School of Mathematical Sciences
Honours in
Pure Mathematics
Applied Mathematics
Statistics
(Fourth Year) 2016
Revised again Wednesday 6 April, 2016 with updated synopsis for M43071
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1. HONOURS CONTACTS ........................................................................................................................ 3
2. ENTRY REQUIREMENTS AND ENROLMENT .................................................................................. 4
3. HONOURS UNIT REQUIREMENTS .................................................................................................... 5
4. LIST OF HONOURS LECTURE TOPICS.............................................................................................. 5
4.0
4.1
4.2
4.3
5.
WRITING AND PRESENTING FOR HONOURS ......................................................................... 7
PURE MATHEMATICS LECTURE TOPICS ................................................................................ 7
STATISTICS LECTURE TOPICS .................................................................................................. 8
APPLIED MATHEMATICS LECTURE TOPICS .......................................................................... 9
INTERMEDIATE REPORTS AND PROJECTS .............................................................................. 10
MARKING GUIDELINES FOR HONOURS PROJECTS ............................................................... 12
6.
DETAILS OF TOPICS....................................................................................................................... 14
6.0 WRITING AND PRESENTING FOR HONOURS ....................................................................... 14
M40110 WRITING AND PRESENTING FOR HONOURS ............................................................ 14
6.1 PURE MATHEMATICS ................................................................................................................. 15
M40011 MEASURE THEORY ......................................................................................................... 15
M41041 DIFFERENTIAL GEOMETRY .......................................................................................... 17
M41051 TOPOLOGY ........................................................................................................................ 18
M41072 TOPICS IN GRAPH THEORY ........................................................................................... 19
M41061 ENUMERATIVE AND PROBABILISTIC COMBINATORICS ....................................... 19
M41022 PARTIAL DIFFERENTIAL EQUATIONS ........................................................................ 20
M41082 ADVANCED TOPICS IN ANALYSIS ............................................................................... 21
6.2
STATISTICS .............................................................................................................................. 22
M42011 STOCHASTIC CALCULUS AND MATHEMATICAL FINANCE .................................. 22
M42021 STOCHASTIC PROCESSES I ............................................................................................ 23
M42032 STOCHASTIC PROCESSES II .......................................................................................... 23
M42052 MARKOV CHAINS AND MIXING TIMES ...................................................................... 24
6.3
APPLIED MATHEMATICS ..................................................................................................... 26
M43011 ADVANCED COMPUTATIONAL MATHEMATICS ...................................................... 26
M43031 MATHEMATICS IN BIOLOGY AND MEDICINE .......................................................... 27
M43051 GENERAL RELATIVITY .................................................................................................. 28
M43021 ADVANCED METHODS FOR APPLIED MATHEMATICS ........................................... 29
M43042 ADVANCED TOPICS IN FLUID DYNAMICS ................................................................ 30
M43062 OPTIMISATION ................................................................................................................. 30
M43071 COMPUTATIONAL VISUALISATION AND NUMERICAL HYDRODYNAMICS ..... 31
7. DETAILS OF EXTERNAL COURSES ............................................................................................... 32
8. HONOURS YEAR PRIZES.................................................................................................................. 33
LEO GLEESON PRIZE ..................................................................................................................... 33
CARL MOPPERT PRIZE IN MATHEMATICS ............................................................................... 33
PURE MATHEMATICS PRIZE........................................................................................................ 33
STATISTICS PRIZE .......................................................................................................................... 33
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1. Honours contacts
Honours Coordinator
Dr Jerome Droniou
Room 415, Maths Building, 9 Rainforest Walk
P: 9905 4489
E: jerome.droniou@monash.edu
Deputy Honours Coordinator
Dr Alina Donea
Room 319, Maths Building, 9 Rainforest Walk
P: 9905 4488 E: alina.donea@monash.edu
Mathematics Advisor (honours)
A/Prof. Burkard Polster
Room 411, Maths Building, 9 Rainforest Walk
P: 9905 4493
E: burkard.polster@monash.edu
Applied Mathematics Advisor (honours)
Dr Simon Clarke
Room 327, Maths Building, 9 Rainforest Walk
P: 9905 4421
E: simon.clarke@monash.edu
Statistics Advisor (honours)
Dr Andrea Collevecchio
Room 456 Maths Building, 9 Rainforest Walk
P: 9905 4404
E: andrea.collevecchio@monash.edu
Administrative Officer (honours)
Ms Linda Mayer
Room 406, Maths Building, 9 Rainforest Walk
P: 9905 4434
E: linda.mayer@monash.edu
School Manager
Mrs Gertrude Nayak
Room 422, Maths Building, 9 Rainforest Walk
P: 9905 4437
E: gertrude.nayak@monash.edu
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2. Entry Requirements and Enrolment
The normal minimum requirement for honours (fourth year) in mathematics is:
i. an average of at least 70 in 24 points of relevant third year units, or equivalent*
ii. a willing supervisor for your honours project before the end of the first week of
December of the year before commencing your honours degree.
*A list of relevant units that can be counted in this average is available from the School’s
Honours Coordinator.
How to apply for Honours 2016
1.
Complete the hardcopy application form available from mid-September 2015 from
http://www.monash.edu/science/current/honours/how-to-apply2
2.
IMPORTANT: AS EARLY AS POSSIBLE go to the Science Student Services Office to obtain
your ELIGIBILITY CHECK on page 3 of your Honours application.
3.
AS EARLY AS POSSIBLE go to the School's Honours Cooordinator, Dr Jerome Droniou to
consult with him about your Honours application and obtain his approval and signature on
Section D of your Honours Application. The School will retain a copy of your Honours
Application.
4.
Return your completed Honours Application form to the Science Student Services Office by the
Deadline of Friday 13 November 2015.
5. Successful applicants will be notified by the Faculty of Science during mid December 2015. You
will receive an email from the School advising you to contact the Honours Coordinator, Dr
Jerome Droniou and the relevant Honours discipline coordinators to discuss your choice of
lecturing topics and honours project*. Once both have approved your selections, you need to
complete the School's Internal Lecture Topic / Project enrolment form (which will be provided
to you by the school via an email) and submit accordingly as instructed in this email
correspondence
*IMPORTANT: the Administrative Officer (honours) must be put in CC of any email
sent to the discipline advisor or the honours coordinator.
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3. Honours Unit Requirements
Each honours program in the areas of:
Pure Mathematics, Applied Mathematics and Statistics honours:
and whether full-time or part-time
consists of two enrolled units through the Faculty of Science, these are:
MTH4200 [FY] /MTH4100 [FY]
•
MTH4200 Mathematics honours coursework unit - [FY] (24 credit points):
which consists of six lecture topics (4 credit points each) and for Statistics it
may be 4 x 6 credit point lecture topics or 2 x 6 credit point lecture topics
plus 3 x 4 credit point lecture topics.
• MTH4100 Mathematics honours project unit - [FY] (24 credit points):
which
consists of one lecture topic (4 credit points), plus EITHER a research project
Option A (20 credit points) OR a research project Option B (12 credit points),
plus 2 modules (each worth 4 credit points).
4. List of Honours lecture Topics
The lecture topics fall into three main groups – Pure Mathematics, Applied Mathematics
and Statistics.
You can select lecture topics from any group, as long as the requirements of your enrolled
honours units are met (refer to previous Section 2 Honours unit requirements). It is your
own responsibility to ensure that a sufficient number of lecture topics are taken. Honours
topics from other Schools or outside Monash via AMSI Advanced Collaborative
Environment (ACE) may also be available in other areas, and you can take these with the
permission of the honours coordinator. Up to two third-year mathematics lecture units
may be also counted towards honours in Pure Mathematics, Statistics and Applied
Mathematics, with the permission of the honours coordinator.
In addition, ALL Honours students are eligible to attend the AMSI Summer School
organised through the Australian Mathematical Sciences Institute (AMSI). Upon approval
of the honours coordinator, each student can have one of the honours courses offered
there count towards their honours degree. An AMSI course will count as equivalent to one
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honours lecture topic being offered by the School of Mathematical Sciences. The AMSI
summer school is also open to prospective honours students.
Prospective honours students or students still enrolled in honours in semester 1 following
the Summer School are eligible to receive support for travel and accommodation. The link
to the AMSI summer school is given in Section 7.
Students enrolled in Statistics can also select topics given in the Key Centre for Statistical
Sciences (KCSS), still with approval of the honours coordinator and statistics honours
advisor.
IMPORTANT: if more than the total required number of points (using 4 points for third-year lecture
units) are taken for the coursework unit (MTH4200), then the best weighted 24 points will be used to
determine the final mark for that overall unit.
SELECTION OF HONOURS TOPICS:
Before the start of the teaching semester, you will be asked to select your honours topics.
This selection must be approved by your supervisor, honours discipline advisor and
honours coordinator.
Once you have selected, with your supervisor's approval, your topic choices, send an email
to the Honours Coordinator and the relevant Honours Discipline Advisor (and CC to
your Supervisor and the Administrative officer (honours)) with the list of intended
honours topics - including 3rd level mathematics units, AGR honours courses, AMSI
summer school topics and KCSS topics.
Not all of the lecture topics listed below will necessarily be offered next year; for example,
topics that attract an insufficient number of students may be cancelled. It is therefore
important that you inform the School of your topics choice as soon as you can. A topic not
offered as a lecture course may be, with approval of the honours coordinator, used as the
basis of a reading course.
Your choice of lecture topics appears only in the School records; you are allowed to change
it up to the census date. Make sure to seek approval for any change, and to inform the
Administrative Officer (honours) of the change.
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4.0 Writing and Presenting for Honours
Semester One
M40110 Writing and Presenting for Honours
Please note, this unit is compulsory for honours students in all 3 disciplines of the
School!
4.1 Pure Mathematics lecture topics
Semester One
M40011 Measure Theory
M41041 Differential Geometry
M41051 Topology
M41072 Topics in Graph Theory
Semester Two
M41061 Combinatorial Enumeration
M41022 Partial Differential Equations
M41082 Advanced Topics in Analysis
As a student in the Mathematics Honours program you may also choose from the topics
on offer through the AMSI Summer School to count as credit towards your honours
degree. Either one 4 week course or a two 2 week course can count as 4pts (see Section 7).
You may also take external honours units via AMSI Advanced Collaborative Environment
(ACE) to count as 4pt credit towards your honours degree. Teaching is accessed remotely
via a specifically booked room and a school laptop. (see Section 7).
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4.2 Statistics lecture topics
Semester One
M42011 Stochastic Calculus and Mathematical Finance - 6 pt
M42021 Stochastic Processes I - 4 pt
Semester Two
M42032 Stochastic Processes II - 6 pt
M42052 Markov Chains and Mixing Times
As a student in the Statistics Honours program you may also choose from the topics on
offer through the AMSI Summer School to count as credit towards your honours degree.
Either one 4 week course or a two 2 week course can count as 4pts (see Section 7).
In addition, it may also be possible to take Courses from the Key Centre for Statistical Sciences
(KCSS) to counts as 4 pt credit towards your honours degree. Students should contact the
Statistics Honours Advisor (see Section 1).
Finally, you may also take external honours units via AMSI Advanced Collaborative
Environment (ACE) to count as 4 pt credit towards your honours degree. Teaching is
accessed remotely via a booked room in the school and a laptop (see Section 7).
Please note, further to this 2 additional OFF CAMPUS units may possibly also be taken.
Students should contact the Statistics Honours Advisor to see if these additional OFF
CAMPUS units are on offer in the current year (see Section 1).
IMPORTANT. If combinations of 4 pt and 6 pt lecture topics are taken for MTH4200 in
the Statistics Honours program then the best weighted mark for 24 points of workload
will be used to determine the final mark for that unit.
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4.3 Applied Mathematics lecture topics
Applied Mathematics Core topics:
Semester One
M40110 Writing and Presenting for Mathematics
M43011 Advanced Computational Mathematics
M43021 Advanced Methods for Applied Mathematics
Elective topics:
Students in the Applied Mathematics honours program must also complete the equivalent
of at least four approved elective lecture topics, including at least two in an approved area
of “applied mathematics” directly below (as approved by the Applied Mathematics
Honours advisor).
Applied Mathematics Elective Topics
Semester One
M43031 Mathematics in Biology and Medicine
M43051 General Relativity
Semester Two
M43042 Advanced Topics in Fluid Dynamics
M43062 Optimisation
M43071 Computational Visualisation and Numerical Hydrodynamics
The electives may also include honours lecture topics offered for the Mathematics and
Statistics honours programs, as listed in this handbook, provided the appropriate prerequisites have been completed.
Up to two approved ASP, ATM or MTH units at level-3 may also be taken but will be
counted as 4 pts at honours level, for example MTH3020, MTH3051, MTH3140,
MTH3360, ASP3051 and ATM3050 if they have not already been taken.
The electives may also include approved honours lecture topics that are: given at the AMSI
Summer School; or formal coursework topics given by other schools in the faculty (with
their approval); or are offered by other universities via AMSI Advanced Collaborative
Environment (ACE). In those cases the point weighting will be determined by the Applied
Mathematics honours advisor, usually based upon that used by the relevant school or
university.
Applied Mathematics honours program requirements:
Students in the Applied Mathematics honours program must complete the equivalent of 48
points of approved coursework and research, including:
•
The core topics M40110, M43011 and M43022 (12 pts total)
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5.
•
At least 16 pts of other lecture topics (including at least two in an approved
area of “applied mathematics” and no more than two third-year lecture units,
each weighted as 4 points) as approved by the Applied Mathematics honours
advisor. If more than 16 pts are completed then the best weighted 16 pts will be
used to determine the final mark for this component.
•
An intermediate report (4 pts) in their first semester and a full-year honours
research project (worth 16 pts). In addition to the written component, both the
intermediate report and the honours project include the completion of an
assessed research outline (to be completed by week 4) and assessed oral
presentation(s).
Intermediate Reports and Projects
As outlined in Section 3, the requirements for completing the honours year in the School
of Mathematical Sciences include a research project (project unit) and various lecture topics
(coursework unit).
The research project is offered in two forms, each worth 20 credit points. It is important to
note that these two formats are equally weighted; neither is considered better/harder/more
prestigious than the other in any way.
Option A:
This type of project must contain a research objective that has not already been developed
in the literature. This does not mean that the research project needs to result in publishable
material. It can consist of a minor variation of a known result that has not been published
before, for example, a generalization/extension of a result that an expert could establish
easily on their own, but instead leaves as an exercise for the readers of a paper to complete.
The originality of the thesis can also be in terms of new insights into a possibly wellestablished area, rather than a genuinely novel result.
This objective will be clearly stipulated in the project handbook, and discussed between the
supervisor and the student during the initial design of the project; it must also be validated
by the honours coordinator. The final mark will be based on the degree to which the
student is able to reach the objective by the end of the project, or is able to detail scientific
reasons for why the planned objective was not achieved.
Option B:
This option has two components: a reading project, and research skill modules. The
reading project is worth 12 credit points, and the student must select 2 modules (each
worth 4 credit points) in consultation with the supervisor, among a list of offered modules;
see below. The reading project consists in reading, understanding and re-writing classical
results from the literature (book chapter(s) or article(s)). The modules are designed to teach
skills that are essential in a research environment.
For each of these forms, the “project” part contains an intermediate report component,
worth 4 credit points, which must be completed in the first semester. Its recommended
length is 15-25 pages. It comprises a survey of the literature, as well as a discussion of any
mathematical techniques necessary to complete the project, but which are unfamiliar to the
student. At the end of the semester the student will be required to give a 25 minute talk on
their intermediate report and goals for the rest of the project.
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The final project is due at the end of the final semester and includes a 25 minute talk. The
recommended length of the written project is 40-50 pages. Examples of well presented
projects and further guidelines can be obtained from the honours advisors (see Section 1)
or your supervisor.
The breakdown of the assessment for the project unit (MTH4100, 24 credit points) is as
follows:
- One lecture topic (worth 4 credit points): 16.7%
- Research project (worth 20 credit points): 83.3%, made of:
* For Option A: intermediate report 16.7%, final report 83.3%.
* For Option B intermediate report 16.7%, final report 33.3%, each module
16.7%
The marks for both the intermediate report component and the final report is an average
of the mark for the oral presentation (weighted 20%) and of the written document
(weighted 80%). The oral presentations are assessed by all staff in the School of
Mathematics who attend the talks; the intermediate report written document is assessed by
your supervisor; the final report is assessed by your supervisor and two other markers.
Please note that in regards to the submission of the intermediate report and the project, the
School of Mathematical Sciences adopts the Faculty of Science policy on late submission of
assessment in honours as found in Chapter 3.5 of the “Faculty of Science Honours rogram
Policy, Procedures and Guidelines for Good Practice” viewable at:
http://monash.edu/science/current/honours/docs/honours-policy.pdf.
This means a penalty of 5% per day late will be applied for the intermediate report and
honours project unless an extension is granted through the special consideration process. If
you feel you cannot submit your work by the specified submission date, due to
circumstances outside of your control, you must immediately contact your project
supervisor and the honours coordinator to explain and seek a formal extension.
The time you spend on the intermediate report and the project should be similar to the
time spent on the equivalent (credit points-wise) number of lecture topics. It is therefore
clear that the intermediate report and project should be commenced at the beginning of
your first semester. You should choose a project topic and find a supervisor well before the
end of the first week of December of the year before you commence. It is also advisable to
meet your supervisor regularly, say once a week, as soon as the teaching semester begins. A
guide to the good practice and the responsibilities of supervisors and students in
undertaking honours research can be found in the afore-mentioned “Faculty of Science
Honours Program Policy, Procedures and Guidelines for Good Practice”
The School offers a comprehensive range of project topics in advance. However, we will
also attempt to satisfy your own interests, provided a supervisor with the appropriate
expertise is available. Further information may be obtained when you enrol, but it is not
necessary to wait until then. You are welcome to consult the honours coordinator or
honours advisors. If you choose to develop a proposal or undertake one of the prepared
proposals, a final proposal must be submitted by Week 4 of your first semester.
Research skills modules for Option B of the research project
All modules are assessed by a common panel of 2-4 staff (not necessarily comprising the
supervisor).
* Seminar (4pt, 1st Semester, mandatory in 2016):
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Students work in groups to read and understand a few research articles. Each student needs
to present one part of a paper, but all students assigned to one paper need to collaborate to
understand the whole paper. Students regularly present the elements they understood in the
paper, and those they still have to work on. Collaboration also happens during these
presentations, as all other students can then discuss and give ideas. The main element in
this module is therefore collaborative work.
* Extension to seminar (4pt, 2nd Semester):
The purpose of this module is to extend results covered in the seminar, either by
investigating generalisations, counter-examples or computational illustrations. This does
not have to result in novel mathematical results, it just must be new to students.
* Computer assignment (4pt, 1st or 2nd Semester):
This module aims at developing computer skills which are essentials to today's research in
mathematics. Students will have to write a program to illustrate some mathematical result.
This result will probably be related to their project's field, but the computational work must
not be directly part of the project itself.
* Research proposal (4pt, 2nd Semester):
Applying for a research grant is part of working in a research environment, but draws on
skills which are generally useful in many work environments: how to present in a
convincing way technical and specialised proposal to non-specialists? In this module,
students need to write a mock grant application on a research proposal. It must be deep
enough to convince specialists, but also appealing and clear to non-specialists. A budget
must be included.
MARKING GUIDELINES FOR HONOURS PROJECTS
H1 (80 - 100)
An ‘upper H1’ (90 - 100) student has strengths in all of the following areas:
• outstanding command of expression and logical argument in a skillfully structured
manuscript;
• superior evaluation and integration of existing literature;
• evidence of significant insight and original thought in dealing with the critical issues;
• sophisticated understanding of research methods;
• thoughtful and appropriate choice of data analysis (where appropriate) and outstanding
presentation and reporting of results;
• clear and coherent interpretation of the thesis results, and/or the results of other studies;
• comprehensive understanding of the importance of the results in the context of research
topic.
A ‘lower H1’ (80 -90) student displays many of the above strengths but is less well-balanced
in overall quality.
H2A (70 - 79)
The project is characterised by most of the following:
• the manuscript is well written, logically argued and generally well structured;
• the evaluation and integration of the existing literature is very sound without being
outstanding;
• reasonable insight and some evidence of original thought in dealing with the critical
issues
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• evidence of a solid understanding of research methods;
• choice of data analysis (where appropriate) that is appropriate (although less well justified
than might be expected of H1 standard), and clear presentation of results;
• generally sound but pedestrian interpretation of results and their importance to the
research topic.
H2B (60 - 69)
The project is characterised by most of the following:
• generally competently written, although some problems exist in the logical organisation of
the text and the way it is expressed;
• provides an adequate coverage of the literature, although it tends to be more descriptive
than evaluative, and arguments are often disjointed;
• occasional evidence of insight into the issues underlying the thesis or essay, but little
evidence of original thinking;
• basic but somewhat limited understanding of the research methods;
• serviceable choice of data analysis (where appropriate), although other approaches may
have been more appropriate;
• the presentation of results lacks clarity and the interpretation of results or other studies is
adequate but limited.
H3 (50 - 59)
The project is characterized by most of the following:
• the work is not well written and shows flaws in the structuring of logical arguments;
• coverage of the necessary literature is weak, with insufficient information provided to
support the arguments made, or conclusions drawn, within the thesis or essay;
• little evidence of insight and ideas tend to be highly derivative;
• knowledge of research methods is deficient;
• data analysis techniques (where appropriate) are arbitrary or inappropriate;
• the results are poorly presented;
• interpretations are superficial, demonstrating a weak understanding of the results and
their relevance to the research topic.
Fail (0 - 50)
The project is characterized by most of the following:
• the work is very poorly written and shows a serious inability to structure and present a
logical argument;
• coverage of the necessary literature is inadequate, with little information provided
relevant to the claims made, or conclusions drawn, within the thesis;
• serious misunderstanding of key concepts and issues;
• knowledge of research methods is lacking;
• data analysis techniques (where appropriate) are inappropriate and the results are
presented inadequately;
• an inability to show how the results of the research project relate to the research topic;
serious misinterpretations of results.
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6.
Details of topics
6.0 Writing and Presenting for Honours
M40110 WRITING AND PRESENTING FOR HONOURS
Offered Semester: One
Level: 4 pt Honours Topic
Lecturer(s): Leo Brewin and Simon Clarke
Contact Details: Contact Details: Room 317, 3rd floor, Maths Building, Clayton
campus, (03) 9905 4456, email Leo.Brewin@monash.edu and Room 327, 3rd floor,
Maths building, Clayton campus, (03) 9905 4421, email Simon.Clarke@monash.edu
M40110 is a compulsory topic for ALL honours students in Mathematical Sciences.
There will be an introductory lecture, but most of the work will be a study guide that needs
to be read through, along with practical reports and presentations. It replaces the previous
compulsory unit of Introduction to Computing for Honours.
Objectives: Proposed topic is to give honours students guidance and feedback on
presenting and writing mathematics, prior to their Intermediate report and presentation at
the end of the first semester of study.
Aim: Is also to teach students LATEX, BibTEX and Beamer, however guidance on
writing and presentation style will be hopefully be technology independent.
Will use or adapt standard introductions on LATEX, BibTEX and Beamer.
Follow something like Fowler's Modern English Usage as guide.
Syllabus:
Technical Aspects:
• LATEX and BIBTEX software.
• Structure of BibTeX, LaTeX and Beamer documents.
• Managing and citing references using BibTeX.
• Writing mathematics in LaTeX.
Writing:
• Structure of documents, e.g. reports, papers, theses, presentations.
• What to include and referencing.
• Writing Introductions, Conclusions and Abstracts (repeat, repeat, repeat).
• Equations: formatting, e.g. spacing, inline vs display equations, multiple line
equations and subequations, when to include equation numbers and referencing.
• Tables and figures: guidelines, e.g. should always be discussed in text and captions.
• Table of Contents.
• Appendices.
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Presenting Mathematics:
• Audience awareness.
• Timing.
• Content: how much and what to include.
• Know your audience.
• Handling questions.
Mode of Delivery:
• Self-paced Study Guide with good and bad initial presentations and examples.
Assessment:
• Practical presentations, e.g. 10MT (4 slides of less) with feedback.
• 10 page report on classical paper with feedback.
• Pass/Fail (will be assessed in mid-semester and final reports and presentations).
References:
Fowler's Modern English Usage.
Higham, N.J., Handbook of Writing for the Mathematical Sciences, SIAM,
1998.
Krantz, S.G., A Primer of Mathematical Writing, AMS, 1997.
LATEX tutorial material, http://www.ctan.org/topic/tut-latex
6.1 Pure Mathematics
M40011 MEASURE THEORY
Offered Semester: One
Level: 4 pt Honours Lecture Topic
Lecturer(s): Andrea Collevecchio and Jerome Droniou
Contact Details: Room 456, 4th floor, Maths Building, Clayton campus, (03) 9905
4404, email Andrea.Collevecchio@monash.edu and Room 415 4th floor, Maths
building, Clayton campus, (03) 9905 4489, email Jerome.Droniou@monash.edu
Backround: Measure theory is one of the few theories which permeates all core
mathematical domains (pure, applied and statistics). It appears for example in pure
analysis, for the construction of Fourier transforms on generic groups, in PDE analysis, to
deal with data/solutions jumps or source terms with a lower dimensionality, in group
theory and geometry, where the Haar measure helps characterizing Lie groups, in number
theory, where classical series can be dealt with as integration against a discrete measure, in
dynamical systems, to analyse their ergodicity, in numerical analysis, to provide a link
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between the continuous (e.g. PDE) theory and the discrete theory, in probability and
statistics, whose theory is entirely founded on measures.
Syllabus: We propose to develop Lebesgue integration and probability theory from the
core elements of measure theory. The initial background will be kept to a minimum. In
particular, it is only required knowledge of real analysis and elementary probability theory.
On the other hand, the topics covered in this course will be fundamental for the
understanding of advanced courses, as described above. The hardest proofs (e.g.
construction of the Lebesgue measure) will be omitted, to concentrate on applications and
illustrations directly useful to students for their project and other topics.
Material susceptible to be covered includes:
• Measurable sets and measurable functions: basic definitions, illustrations, pointwise
limits of measurable functions.
• Measure theory: definition of a sigma-algebra, of measures. Exemples: Lebesgue
measure (construction omitted), discrete measure, probabilities.
• Lebesgue integration: definition of the integral, convergence theorems, application
to continuity and differentiation under the integral sign.
• Lebesgue spaces (L^p): completeness of the spaces, Riesz representation theorem.
• Product measures (Fubini theorem).
• Probability: Borel-Cantelli lemmata, conditional expectation and Radon-Nykodim
theorem, Martingales.
In particular martingale convergence theorem, L^p-bounded martingales, uniform
integrability.
Examples of possible applications and illustrations used throughout the unit:
• Convergence, continuity and differentiability theorems applied to series: how
to trivialise the (difficult) theorems from Real Analysis.
• Representing source terms in DE/PDEs with lower dimensionality using
measures.
• Weak derivatives and principles for constructing Sobolev spaces to deal with
DE/PDEs with irregular data (no PDE theory here, of course!)
• Probability: Kolmogorov zero-one law, applications of martingales.
Reference books
- A.Torchinsky "Real Variables"
- H.L. Royden "Real Analysis". Third edition.
- D. Williams "Probability theory with martingales".
- W. Rudin "Real and complex analysis".
- M. Capinski & E. Kopp "Measure, Integral and Probability"
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M41041 DIFFERENTIAL GEOMETRY
Offered Semester: One
Level: 4pt Honours Lecture Topic
Lecturer(s): Todd Oliynyk
Contact Details: Contact Details: Room 419, 4th floor, Maths Building, Clayton
campus, (03) 9905 4433, email Todd.Oliynyk@monash.edu
Aims: Manifolds are “topological spaces with local coordinates” (e.g. surfaces) i.e. spaces
locally homeomorphic to the euclidean m-space. A differentiable structure on such a
manifold makes it possible to generalize to these “curved spaces” many basic concepts of
calculus, connected with differentiation or integration of functions and maps defined on
the “flat” euclidean m-space.
This is an introductory course on differentiable manifolds and related basic concepts,
which are the common ground for differential geometry, differential topology, global
analysis, (i.e calculus on manifolds including geometric theory of integration) and modern
mathematical physics.
Syllabus: Smooth manifolds; coordinate systems. Tangent vectors; tangent and cotangent
bundles of a manifold, tensor fields; connections and covariant differentiation. Metrics.
Geodesics. Curvature tensor. Relation between geometry and topology of manifolds.
Applications to general relativity.
Prerequisites: MTH2010 Multivariable Calculus and MTH2021 Linear Algebra with
Applications. M41051 Topology is recommended as a co-requisite.
Recommended reading:
Informal lecture notes may be provided.
Do Carmo, M., “Introduction to Riemannian geometry”, Birkhäuser, 1992.
References:
Spivak, M.A., “A comprehensive introduction to differential geometry”, Vol. I, Publish or
Perish Inc., 1979 (1970).
Spivak, M., “Calculus on manifolds”, W.I. Benjamin, Inc., 1965.
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M41051 TOPOLOGY
Offered Semester: One
Level: 4 pt Honours Lecture Topic
Lecturer(s): Jessica Purcell
Contact Details: Contact Details: Room 416, 4th floor, Maths Building, Clayton
campus, (03) 9905 4490, email Jessica.Purcell@monash.edu
Aims: Topology is the geometry of continuity, and as such it is involved in almost all parts
of mathematics. The aim of this unit is to explain the basic concepts of topology, with the
help of examples from geometry, analysis and algebra, and to see how topology interacts
with geometric, algebraic and analytic properties.
An example from classical geometry is the Euler polyhedron formula, which shows that the
quantity (number of vertices) − (number of edges) + (number of faces) is independent of
the way a surface is divided into polygonal regions. A related example in differential
geometry is the Gauss-Bonnet theorem, which shows that the total curvature of a surface
depends only on the number of "holes'' in it.
Syllabus: Open, closed and compact sets. Continuous mappings and homeomorphisms.
Curves, surfaces and other manifolds, in particular spheres and projective spaces. From
the Euler polyhedron formula to homology. The classification of surfaces: genus and
orientability. Fundamental group, quotient spaces and coverings. Genus and the GaussBonnet theorem.
Prerequisites: MTH3160 Analysis and Topology, MTH2140 / MTH3140 Real Analysis
References:
Armstrong, M. A., “Basic topology”, McGraw-Hill, 1979.
Stillwell, J., “Classical topology and combinatorial group theory”, Springer Verlag, 1980.
Thurston, W.P., “Three-dimensional geometry and topology”, vol. 1, Princeton University
Press, 1997.
Weeks, J., “The shape of space”, 2nd edition, Marcel Decker, 2002.
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M41072 TOPICS IN GRAPH THEORY
Offered Semester: One
Level: 4 pt Honours Lecture Topic
Lecturer(s): David Wood
Contact Details: Room 424, 4th floor, Maths Building, Clayton campus, (03) 9905
4417, email David.Wood@monash.edu
Aims: Networks are a pervasive element of modern life. Social networks, computer
networks, and the worldwide web are common examples. Graph theory is the
mathematics of networks. It is a modern discipline with numerous applications within and
beyond mathematics, especially in computer science. This subject is an introduction to
graph theory, with selected key topics studied in depth. Students will gain experience in
creative thinking, problem solving, algorithm execution and analysis, and writing proofs.
This subject is suitable for all mathematically capable students who have studied some third
year mathematics.
Syllabus: graph basics, trees, connectivity, matchings, planar graphs, graph colouring,
graph minors, probabilistic method.
References:
No specific textbook but "Graph Theory" by R. Diestel is appropriate reading.
M41061 ENUMERATIVE AND PROBABILISTIC COMBINATORICS
Offered Semester: Two
Level: 4 pt Honours Lecture Topic
Lecturer(s): Ian Wanless and Jane Gao
Contact Details: Room 448, 4th floor, Maths Building, Clayton campus, (03) 9905
4442, email Ian.Wanless@monash.edu and Room 457, 4th floor, Maths building,
Clayton campus, (03) 9905 4413, email Jane.Gao@monash.edu
Aim: Combinatorial enumeration is a fancy way of saying counting stuff. You may think
that you mastered counting in kindergarten and then moved on. However counting can be
very challenging. What do you do when you run out of fingers and toes? Some objects
require advanced techniques to count them. In the first half of this course we will look at
counting combinatorial objects such as permutations, partitions and compositions, and
graphs. Some cannot be counted exactly or have extremely complicated formulae, and in
such cases we can try approximate counting (usually that means an asymptotic formula). A
key tool will be generating functions, which are power series that count things for you (it's
almost magic).
Probabilistic combinatorics deals with random combinatorial objects such as random
graphs, random colouring of graphs, and random k-SAT. It also refers to using
probabilistic tools to solve classical combinatorial problems. In the second half of this
course we will learn some basic probabilistic tools and techniques and use them to solve
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various types of mathematical problems. We will also study randomised algorithms and
properties of random objects.
Syllabus: ordinary and exponential generating functions; first and second moment
methods, Lovasz local lemma, Poisson paradigm, optional stopping theorem.
Prerequisites: Familiarity with basic graph theory and probability theory is desirable for
the course. Please talk to the lecturer if you are concerned about your background.
References:
H.S.Wilf, Generating functionology,
http://www.math.upenn.edu/~wilf/DownldGF.html
N. Alon, J. H. Spencer, The probabilistic method (Third Edition).
M. Mitzenmacher, E. Upfal, Probability and Computing: Randomized algorithms and
Probabilistic analysis.
M41022 PARTIAL DIFFERENTIAL EQUATIONS
Offered Semester: Two
Level: 4 pt Honours Lecture Topic
Lecturer(s): Julie Clutterbuck and Yann Bernard
Contact Details: Room 409, 4th floor, Maths building, Clayton campus, (03) 9905
4521, email Julie.Clutterbuck@monash.edu and Room 421, 4th floor, Maths
building, Clayton campus, (03) 9905 4462, email Yann.Bernard@monash.edu
Content: Partial Differential Equations are ubiquitous in the modelling of physical
phenomena, such as fluid dynamics, as well as in several areas of mathematics, such as
differential geometry.
We will cover the following topics:
Basic properties of Lebesgue integration: Lp spaces, convergence theorems, regularisation
by convolution Sobolev spaces theory: weak derivatives, continuous and compact
embeddings, trace theorem Elliptic equations in the variational case: weak solutions (for
various boundary conditions), Lax-Milgram theorem, maximum principle, Regularity
results, classical solutions for regular data Maximum principles
Prerequisites: Functional analysis (MTH3160); Partial differential equations (MTH3011)
and measure theory (M340011) are also recommended.
References:
R.A. Adams & J. J.F. Fournier, Sobolev Spaces 2nd ed, Academic Press, 2003
L.C. Evans, Partial Differential Equations, American Mathematical Society, 2010
Assessment:
* 8 small assignments submitted throughout the semester (50% of the final grade).
* A 20 minute final oral exam (50% of the final grade).
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M41082 ADVANCED TOPICS IN ANALYSIS
Offered Semester: Two
Level: 4 pt Honours Lecture Topic
Lecturer(s): Zihua Guo and Andy Hammerlindl
Contact Details: Room 431, 4th floor, Maths building, Clayton campus, (03) 9905
9647, email Zihua.Guo@monash.edu.au and Room 428, 4th floor, Maths building,
Clayton campus, (03) 9905 4416, email Andy.Hammerlindl@monashedu
Aims: This unit will include two 12-lecture advanced modules on areas of analysis, giving
interested students a taste of some areas of modern research in that field.
Syllabus: Details of the planned syllabus of the modules to be offered in any year will be
advised to interested students closer to the start of the semester in which it is taught.
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6.2 Statistics
M42011 STOCHASTIC CALCULUS AND MATHEMATICAL FINANCE
Offered Semester: One
Level: 6 pt Honours Lecture Topic
Lecturer(s): Fima Klebaner
Contact Details: Room 460, 4th floor, Maths Building, Clayton campus, (03) 9905
4409, email Fima.Klebaner@monash.edu
Aims: This unit is a 6 point honours lecture topic and provides an introduction to
Stochastic Calculus and mathematics of financial derivatives. Stochastic calculus is an
extension of calculus to non-differentiable functions. It is a branch of pure mathematics,
which found use in applications. Besides finance it is also used in engineering. We teach
from the book: Klebaner, Fima “Introduction to stochastic calculus with applications”, 2nd
Ed, Imperial College Press, 2005.
Syllabus: Variations and Quadratic variation of functions. Review of Integration and
Probability. Brownian motion. Ito integrals and Ito’s formula. Stochastic Differential
Equations and Diffusions. Calculation of expectations and PDE’s, Feynman-Kac formula.
Martingales and Semimartingales. Change of Probability Measure and Girsanov Theorem.
Fundamental Theorems of Asset Pricing. Change of Numeraire. Application to options.
Prerequisites: Some knowledge of probability is required and some knowledge of financial
mathematics is desirable.
References:
Hull, John, “Options, futures and other derivative securities”, Prentice Hall.
Klebaner, Fima, “Introduction to stochastic calculus with applications”, 2nd edition,
Imperial College Press, 2005.
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M42021 STOCHASTIC PROCESSES I
Offered Semester: One
Level: 4 pt Honours Lecture Topic
Lecturer(s): Andrea Collevecchio
Contact Details: Room 456, 4th floor, Maths Building, Clayton campus, (03) 9905
4404, email Andrea.Collevecchio@monash.edu
Aims: This unit is a 4 point honours lecture topic and provides an introduction to general
theory of stochastic processes and their applications. Stochastic processes constitute a
branch of applied mathematics, which found use in various areas of engineering, resource
management, control, and finance. The main uses of stochastic processes are the
estimation and filtering, stochastic optimization and control.
Syllabus: Basic concepts of the probability theory and stochastic processes. Theorem of
Kolmogorov and examples of random processes. Linear theory of stationary random
processes. Stochastic integrals. Spectral analysis of random processes in linear systems.
White noise. Random Processes with orthogonal and independent increments. Nonstationary linear systems, state and observation model. Kalman filter and its applications to
stochastic optimization and control.
Prerequisites: It would be useful to have MTH2222 and MTH3241 as prerequisites. Some
knowledge of basic probability is desirable.
References:
H. Kramer and M. Leadbetter, Stationary and Related Stochastic Processes. John Wiley &
Sons, NY, 1967.
B. Miller and A. Pankov, Theory of Random Processes, Nayka, Moscow, 2002.
M42032 STOCHASTIC PROCESSES II
Offered Semester: Two
Level: 6 pt Honours Lecture Topic
Lecturer(s): Andrea Collevecchio
Contact Details: Room 456, 4th floor, Maths Building, Clayton campus, (03) 9905
4404, email Andrea.Collevecchio@monash.edu
Aims: This unit is a 6 point honours lecture topic and the aim of the unit is to provide an
introduction to general theory of random walks and Markov processes. The basis of the
course is martingale theory, which constitutes an important part of all modern probability
theory and helps to understand most of the approaches in financial mathematics and
actuarial science.
Syllabus: Martingales. Definitions and examples, stopping times. Optional stopping
theorem. Convergence of martingales. Applications of martingales to analysis of random
walks. Definition and properties of Markov processes, Markov chains in discrete time.
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Markov chains in continuous time, the birth-and-death processes. Linear stochastic
differential equations and Kalman-Bucy filter.
Prerequisites: It would be useful to have MTH2222, MTH3230 and MTH3241 as
prerequisites.
References: S. Ross, Stochastic Processes, John Wiley & Sons, NY, 1996.
R. Elliott, L. Aggoun, and J. Moor, Hidden Markov Models, Springer, 1998.
M42052 MARKOV CHAINS AND MIXING TIMES
Offered Semester: Two
Level: 4 pt Honours Lecture Topic
Lecturer(s): Tim Garoni
Contact Details: Room 445, 4th floor, Maths Building, Clayton campus, (03) 9905
4400, email Tim.Garoni@monash.edu
Aims: Markov chains with large finite state spaces play a fundamental role in a diverse
range of disciplines, including combinatorics, computer science, mathematical physics
(statistical mechanics), and of course probability. This course aims to present an
introduction to the significant developments that have occurred in this cross-disciplinary
field over the past three decades.
The classical theory of Markov chains studied fixed chains, and the goal was to estimate the
rate of convergence to stationarity as time tends to infinity. In modern applications,
however, a different asymptotic analysis is required. The "mixing time" of a Markov chain
is introduced to quantify how many steps are required to reach approximate stationarity.
One then considers families of Markov chains, and the goal is to understand how the
mixing time grows as the size of the state space increases.
This course will emphasise the cross-disciplinary nature of the field. From the probabilistic
side, we will introduce important concepts such as couplings and stationary times, and use
them to study random walks on groups and graphs. This will include a discussion of card
shuffling, and the famous "seven shuffles suffice" theorem. From the mathematical physics
side, we will discuss how to construct and analyse Markov chain Monte Carlo methods for
studying Gibbs distributions. From the combinatorial/computer science side, we will
discuss how to use Markov chains to construct efficient algorithms to solve #P-hard
counting problems.
Syllabus: Foundations of finite Markov chains. Properties of total variation distance,
distance from stationarity, mixing time. Couplings, and the coupling method for upperbounding mixing times. Strong stationary times, random walks on groups, and card
shuffling. Spectral methods, the relaxation time, and combinatorial bounds. The cutoff
phenomenon, and exact sampling via coupling from the past. Lower bounds via the
bottleneck ratio and distinguishing statistics. Metropolization and approximate sampling
from Gibbs distributions. Approximate counting, random sampling of combinatorial
structures, and fully-polynomial randomized approximation schemes.
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Prerequisites: Familiarity with probability and/or combinatorics would be useful, but will
not be assumed.
References:
Markov Chains and Mixing Times, by D. Levin, Y. Peres and E. Wilmer (AMS, 2009)
Reversible Markov Chains and Random Walks on Graphs, by D. Aldous and J. Fill,
(http://www.stat.berkeley.edu/~aldous/RWG/book.html)
Finite Markov Chains and Algorithmic Applications, by Olle Haggstrom (Cambridge
University Press, 2002).
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6.3 Applied Mathematics
Students undertaking the Applied Mathematics honours program must complete the three
core lecture topics, M43011, M43022 (listed below) and M40110 (as described in Section
6.0 – Writing and Presenting for Honours) plus at least 16 points of other approved
lecture topics as described in Section 4 of this handbook.
M43011 ADVANCED COMPUTATIONAL MATHEMATICS
Offered Semester : One
Level: 4 pt Honours Lecture Topic
Lecturer(s): Simon Clark
Contact Details: Room 327, 4th floor, Maths Building, Clayton campus, (03) 9905
4421, email Simon.Clarke@monash.edu
Aims: To appreciate some of the important computational techniques which have broad
applicability to solving research problems in applied mathematics, especially the main
approaches and key factors involved in the numerical approximation of partial differential
equations by finite difference methods.
Syllabus: The main topics covered are:
•
A brief overview of classes of methods for multidimensional problems; nonuniform grids and coordinate transformations; truncation error, stability,
consistency and convergence.
•
Direct and iterative solution of elliptic PDEs, including conjugate gradient and
multigrid approaches.
•
Direct and operator splitting methods for the solution of parabolic PDEs in
multiple dimensions, including FFT and ADI methods.
•
Methods for the accurate solution of the advection equation and hyperbolic
PDEs, including upwind and flux-splitting techniques.
Prerequisites: MTH3011 and satisfactory completion of (or exemption from). MTH3051
recommended.
Textbook (access essential):
LeVeque, Randall, Finite Difference Methods for Ordinary and Partial Differential
Equations, SIAM, 2007.
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M43031 MATHEMATICS IN BIOLOGY AND MEDICINE
Offered Semester: One
Level: 4 pt Honours Lecture Topic
Lecturer(s): Mark Flegg, Pascal Buenzli and TBA
Contact Details: Room 351, 4th floor, Maths Building, Clayton campus, (03) 9902
0596, email Mark.Flegg@monash.edu and Room 353, 4th floor, Maths Building,
Clayton Campus, (03) 9905 1953, email Pascal.Buenzli@monash.edu and TBA
Aims: This course is an introduction to the application of mathematics in biological and
biomedical systems, a rapidly expanding area generating increasing scientific interest
providing quantitative theory to previously heuristic observations.
The course will explore how continuous (ODEs, PDEs), discrete, and stochastic models
(Gillespie, Brownian motion) enable us to explore a wide range of biological phenomena
occurring at several time and length scales.
Several examples will be drawn from the biomedical research literature (e.g. cancer,
osteoporosis). The aim of the course is to introduce the students to both the mathematical
methods and the biological applications and to generate a realisation of the potential of
mathematics in biomedical applications.
Syllabus:
• Introduction to Mathematical Biology, examples, challenges
• Cellular and subcellular scale – Balance equations, enzyme kinetics, receptor–ligand
binding and trafficking, intracellular regulatory networks
• Population and ecological models - bifurcations, oscillations, biological invasions
• Disease modelling – SIR/SEIR/SIS models, heterogeneous and structured population
models. Applications to malaria, HIV, and Hepatitis C
• Patterning and Tissue growth – Turing patterns, fingering, branching patterns
• Stochastic temporal models – chemical master equation, stochastic simulation
algorithms (Gillespie SSA), Fokker-Planck equation. Application to biological switches
• Stochastic spatio-temporal models – stochastic reaction–diffusion and advection–
diffusion processes, random walks, lattice-free simulations, chemotaxis, collective
behaviour.
Prerequisites:
• MTH2032 - Differential Equations with Modelling
• MTH3051 - Introduction to Computational Mathematics
References:
[1] Lauffenburger DA & Linderman J (1996) Receptors: Models for binding, trafficking, and
signalling (Oxford University Press)
[2] Edelstein-Keshet L (2005) Mathematical models in biology (SIAM Classics in Applied
Maths)
[3] Keener J & Sneyd J (2009) Mathematical Physiology; I Cellular Physiology & II Systems
Physiology, 3rd Ed (Springer)
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[4] Murray JD (2002) Mathematical Biology; I An introduction & II Spatial models and
biomedical applications, 3rd Ed (Springer)
M43051 GENERAL RELATIVITY
Offered Semester: One
Level: 4 pt Honours Lecture Topic
Lecturer(s): Leo Brewin
Contact Details: Room 317, 4th floor, Maths Building, Clayton campus, (03) 9905
4456, email Leo.Brewin@monash.edu
Aims: The aim is to study General Relativity: to understand its claims as a gravitational
theory, its relationship with special relativity and Newtonian gravitational theory and to
examine its predictions and their observational verification (or even non-verification). This
topic is also designed to serve as an introductory course for anyone who wants to do
further study in GR.
Syllabus: Differential geometry: Riemannian manifolds, covariant differentiation,
geodesics, curvature tensor. Einstein's equations, black holes, experimental tests of general
relativity.
Prerequisites: MTH2010 Multivariable Calculus and ASP2051 /ASP3051 Relativity and
Cosmology.
Recommended book:
D'Inverno, R., Introducing Einstein's relativity, Oxford, 1992.
References:
Schutz, B.F., “A first course in general relativity”, CUP, 1985.
Wald, R.M., “General relativity”, U. of Chicago Press, 1984.
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M43021 ADVANCED METHODS FOR APPLIED MATHEMATICS
Offered Semester: One
Level: 4 pt Honours Lecture Topic
Lecturer(s): Michael Page and Paul Cally
Contact Details: Room 326, 3rd floor, Maths Building, Clayton campus, (03) 9905
4486, email Michael.Page@monash.edu and Room 322, 3rd floor, Maths building,
Clayton campus, (03) 9905 4471, email Paul.Cally@monash.edu
Aims: To appreciate some of the key principles of approximation of solutions of
differential equations and integrals by asymptotic analysis.
Syllabus: The syllabus will follow the textbook (Bender & Orszag) quite closely, so it is
essential that students have access that that book. Some “lectures” will also be set as
reading.
The main topics covered are:
•
Local analysis of linear ODEs, including irregular singular points and
asymptotic series;
•
Asymptotic expansion of integrals, including stationary phase and steepest
descent;
•
Perturbation series, including singular series and asymptotic matching;
•
Boundary-layer theory, including matching between layers;
•
WKB theory.
Assessment: Continuous assessment (problems from textbook and problem sheets) 50%
Exam 50%
Prerequisites: MTH3060 recommended.
Textbook (access essential):
Bender, C.E., and Orszag, S.A., Advanced Mathematical Methods for Scientists and Engineers,
Springer 1999 (originally McGraw-Hill 1978).
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M43042 ADVANCED TOPICS IN FLUID DYNAMICS
Offered Semester: Two
Level: 4 pt Honours Lecture Topic
Lecturer(s): Anja Slim, Phil Hall and Simon Clarke
Contact Details: Room 347, 3rd floor, Maths Building, Clayton campus, (03) 9905
8619, email Anja.Slim@monash.edu, Room 441, 4th floor, Maths building, Clayton
campus, (03) 9905 4448, email Phil.Hall@monash.edu and Room 327, 3rd floor,
Maths building, Clayton campus, (03) 9905 4421, email Simon.Clarke@monash.edu
Aims: This unit will include two 12-lecture advanced modules on the application of applied
mathematics in fluid dynamics, giving interested students a taste of some areas of modern
research in that field. Depending on the availability of staff or visiting research, possible
topics for modules could include: “Hydrodynamic stability”, Nonlinear waves”,
“Biological fluid dynamics”, Geophysical fluid dynamics” or “Advanced boundary-layer
theory” for example.
Syllabus: Details of the planned syllabus of the modules to be offered in any year will be
advised to interested students closer to the start of the semester in which it is taught.
Pre-requisites: MTH3360 Fluid dynamics and MTH3011 Partial differential equations
recommended.
M43062 OPTIMISATION
Offered Semester: Two
Level: 4 pt Honours Lecture Topic
Lecturer(s): Andreas Ernst and Davaa Baatar
Contact Details: Room 346, 3rd floor, Maths building, Clayton campus, (03) 9905
4455, email Andreas.Enrst@monash.edu and Room 350, 3rd floor, Maths building,
Clayton campus, (03) 9905 0802, email davaatseren.baatar@monash.edu
Aims: To introduce students to the theory, techniques and applications of linear and
integer programming. In this course we cover linear programming, duality, column
generation, polyhedral analysis, cutting plane, branch and bound and cut method,
decomposition techniques and heuristic algorithms.
Syllabus:
• Linear models with continuous variables, degeneracy, initial solution, Bland’s rule
• Duality, shadow price, Column Generation rule
• Integer models and complexity
• Polyhedral analysis, Totally unimodularity
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• Valid inequalities, Cutting planes, Branch and Bound Method
• Decomposition methods: Lagrangian relaxation, Bender's Decomposition,
Danzig–Wolfe decomposition
• Dynamic programming, Bellman’s principle
• MetaHeuristics
Prerequisites: Linear algebra
M43071 COMPUTATIONAL VISUALISATION AND NUMERICAL
HYDRODYNAMICS
Offered Semester: Two
Level: 4 pt Honours Lecture or Reading Topic
Lecturer(s): Alina Donea and Hans De Sterck
Contact Details: Room 319, 3rd floor, Maths Building, Clayton campus, (03) 9905
4488, email Alina.Donea@monash.edu and Room 329, 3rd floor, Maths building,
Clayton campus, (03) 9905 6294, email Hans.deStercke@monash.edu
Advanced methods of numerical simulation and visualization are of great importance
for understanding complex physical phenomena in fluid dynamics and astrophysics and
for designing advanced engineering solutions to fluid flow problems. We will first
consider the mathematics of computer vision techniques for visualizing fluid
flows with applications to visualization of big data sets in hydrodynamics and solar
imaging. We will then discuss finite volume numerical methods for hyperbolic
conservation laws that model compressible fluids. We will investigate the
mathematical structure of hyperbolic conservation laws and formulate accurate and
efficient numerical simulation methods that have broad applicability in areas such as
astrophysics and aerospace engineering.
Aims: The first part of the unit will be devoted to the study of mathematical principles
of visualization of images and three-dimensional flows and structures. We will review
computer vision techniques in a practical manner according to the research interest of
students. The second part of the unit will cover finite volume numerical methods for
hyperbolic conservation laws, in particular, the Euler equations of compressible gas
dynamics, and related systems of equations.
Prerequisites: MTH2032, MTH3360 (or speak to the lecturers). Computer skills: some
previous experience with numerical computing is expected.
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Syllabus:
PART 1:
•
Understand the fundamentals of image processing, which is needed in almost all
computer vision applications.
•
In recent years, visualisation has become an important part of scientific and
engineering. In parallel to the continuously increasing computing power available, the
amount of data to be processed has grown quickly. As is the case for the Internet, there
is a growing gap between the quantity of information that is theoretically available and
the ability to efficiently process and handle this information. The extraction of the
essential contents of the information and its reduction to a reasonable size thus
becomes a more and more important task.
•
Applications to hydrodynamics and solar images from satellites. Solar physics will
be discussed based on the processed images.
PART 2:
• Develop an understanding of the mathematical properties of the hyperbolic
conservation laws that describe compressible fluids.
• Finite volume methods for hyperbolic conservation laws, with applications to
compressible gas dynamics and magneto hydrodynamics.
• Implementation of the methods in your choice of MATLAB / Python / C++ /
Fortran /... .
Assessment: Assignments 50%, Exam 50%.
References:
PART 1: Lecture Notes, & “Computer Vision: Algorithms & Applications,” Richard
Szeliski.
PART 2: Lecture notes by Hans De Sterck
7. Details of External Courses
In 2016, the AMSI Summer School will be hosted by RMIT University between 4- 29
January 2016. To receive more information about the Summer School please visit:
http://ss16.amsi.org.au/
The 2016 AMSI Advanced Collaborative Environment (ACE) honours topics will be
published on the following link during February 2016:
http://highered.amsi.org.au/agr-honours-courses/
If you are interested in taking either one of these AMSI summer semester topics over
summer 2016 OR AMSI Advanced Collaborative Environment (ACE) honours topics
within the 2016 teaching semesters, please speak to the Honours Coordinator (see Section 1).
The Key Centre For Statistical Sciences (KCSS) also offers external courses which you
may, please visit :
http://www.latrobe.edu.au/mathematics-and-statistics/your-study-experience/keycentre-of-statistical-science
If you are interested in taking one of these KCSS units, please contact the Statistics
Honours Advisor (see Section 1).
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8. Honours year prizes
All students enrolled in honours in the School of Mathematical Sciences are eligible for the
following prizes:
LEO GLEESON PRIZE
Prize for the best honours student completing the Applied Mathematics honours program.
CARL MOPPERT PRIZE IN MATHEMATICS
Prize for the best all-round mathematics honours student.
PURE MATHEMATICS PRIZE
Prize for the outstanding honours student in pure mathematics.
STATISTICS PRIZE
Prize for the best honours student in statistics.
S:\ SchoolAdmin\honours\2016_honours_Course_handbook
Revised Wednesday 6 April, 2016 5.00 pm
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