline https://www.sydney.edu.au/units Session: Semester 1 Classes: Refer to the unit of study outline https://www.sydney.edu.au/units Prohibitions: MATH1001 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: Refer to the unit of study outline https://www.sydney.edu.au/units 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. Students are strongly recommended to complete MATH1021 Calculus Of One Variable or MATH1921 Calculus Of One Variable (Advanced) before commencing MATH1023 Multivariable Calculus and Modelling or MATH1923 Multivariable Calculus and Modelling (Adv).

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

Credit points: 3 Teacher/Coordinator: Refer to the unit of study outline https://www.sydney.edu.au/units Session: Semester 1,Semester 1 Classes: Refer to the unit of study outline https://www.sydney.edu.au/units Prohibitions: MATH1001 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: Refer to the unit of study outline https://www.sydney.edu.au/units 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: Refer to the unit of study outline https://www.sydney.edu.au/units Session: Intensive January,Semester 1,Semester 2 Classes: Refer to the unit of study outline https://www.sydney.edu.au/units 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: Refer to the unit of study outline https://www.sydney.edu.au/units 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.

Students are strongly recommended to complete MATH1021 or MATH1921 Calculus Of One Variable (Advanced) before commencing MATH1023 Multivariable Calculus and Modelling or MATH1923 Multivariable Calculus and Modelling (Adv).

Students are strongly recommended to complete MATH1021 or MATH1921 Calculus Of One Variable (Advanced) before commencing MATH1023 Multivariable Calculus and Modelling or MATH1923 Multivariable Calculus and Modelling (Adv).

Textbooks

Multivariable Calculus and Modelling (Course Notes for MATH1023)

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

Credit points: 3 Teacher/Coordinator: Refer to the unit of study outline https://www.sydney.edu.au/units Session: Semester 2 Classes: Refer to the unit of study outline https://www.sydney.edu.au/units 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: Refer to the unit of study outline https://www.sydney.edu.au/units 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. Students are strongly recommended to complete MATH1021 or MATH1921 Calculus Of One Variable (Advanced) before commencing MATH1023 Multivariable Calculus and Modelling or MATH1923 Multivariable Calculus and Modelling (Adv).

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

Credit points: 3 Teacher/Coordinator: Refer to the unit of study outline https://www.sydney.edu.au/units Session: Semester 2 Classes: Refer to the unit of study outline https://www.sydney.edu.au/units 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: Refer to the unit of study outline https://www.sydney.edu.au/units 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.

#### 2000-level units of study

###### Program selective (Mathematics major)

**DATA2902 Data Analytics: Learning from Data (Adv)**

Credit points: 6 Teacher/Coordinator: Refer to the unit of study outline https://www.sydney.edu.au/units Session: Semester 2 Classes: Refer to the unit of study outline https://www.sydney.edu.au/units Prerequisites: A mark of 65 or above in (DATA1X01 or ENVX1002 or [MATH1X05 and MATH1XXX (excluding MATH1X05)] or BUSS1020 or ECMT1010) Prohibitions: STAT2012 or STAT2912 or DATA2002 Assumed knowledge: Successful completion of a first-year or second-year unit in statistics or data science including a substantial coding component. The content from STAT2X11 will help but is not considered essential. Students who are not comfortable using the R software for statistical analysis should familiarise themselves before attempting the unit, e.g. taking OLET1632: Shark Bites and Other Data Stories Assessment: Refer to the unit of study outline https://www.sydney.edu.au/units Mode of delivery: Normal (lecture/lab/tutorial) day

Technological advances in science, business, and engineering have given rise to a proliferation of data from all aspects of our life. Understanding the information presented in these data is critical as it enables informed decision making into many areas including market intelligence and science. DATA2902 is an intermediate unit in statistics and data sciences, focusing on learning advanced data analytic skills for a wide range of problems and data. In this unit, you will learn how to ingest, combine and summarise data from a variety of data models which are typically encountered in data science projects as well as reinforce your programming skills through experience with statistical programming language. You will also be exposed to the concept of statistical machine learning and develop the skills to analyse various types of data in order to answer a scientific question. From this unit, you will develop knowledge and skills that will enable you to embrace data analytic challenges stemming from everyday problems.

**STAT2911 Probability and Statistical Models (Adv)**

Credit points: 6 Teacher/Coordinator: Refer to the unit of study outline https://www.sydney.edu.au/units Session: Semester 1 Classes: Refer to the unit of study outline https://www.sydney.edu.au/units Prerequisites: (MATH1X21 or MATH1931 or MATH1X01 or MATH1906 or MATH1011) and a mark of 65 or greater in (DATA1X01 or MATH10X5 or MATH1905 or STAT1021 or ECMT1010 or BUSS1020) Prohibitions: STAT2011 Assessment: Refer to the unit of study outline https://www.sydney.edu.au/units Mode of delivery: Normal (lecture/lab/tutorial) day

This unit is essentially an advanced version of STAT2011, with an emphasis on the mathematical techniques used to manipulate random variables and probability models. Common distributions including the Poisson, normal, beta and gamma families as well as the bivariate normal are introduced. Moment generating functions and convolution methods are used to understand the behaviour of sums of random variables. The method of moments and maximum likelihood techniques for fitting statistical distributions to data will be explored. The notions of conditional expectation and prediction will be covered as will be distributions related to the normal: chi^2, t and F. The unit has weekly computer classes where you will learn to use a statistical computing package to perform simulations and carry out computer intensive estimation techniques like the bootstrap method.

Textbooks

Mathematical Statistics and Data Analysis (3rd edition), J A Rice

###### Program selective set one (Statistics major)

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

Credit points: 6 Teacher/Coordinator: Refer to the unit of study outline https://www.sydney.edu.au/units Session: Semester 1 Classes: Refer to the unit of study outline https://www.sydney.edu.au/units 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: Refer to the unit of study outline https://www.sydney.edu.au/units 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

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

Credit points: 6 Teacher/Coordinator: Refer to the unit of study outline https://www.sydney.edu.au/units Session: Semester 1 Classes: Refer to the unit of study outline https://www.sydney.edu.au/units 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: Refer to the unit of study outline https://www.sydney.edu.au/units 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

###### Program selective set two (Statistics major)

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

Credit points: 6 Teacher/Coordinator: Refer to the unit of study outline https://www.sydney.edu.au/units Session: Semester 1 Classes: Refer to the unit of study outline https://www.sydney.edu.au/units 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: Refer to the unit of study outline https://www.sydney.edu.au/units 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

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

Credit points: 6 Teacher/Coordinator: Refer to the unit of study outline https://www.sydney.edu.au/units Session: Semester 1 Classes: Refer to the unit of study outline https://www.sydney.edu.au/units 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: Refer to the unit of study outline https://www.sydney.edu.au/units 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

**MATH2923 Analysis (Advanced)**

Credit points: 6 Teacher/Coordinator: Refer to the unit of study outline https://www.sydney.edu.au/units Session: Semester 2 Classes: Refer to the unit of study outline https://www.sydney.edu.au/units 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: Refer to the unit of study outline https://www.sydney.edu.au/units 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 framework 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

**MATH2988 Number Theory and Cryptography Adv**

Credit points: 6 Teacher/Coordinator: Refer to the unit of study outline https://www.sydney.edu.au/units Session: Semester 2 Classes: Refer to the unit of study outline https://www.sydney.edu.au/units Prerequisites: MATH1902 or MATH1904 or (a mark of 65 or above in MATH1002 or MATH1004 or MATH1064) Prohibitions: MATH2068 or MATH2088 Assessment: Refer to the unit of study outline https://www.sydney.edu.au/units Mode of delivery: Normal (lecture/lab/tutorial) day

This unit of study is an advanced version of MATH2088, 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.

**MATH2970 Optimisation and Financial Mathematics Adv**

Credit points: 6 Teacher/Coordinator: Refer to the unit of study outline https://www.sydney.edu.au/units Session: Semester 2 Classes: Refer to the unit of study outline https://www.sydney.edu.au/units Prerequisites: [MATH19X1 or MATH1906 or (a mark of 65 or above in MATH1021 or MATH1001)] and [MATH1902 or (a mark of 65 or above in MATH1002)] Prohibitions: MATH2070 Assumed knowledge: MATH19X3 or MATH1907 or a mark of 65 or above in MATH1003 or MATH1023 Assessment: Refer to the unit of study outline https://www.sydney.edu.au/units Mode of delivery: Normal (lecture/lab/tutorial) day

Note: Students may enrol in both MATH2970 and MATH3975 in the same semester

The content of this unit of study parallels that of MATH2070, but students enrolled at Advanced level will undertake more advanced problem solving and assessment tasks, and some additional topics may be included.

###### Program selective (Financial Mathematics and Statistics major)

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

Credit points: 6 Teacher/Coordinator: Refer to the unit of study outline https://www.sydney.edu.au/units Session: Semester 1 Classes: Refer to the unit of study outline https://www.sydney.edu.au/units 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: Refer to the unit of study outline https://www.sydney.edu.au/units 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

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

Credit points: 6 Teacher/Coordinator: Refer to the unit of study outline https://www.sydney.edu.au/units Session: Semester 1 Classes: Refer to the unit of study outline https://www.sydney.edu.au/units 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: Refer to the unit of study outline https://www.sydney.edu.au/units 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

###### Program selective (Data Science major)

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

Textbooks

As set out in the Intermediate Mathematics Handbook

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

Textbooks

As set out in the Intermediate Mathematics Handbook

**STAT2911 Probability and Statistical Models (Adv)**

Credit points: 6 Teacher/Coordinator: Refer to the unit of study outline https://www.sydney.edu.au/units Session: Semester 1 Classes: Refer to the unit of study outline https://www.sydney.edu.au/units Prerequisites: (MATH1X21 or MATH1931 or MATH1X01 or MATH1906 or MATH1011) and a mark of 65 or greater in (DATA1X01 or MATH10X5 or MATH1905 or STAT1021 or ECMT1010 or BUSS1020) Prohibitions: STAT2011 Assessment: Refer to the unit of study outline https://www.sydney.edu.au/units Mode of delivery: Normal (lecture/lab/tutorial) day

This unit is essentially an advanced version of STAT2011, with an emphasis on the mathematical techniques used to manipulate random variables and probability models. Common distributions including the Poisson, normal, beta and gamma families as well as the bivariate normal are introduced. Moment generating functions and convolution methods are used to understand the behaviour of sums of random variables. The method of moments and maximum likelihood techniques for fitting statistical distributions to data will be explored. The notions of conditional expectation and prediction will be covered as will be distributions related to the normal: chi^2, t and F. The unit has weekly computer classes where you will learn to use a statistical computing package to perform simulations and carry out computer intensive estimation techniques like the bootstrap method.

Textbooks

Mathematical Statistics and Data Analysis (3rd edition), J A Rice

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**DATA3888 Data Science Capstone**

Credit points: 6 Teacher/Coordinator: Refer to the unit of study outline https://www.sydney.edu.au/units Session: Semester 1 Classes: Refer to the unit of study outline https://www.sydney.edu.au/units Prerequisites: DATA2001 or DATA2901 or DATA2002 or DATA2902 or STAT2912 or STAT2012 Assessment: Refer to the unit of study outline https://www.sydney.edu.au/units Mode of delivery: Normal (lecture/lab/tutorial) day

In our ever-changing world, we are facing a new data-driven era where the capability to efficiently combine and analyse large data collections is essential for informed decision making in business and government, and for scientific research. Data science is an emerging interdisciplinary field with its focus on high performance computation and quantitative expression of the confidence in conclusions, and the clear communication of those conclusions in different discipline context. This unit is our capstone project that presents the opportunity to create a public data product that can illustrate the concepts and skills you have learnt in this discipline. In this unit, you will have an opportunity to explore deeper disciplinary knowledge; while also meeting and collaborating through project-based learning. The capstone project in this unit will allow you to identify and place the data-driven problem into an analytical framework, solve the problem through computational means, interpret the results and communicate your findings to a diverse audience. All such skills are highly valued by employers. This unit will foster the ability to work in an interdisciplinary team, to translate problem between two or more disciplines and this is essential for both professional and research pathways in the future.

**FMAT3888 Projects in Financial Mathematics**

Credit points: 6 Teacher/Coordinator: Refer to the unit of study outline https://www.sydney.edu.au/units Session: Semester 2 Classes: Refer to the unit of study outline https://www.sydney.edu.au/units Prerequisites: (MATH2070 or MATH2970) and (STAT2011 or STAT2911) Assumed knowledge: STAT2X11, MATH2X70 Assessment: Refer to the unit of study outline https://www.sydney.edu.au/units Mode of delivery: Block mode

Mathematics and statistics are powerful tools in finance and more generally in the world at large. To really experience the power of mathematics and statistics at work, students need to identify and explore interdisciplinary links. Engagement with other disciplines also provides essential foundational skills for using mathematical and statistical ideas in financial contexts and in the world beyond. In this unit you will commence by working on a group project in an area of financial mathematics or statistics. From this project you will acquire skills of teamwork, research, wring 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.

**MATH3888 Projects in Mathematics**

Credit points: 6 Teacher/Coordinator: Refer to the unit of study outline https://www.sydney.edu.au/units Session: Semester 2 Classes: Refer to the unit of study outline https://www.sydney.edu.au/units 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: Refer to the unit of study outline https://www.sydney.edu.au/units 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.

**MATH3961 Metric Spaces (Advanced)**

Credit points: 6 Teacher/Coordinator: Refer to the unit of study outline https://www.sydney.edu.au/units Session: Semester 1 Classes: Refer to the unit of study outline https://www.sydney.edu.au/units Prerequisites: A mark of 65 or greater in 12 credit points of 2000-level Mathematics units Prohibitions: MATH4061 Assumed knowledge: Real analysis and vector spaces. For example (MATH2922 or MATH2961) and (MATH2923 or MATH2962) Assessment: Refer to the unit of study outline https://www.sydney.edu.au/units Mode of delivery: Normal (lecture/lab/tutorial) day

Topology, developed at the end of the 19th Century to investigate the subtle interaction of analysis and geometry, is now one of the basic disciplines of mathematics. A working knowledge of the language and concepts of topology is essential in fields as diverse as algebraic number theory and non-linear analysis. This unit develops the basic ideas of topology using the example of metric spaces to illustrate and motivate the general theory. Topics covered include: Metric spaces, convergence, completeness and the Contraction Mapping Theorem; Metric topology, open and closed subsets; Topological spaces, subspaces, product spaces; Continuous mappings and homeomorphisms; Compactness Connectedness Hausdorff spaces and normal spaces. You will learn methods and techniques of proving basic theorems in point-set topology and apply them to other areas of mathematics including basic Hilbert space theory and abstract Fourier series. By doing this unit you will develop solid foundations in the more formal aspects of topology, including knowledge of abstract concepts and how to apply them. Applications include the use of the Contraction Mapping Theorem to solve integral and differential equations.

**MATH3962 Rings, Fields and Galois Theory (Adv)**

Credit points: 6 Teacher/Coordinator: Refer to the unit of study outline https://www.sydney.edu.au/units Session: Semester 1 Classes: Refer to the unit of study outline https://www.sydney.edu.au/units Prerequisites: MATH2961 or MATH2922 or a mark of 65 or greater in (MATH2061 or MATH2022) Prohibitions: MATH3062 or MATH4062 Assessment: Refer to the unit of study outline https://www.sydney.edu.au/units Mode of delivery: Normal (lecture/lab/tutorial) day

This unit of study investigates the modern mathematical theory that was originally developed for the purpose of studying polynomial equations. The philosophy is that it should be possible to factorize 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 generalizes 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.

The basic theoretical tool needed for this program is the concept of a ring, which generalizes 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.

**MATH3963 Nonlinear ODEs with Applications (Adv)**

Credit points: 6 Teacher/Coordinator: Refer to the unit of study outline https://www.sydney.edu.au/units Session: Semester 1 Classes: Refer to the unit of study outline https://www.sydney.edu.au/units Prerequisites: A mark of 65 or greater in 12 credit points of MATH2XXX units of study Prohibitions: MATH3063 or MATH4063 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: Refer to the unit of study outline https://www.sydney.edu.au/units 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 of different nature. 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.

Textbooks

Differential Dynamical Systems, J D Meiss (online)

**MATH3968 Differential Geometry (Advanced)**

Credit points: 6 Teacher/Coordinator: Refer to the unit of study outline https://www.sydney.edu.au/units Session: Semester 2 Classes: Refer to the unit of study outline https://www.sydney.edu.au/units Prerequisites: A mark of 65 or greater in 12 credit points of MATH2XXX units of study Prohibitions: MATH4068 Assumed knowledge: (MATH2921 and MATH2922) or MATH2961 Assessment: Refer to the unit of study outline https://www.sydney.edu.au/units 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. 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.

**MATH3969 Measure Theory and Fourier Analysis (Adv)**

Credit points: 6 Teacher/Coordinator: Refer to the unit of study outline https://www.sydney.edu.au/units Session: Semester 2 Classes: Refer to the unit of study outline https://www.sydney.edu.au/units Prerequisites: A mark of 65 or greater in 12 credit points of MATH2XXX units of study Prohibitions: MATH4069 Assumed knowledge: Real analysis and vector spaces. For example MATH2X21 and MATH2X23 Assessment: Refer to the unit of study outline https://www.sydney.edu.au/units Mode of delivery: Normal (lecture/lab/tutorial) day

Measure theory is the study of such fundamental ideas as length, area, volume, arc length and surface area. It is the basis for the integration theory used in advanced mathematics since it was developed by Henri Lebesgue in about 1900. Moreover, it is the basis for modern probability theory. The course starts by setting up measure theory and integration, establishing important results such as Fubini's Theorem and the Dominated Convergence Theorem which allow us to manipulate integrals. This is then applied to Fourier Analysis, and results such as the Inversion Formula and Plancherel's Theorem are derived. The Radon-Nikodyn Theorem provides a representation of measures in terms of a density. Probability theory is then discussed with topics including distributions and conditional expectation.

**MATH3971 Convex Analysis and Optimal Control (Adv)**

*This unit of study is not available in 2022*

Credit points: 6 Teacher/Coordinator: Prof Georg Gottwald Session: Semester 1 Classes: Lecture 3hours/week, tutorial 1hr/week Prerequisites: A mark of 65 or above in 12cp from (MATH2XXX or STAT2XXX or DATA2X02) Prohibitions: MATH4071 Assumed knowledge: MATH2X21 and MATH2X23 and STAT2X11 Assessment: Assignment (15%), assignment (15%), exam (70%) Mode of delivery: Normal (lecture/lab/tutorial) day

The questions how to maximise your gain (or to minimise the cost) and how to determine the optimal strategy/policy are fundamental for an engineer, an economist, a doctor designing a cancer therapy, or a government planning some social policies. Many problems in mechanics, physics, neuroscience and biology can be formulated as optimistion 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 straight forward examples of such problems. You will also learn about important classes of optimisation problems arising in finance, economics, engineering and insurance.

**MATH3974 Fluid Dynamics (Advanced)**

Credit points: 6 Teacher/Coordinator: Refer to the unit of study outline https://www.sydney.edu.au/units Session: Semester 1 Classes: Refer to the unit of study outline https://www.sydney.edu.au/units Prerequisites: An average mark of 65 or more in (12 credit points of MATH2XXX) Prohibitions: MATH4074 Assumed knowledge: [MATH2961 and MATH2965] or [MATH2921 and MATH2922] Assessment: Refer to the unit of study outline https://www.sydney.edu.au/units Mode of delivery: Normal (lecture/lab/tutorial) day

This unit of study provides an introduction to fluid dynamics, starting with a description of the governing equations and the simplifications gained by using stream functions or potentials. It develops elementary theorems and tools, including Bernoulli's equation, the role of vorticity, the vorticity equation, Kelvin's circulation theorem, Helmholtz's theorem, and an introduction to the use of tensors. Topics covered include viscous flows, lubrication theory, boundary layers, potential theory, and complex variable methods for 2-D airfoils. The unit concludes with an introduction to hydrodynamic stability theory and the transition to turbulent flow.

**MATH3975 Financial Derivatives (Advanced)**

Credit points: 6 Teacher/Coordinator: Refer to the unit of study outline https://www.sydney.edu.au/units Session: Semester 2 Classes: Refer to the unit of study outline https://www.sydney.edu.au/units Prerequisites: A mark of 65 or above in 12cp from (MATH2XXX or STAT2XXX or DATA2X02) Prohibitions: MATH3933 or MATH3015 or MATH3075 Assessment: Refer to the unit of study outline https://www.sydney.edu.au/units Mode of delivery: Normal (lecture/lab/tutorial) day

Note: MATH2X70 and MATH3975 may be taken in the same semester

This unit will introduce you to the mathematical theory of modern finance with the special emphasis on the valuation and hedging of financial derivatives, such as: forward contracts and options of European and American style. You will learn about the concept of arbitrage and how to model risk-free and risky securities. Topics covered by this unit include: the notions of a martingale and a martingale measure, the fundamental theorems of asset pricing, complete and incomplete markets, the binomial options pricing model, discrete random walks and the Brownian motion, the Black-Scholes options pricing model and the valuation and hedging of exotic options. Students completing this unit have been highly sought by the finance industry, which continues to need graduates with quantitative skills. Students enrolled in this unit at advanced level will have to undertake more challenging assessment tasks, but lectures in the advanced level are held concurrently with those of the corresponding mainstream unit.

**MATH3976 Mathematical Computing (Advanced)**

Credit points: 6 Teacher/Coordinator: Refer to the unit of study outline https://www.sydney.edu.au/units Session: Semester 1 Classes: Refer to the unit of study outline https://www.sydney.edu.au/units Prerequisites: A mark of 65 or above in [(12cp of MATH2XXX) or (6cp of MATH2XXX and 6cp of STAT2XXX or DATA2X02)] Prohibitions: MATH3076 or MATH4076 Assumed knowledge: Strong skills in linear algebra and the theory and methods of ordinary and partial differential equations for example (MATH2961 and MATH2965) or (MATH2921 and MATH2922) Assessment: Refer to the unit of study outline https://www.sydney.edu.au/units 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.

**MATH3977 Lagrangian and Hamiltonian Dynamics (Adv)**

Credit points: 6 Teacher/Coordinator: Refer to the unit of study outline https://www.sydney.edu.au/units Session: Semester 2 Classes: Refer to the unit of study outline https://www.sydney.edu.au/units Prerequisites: A mark of 65 or greater in 12 credit points of MATH2XXX units of study Prohibitions: MATH4077 Assessment: Refer to the unit of study outline https://www.sydney.edu.au/units Mode of delivery: Normal (lecture/lab/tutorial) day

Lagrangian and Hamiltonian dynamics are reformulations of classical Newtonian mechanics into a mathematically sophisticated framework using arbitrary coordinate systems. This formulation of classical mechanics 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 simplify apparently complicated dynamical problems. Connections between geometry and different physical theories beyond classical mechanics are explored. Students will be expected to describe and solve mechanical systems of some complexity including planetary motion and to investigate stability. Hamilton-Jacobi theory will be used to solve problems ranging from geodesic motion (shortest path between two points) on curved surfaces to relativistic motion in the vicinity of black holes.

**MATH3978 PDEs and Waves (Advanced)**

Credit points: 6 Teacher/Coordinator: Refer to the unit of study outline https://www.sydney.edu.au/units Session: Semester 2 Classes: Refer to the unit of study outline https://www.sydney.edu.au/units Prerequisites: A mark of 65 or greater in 6cp from (MATH2X21 or MATH2X65 or MATH2067) and a mark of 65 or greater 6cp from (MATH2X22 or MATH2X61) Prohibitions: MATH3078 or MATH4078 Assumed knowledge: [MATH2X61 and MATH2X65] or [MATH2X21 and MATH2X22] Assessment: Refer to the unit of study outline https://www.sydney.edu.au/units 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.

**MATH3979 Complex Analysis (Advanced)**

Credit points: 6 Teacher/Coordinator: Refer to the unit of study outline https://www.sydney.edu.au/units Session: Semester 1 Classes: Refer to the unit of study outline https://www.sydney.edu.au/units Prerequisites: A mark of 65 or above in 12cp of MATH2XXX Prohibitions: MATH4079 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: Refer to the unit of study outline https://www.sydney.edu.au/units Mode of delivery: Normal (lecture/lab/tutorial) day

This unit continues the study of functions of a complex variable introduced in the second year unit Analysis (MATH2023/2923). It is aimed at highlighting certain topics from analytic function theory that have wide applications and intrinsic beauty. By learning about the analysis of functions of a complex variable, you will acquire a very important background for mathematical areas such as dynamics, algebraic and differential geometry, and number theory; and advanced theoretical physics such as quantum mechanics, string theory, and quantum field theory. 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. 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.

**STAT3921 Stochastic Processes (Advanced)**

Credit points: 6 Teacher/Coordinator: Refer to the unit of study outline https://www.sydney.edu.au/units Session: Semester 1 Classes: Refer to the unit of study outline https://www.sydney.edu.au/units Prerequisites: STAT2X11 Prohibitions: STAT3011 or STAT3911 or STAT3021 or STAT3003 or STAT3903 or STAT3005 or STAT3905 or STAT4021 Assumed knowledge: Students are expected to have a thorough knowledge of basic probability and integral calculus and to have achieved at credit level or above Assessment: Refer to the unit of study outline https://www.sydney.edu.au/units Mode of delivery: Normal (lecture/lab/tutorial) day

A stochastic process is a mathematical model of time-dependent random phenomena and is employed in numerous fields of application, including economics, finance, insurance, physics, biology, chemistry and computer science. This unit will establish basic properties of discrete-time Markov chains including random walks and branching processes. This unit will derive key results of Poisson processes and simple continuous-time Markov chains. This unit will investigate simple queuing theory. This unit will also introduce basic concepts of Brownian motion and martingales. Throughout the unit, various illustrative examples are provided in modelling and analysing problems of practical interest. By completing this unit, you will develop a solid mathematical foundation of stochastic processes for further studies in advanced areas such as stochastic analysis, stochastic differential equations, stochastic control, financial mathematics and statistical inference. Students who undertake STAT3921/4021 will be expected to have a deeper, more sophisticated understanding of the theory and to be able to work with more complicated applications than students who complete the regular STAT3021 unit.

**STAT3922 Applied Linear Models (Advanced)**

Credit points: 6 Teacher/Coordinator: Refer to the unit of study outline https://www.sydney.edu.au/units Session: Semester 1 Classes: Refer to the unit of study outline https://www.sydney.edu.au/units Prerequisites: STAT2X11 and [a mark of 65 or greater in (STAT2X12 or DATA2X02)] Prohibitions: STAT3912 or STAT3012 or STAT3022 or STAT4022 Assessment: Refer to the unit of study outline https://www.sydney.edu.au/units Mode of delivery: Normal (lecture/lab/tutorial) day

This unit will introduce the fundamental concepts of analysis of data from both observational studies and experimental designs using classical 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, randomisation and ideas of factorial designs. Throughout the course you will use the R statistical package to give analyses and graphical displays. This unit is essentially an Advanced version of STAT3012, with additional emphasis on the mathematical techniques underlying applied linear models together with proofs of distribution theory based on vector space methods.

**STAT3923 Statistical Inference (Advanced)**

Credit points: 6 Teacher/Coordinator: Refer to the unit of study outline https://www.sydney.edu.au/units Session: Semester 2 Classes: Refer to the unit of study outline https://www.sydney.edu.au/units Prerequisites: STAT2X11 and a mark of 65 or greater in (DATA2X02 or STAT2X12) Prohibitions: STAT3913 or STAT3013 or STAT3023 Assessment: Refer to the unit of study outline https://www.sydney.edu.au/units Mode of delivery: Normal (lecture/lab/tutorial) day

In today's data-rich world more and more people from diverse fields are needing to perform statistical analyses and indeed more and more tools for doing so are 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 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 various aspects of distribution theory which are necessary for 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. 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.

**STAT3888 Statistical Machine Learning**

Credit points: 6 Teacher/Coordinator: Refer to the unit of study outline https://www.sydney.edu.au/units Session: Semester 2 Classes: Refer to the unit of study outline https://www.sydney.edu.au/units Prerequisites: STAT2X11 and (DATA2X02 or STAT2X12) Prohibitions: STAT3914 or STAT3014 Assumed knowledge: STAT3012 or STAT3912 or STAT3022 or STAT3922 Assessment: Refer to the unit of study outline https://www.sydney.edu.au/units Mode of delivery: Normal (lecture/lab/tutorial) day

Data Science is an emerging and inherently interdisciplinary field. A key set of skills in this area fall under the umbrella of Statistical Machine Learning methods. This unit presents the opportunity to bring together the concepts and skills you have learnt from a Statistics or Data Science major, and apply them to a joint project with NUTM3888 Metabolic Cybernetics where Statistics and Data Science students will form teams with Nutrition students to solve a real world problem using Statistical Machine Learning methods. The unit will cover a wide breadth of cutting edge supervised and unsupervised learning methods will be covered including principal component analysis, multivariate tests, discrimination analysis, Gaussian graphical models, log-linear models, classification trees, k-nearest neighbours, k-means clustering, hierarchical clustering, and logistic regression. In this unit, you will continue to understand and explore disciplinary knowledge, while also meeting and collaborating through project-based learning; identifying and solving problems, analysing data and communicating your findings to a diverse audience. All such skills are highly valued by employers. This unit will foster the ability to work in an interdisciplinary team, and this is essential for both professional and research pathways in the future.

**STAT3925 Time Series (Advanced)**

Credit points: 6 Teacher/Coordinator: Refer to the unit of study outline https://www.sydney.edu.au/units Session: Semester 1 Classes: Refer to the unit of study outline https://www.sydney.edu.au/units Prerequisites: STAT2X11 and (MATH1X03 or MATH1907 or MATH1X23 or MATH1933) Prohibitions: STAT4025 Assessment: Refer to the unit of study outline https://www.sydney.edu.au/units Mode of delivery: Normal (lecture/lab/tutorial) day

This unit will study basic concepts and methods of time series analysis applicable in many real world problems applicable 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 gain through this unit of study will form a strong foundation to work in a financial industry or in a related research organization.

**STAT3926 Statistical Consulting (Advanced)**

Credit points: 6 Teacher/Coordinator: Refer to the unit of study outline https://www.sydney.edu.au/units Session: Semester 1 Classes: Refer to the unit of study outline https://www.sydney.edu.au/units Prerequisites: At least 12cp from STAT2X11 or STAT2X12 or DATA2X02 or STAT3XXX Prohibitions: STAT4026 Assessment: Refer to the unit of study outline https://www.sydney.edu.au/units Mode of delivery: Normal (lecture/lab/tutorial) day

In our ever-changing world, we are facing a new data-driven era where the capability to efficiently combine and analyse large data collections is essential for informed decision making in business and government, and for scientific research. Statistics and data analytics consulting provide an important framework for many individuals to seek assistant 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.