## Master of Research

### Mathematics Units at 700 level (the first year)

All students must enrol in MRES700 Research Communications.

See MRES700 Research Communications

Mathematics students must enrol in **MATH700 Research Frontiers in Mathematics** and six units units of Advanced Disciplinary content, subject to Academic Approval.

**MATH700 Research Frontiers in Mathematics**

This unit is designed to engage students with current research in Mathematics. It will introduce students to a number of the current open research questions across the range of the broad discipline. It is the first of a pair of such units, with the second appearing in the second year of the MRES programme. This unit addresses research across the breadth of the discipline, while the second unit will focus on more particular issues related to the student's project area. Activities may include such things as seminar attendance, directed reading of research papers, the discussion and critiquing of research topics and introduction to new practical techniques with preparatory reading, hands-on experience and a final report. Presentation of a seminar and a written report based on the topics examined are required for completion of this unit.

The courses offered by the Department of Mathematics in the MRes program differ from year to year, depending upon the availability and interests of members of staff. Please check with the Department for details of current offerings.

**Past Course Offerings**

In past years the following courses have been offered

**MATH701 Analysis**

This unit provides an advanced introduction to the key areas of research interest in modern analysis. We will study Lebesgue integration, positive Borel measures, and the all important function spaces Lp. Then we will study the elementary Hilbert space theory and
Banach space techniques. This will provide familiarity with some of the major theorems which make up the analysis toolbox: Monotone and Dominated Convergence theorems; Fatou's lemma; Egorov's theorem; Lusin's theorem; Radon-Nikodym theorem; Fubini-Tonelli theorems about product measures and integration
on product spaces; Uniform Boundedness; Fundamental Theorem of Calculus for Lebesgue Integrals; Minkowski's Inequality; Holder's Inequality; Jensen's Inequality; and Bessel's Inequality.

**MATH702 Algebra**

This unit provides an advanced introduction to key areas of research interest in modern algebra. It will centre around the theory and applications of modules over a ring. Modules are a common generalisation of the notions of vector space over a field, of abelian
group, of group representation, and of square matrix. We will see how to extend some of the theory of these notions developed in undergraduate years to the setting of modules. An important recurring idea will be that of a structure theorem, such as the undergraduate-level result that every finitely-generated
abelian group is a direct sum of cyclic groups. We shall see various structure theorems for the various algebraic notions studied, with an important example being the Wedderburn theorem for semi-simple rings. Applications to representation theory will be particularly emphasised.

**MATH703 Topology**

This unit provides an advanced introduction to the key areas of research interest in modern topology. Topology is the study of continuity. The definition of a topological space was conceived in order to say what it means for a function between such spaces to be
continuous. There are several ways of defining topological structure and the proofs that these are equivalent abstract many concrete results about specific kinds of spaces. Different ways of expressing continuity are obtained. Sequences are not adequate for general topological spaces, they need to be
replaced by nets or filters, and we discuss convergence of those. Particular properties of topological spaces are analysed in detail: these include separation properties, compactness, connectedness, countability conditions, local properties, metrizability, and so on. Applications to basic calculus are
emphasised. We then introduce algebraic topology by discussing the Poincaré or fundamental group of a space.

**MATH704 Research Topics in Mathematics 1**This unit is study of a current topic of Mathematical research. In addition to mastering the material delivered in lectures, the student will be required to undertake independent reading and write a short report on what they have learned. This
will be written using LATEX, with instruction in this typesetting language and BibTEX as part of the unit's curriculum.

**MATH705 Research Topics in Mathematics 2**

This unit is study of a current topic of Mathematical research. As a preparation for life as a researcher, part of the assessment in this unit will be based on oral presentations by the student. The unit will be based around a mixture of
standard lectures and student seminars.

**MATH706 Research Topics in Mathematics 3**

This unit is study of a current topic of Mathematical research. As a preparation for life as a researcher, part of the assessment in this unit will be based on oral presentations by the student. The unit will be based around a mixture of
standard lectures and student seminars.

**MATH707 Advanced Methods in Mathematics 1**A selection of topics in Mathematics which would provide tools for Mathematicians and researchers in other disciplines. Topics would be drawn from a variety of mathematical areas, and would be tailored to the current student cohort.

**MATH708 Advanced Methods in Mathematics 2**A selection of topics in Mathematics which would provide tools for Mathematicians and researchers in other disciplines. Topics would be drawn from a variety of mathematical areas, and would be tailored to the current student cohort.

### Outline of 800 level (the second year)

**Activity 1: Research Frontiers 2**

: In MATH700 the student began involvement in the process of reviewing and reporting on their seminar attendance and reading. These activities continue, but are specifically focused on relating the area of the student's specialization with current Mathematical research. The student's supervision panel will prepare and oversee the student's engagement in a program consisting of attendance at a selection of 4 or 5 research seminars related to the students area of interest or attendance at a relevant conference in or near Sydney or devise a series of appropriate directed readings of equivalent length. This will build on the directed reading the student has done in MATH700. A series of student seminars to be held in the first semester to provide a venue for each student to report to their fellow Mathematics MRES students on a variety of topics, including the relationship of their proposed project area to current Mathematical research and what they have learned at the seminars and conference lectures they have attended.*Activity*: A substantial written report pertaining to an area of current research in Mathematics. The report should be of a length equivalent to a 3-4,000 word essay. The content should be distinct from the student's literature review. The report is to be graded by a member of the student's supervision panel and will count for 10% of the overall SNG.**Output**: Report due at the end of June.**Key date**

**Activity 2: Literature Review**

: Students will attend introductory classes conducted by members of the Mathematics Department explaining the review of scholarly literature in their disciplinary context and its relationship to individual research project (2-4 sessions). The Department of Mathematics will provide instruction in using MathSciNet and the Zentralblatt MATH Database. This will be integrated into their on-going use of LATEX, which they already encountered in MATH700. Under the guidance of his or her supervisor, each student will then do a significant preliminary survey of the literature relevant to his or her proposed research area.**Activity**: Students will be able to demonstrate knowledge of their research topic by situating their individual research project within a broader field or paradigm, including a structured review of associated issues, debates and methodology. Students will identify gaps in the literature, which their research project seeks to redress.**Outcome**: A draft Literature Review or survey as appropriate to disciplinary convention, submitted to the student's supervision panel for comment, by the end of July. The Literature Review should be one quarter to one third of the length of the final thesis, that is, approximately 5-6,000 words. Final version to be incorporated and assessed as part of thesis (Activity 5).**Output**: Draft literature review to go to the supervision panel by the end of July.**Key date**

**Activity 3: Research Methods**

: Discipline-based Research Methods Workshop (8 to 12 sessions), built primarily around student presentations explaining and justifying their choice of methodology in comparison to alternative methods. Members of the Department of Mathematics will oversee these presentations.**Activity**

**Activity 4: Research Planning**

: Students will attend two or more sessions in project management. (Please note: this component of the program will be centrally taught). The Mathematics Department will run two or more sessions on research planning relevant to the students; projects. Students will then work with their Supervision Panel or Individual Supervisor to produce a detailed research plan, to be submitted in written form according to the following recommended schedule:**Activity**- Initial idea late year 1 (in order to identify possible supervision);

- Meeting with supervision team early year 2 (Jan/Feb) to discuss research plan;

- Draft Major (four year) Research Project concept by end of March. This project should be of the size and scope of a PhD. The MRES project will be a pilot of this larger PhD-scale Major Research Project. Final MRES plan by end of April.

- Final Major Project/PhD plan by November.

: Students will develop a major research plan, for a project of up to four years scope. A pilot project will be completed during the MRES year, as a preliminary to a new but related three-year research project undertaken for PhD.**Outcome**: Submitted Written Plan (assessable as Satisfactory/Unsatisfactory). The final Major Project/PhD Plan should include: research question, methodology, budget, ethics (if appropriate), timeline, and chapter breakdown. Presentation to the Mathematics Department of the MRES plan at the end of April (assessable as Satisfactory/Unsatisfactory.)**Output**

**Activity 5: Thesis**

: Each student will complete a significant individual research project of their own design, as outlined in Activity 4.**Activity**: Each student will complete a thesis equivalent to 15,000-20,000 words, subject to disciplinary standards (worth 90% of second year assessment). The thesis will be due October 10. The thesis will be externally examined.**Output**

**Supervision**

- Each student will be integrated into the research team consisting of academic staff, honorary associates, post-docs, PhD students, within the Mathematics Department whose research field matches the student's specialization. This research team will have day-to-day collective responsibility for mentoring and supporting the development of the student. Each group of students will have a supervision panel formed from with this team that will meet regularly with each student and the student's principal supervisor(s) to monitor the student's progress and to ensure that the student is interacting effectively with the team. The supervision panels will give regular feedback to the departmental MRES coordinator so that they can monitor progress and performance.

### Important dates for the second year

**Late in Year 1**

Initial idea for research project.

**January/February**

Meet with supervision panel to discuss research plan.

**First semester**

Student seminars for research frontiers

**End of March**

Draft major (four year) research project concept, size and scope of PhD.

**End of April**

Presentation to Dept of MRES plan

**Both semesters**

8–12 sessions of research methods workshops.

**End of June**

Report for research frontiers A substantial written report pertaining to an area of current research in Mathematics. The report should be of a length equivalent to a 3-4,000 word essay. This will be due at the end of June. The content should be distinct from the student's
literature review. The report is to be graded by a member of the student's supervision panel and will count for 10% of the overall SNG.

**End of July**

Draft literature review. A draft Literature Review or survey as appropriate to disciplinary convention, submitted to the student's supervision panel for comment, by the end of July. The Literature Review should be one quarter to one third of the length of the final
thesis, that is, approximately 5-6,000 words. Final version to be incorporated and assessed as part of thesis.

**October 10**

Each student will complete a thesis equivalent to 15K-20K words, subject to disciplinary standards (worth 90% of second year assessment).

**End of November**

Final major PhD research plan.