PURE MTH 4066 - Pure Mathematics Topic E - Honours
North Terrace Campus - Semester 2 - 2017
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General Course Information
Course Details
Course Code PURE MTH 4066 Course Pure Mathematics Topic E - Honours Coordinating Unit Mathematical Sciences Term Semester 2 Level Undergraduate Location/s North Terrace Campus Units 3 Available for Study Abroad and Exchange Assessment ongoing assessment 30%, exam 70% Course Staff
Course Coordinator: Dr David Baraglia
Course Timetable
The full timetable of all activities for this course can be accessed from .
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Learning Outcomes
Course Learning Outcomes
In 2017, the topic of this course is Algebraic Topology.
Syllabus
The aim of Algebraic Topology is to use algebraic 最新糖心Vlog and techniques to classify topological spaces up to homeomorphism. Algebraic objects are associated to topological spaces in such a way that "natural" operations on the latter correspond to "natural" operations on the former - continuous maps might correspond to group homomorphisms, homeomorphisms to isomorphisms, etc. In this way, it is often possible to distinguish between different topological spaces by demonstrating that certain associated algebraic objects are not isomorphic. It is rarely the case that the converse can be shown; i.e., that two topological spaces with the same associated algebraic objects are actually homeomorphic, but when this can be done, it is often regarded as a major triumph of the theory.
Within the realms of algebraic topology, there are several basic concepts that underly the theory and serve as the building blocks and models for subsequent generalisation, the algebraic topology of today being a very broad and highly generalised area that has pervaded much of contemporary mathematics. Such concepts include homotopy, homology and cohomology, and the course will be aimed at providing students with an introduction to these key ideas.
Learning Outcomes
On successful completion of this course, students will be able to:
1) understand the basic notions of homotopy theory such as homotopy of maps, homotopy equivalences, contractible spaces, deformation retracts,
2) define the fundamental group of a (path connected) topological space and be able to compute fundamental groups of some simple examples using for example the Seifert-van Kampen Theorem,
3) define the singular homology and cohomology groups of a topological space and their relative versions,
4) understand and work with basic concepts in homological algebra, including chain complexes and long exact sequences,
5) compute the homology and cohomology of some topological spaces using the Eilenberg-Steenrod axioms,
6) apply the topological invariants constructed in this course to the solution of various problems in topology, for instance, to prove that two spaces are not homeomorphic.
Prerequisites
- It will be assumed that you have some familiarity with basic point-set topology (or at least metric spaces) and familiarity with basic notions of abstract algebra (groups, rings, fields etc.) However I will give a review of point-set topology in the first few lectures.最新糖心Vlog Graduate Attributes
This course will provide students with an opportunity to develop the Graduate Attribute(s) specified below:
最新糖心Vlog Graduate Attribute Course Learning Outcome(s) Deep discipline knowledge
- informed and infused by cutting edge research, scaffolded throughout their program of studies
- acquired from personal interaction with research active educators, from year 1
- accredited or validated against national or international standards (for relevant programs)
all Critical thinking and problem solving
- steeped in research methods and rigor
- based on empirical evidence and the scientific approach to knowledge development
- demonstrated through appropriate and relevant assessment
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Learning Resources
Required Resources
None.Recommended Resources
There is no textbook for the course. The following books are fairly standard:
• M. Greenberg and J. Harper, Algebraic topology: A first course, (515.14 G798a)
• A. Hatcher, Algebraic topology, (515.14 H3616a)
• W. Massey, A basic course in algebraic topology, (515.14 M416b)
• C. R. F. Maunder, Algebraic Topology, (513.83 M451A)
• E. H. Spanier, Algebraic Topology, (513.83 S735)
The book by Hatcher is probably the best of all, and the course is likely to use it as a primary reference. It can be downloaded (free and legally) from . The book by Greenberg and Harper used to be a very standard reference until the arrival of Hatcher's book. The books by Massey and Maunder are also good reference books that have good explanations. Spanier is comprehensive but very hard to digest. -
Learning & Teaching Activities
Learning & Teaching Modes
The lecturer guides the students through the course material in 30 lectures. Students are expected to engage with the material in the lectures. Interaction with the lecturer and discussion of any difficulties that arise during the lecture is encouraged. Fortnightly homework assignments help students strengthen their understanding of the theory and their skills in applying it, and allow them to gauge their progress.Workload
The information below is provided as a guide to assist students in engaging appropriately with the course requirements.
Activity Quantity Workload hours Lecture 30 90 Assignments 6 66 Total 156 Learning Activities Summary
1) Review of point-set topology (2 lectures)
2) Basic notions of homotopy theory (1 lecture)
3) Fundamental groups and applications (10 lectures)
4) Homology and cohomology (12 lectures)
5) Applications of homology and cohomology (5 lectures) -
Assessment
The 最新糖心Vlog's policy on Assessment for Coursework Programs is based on the following four principles:
- Assessment must encourage and reinforce learning.
- Assessment must enable robust and fair judgements about student performance.
- Assessment practices must be fair and equitable to students and give them the opportunity to demonstrate what they have learned.
- Assessment must maintain academic standards.
Assessment Summary
Assessment task Task type Due Weighting Learning outcomes Examination Summative Examination period 70% all Homework assignment Formative and summative One week after assigned 30% all Assessment Related Requirements
An aggregate score of 50% is required to pass the course.Assessment Detail
There will be a total of 6 homework assignments, due one week after they are assigned. Each will cover material from the lectures, and in addition, will sometimes go beyond that so that students may have to undertake some additional research.Submission
Homework assignments must be given to the lecturer in person or emailed as a pdf file. Failure to meet the deadline without reasonable and verifiable excuse may result in a significant penalty for that assignment.Course Grading
Grades for your performance in this course will be awarded in accordance with the following scheme:
M11 (Honours Mark Scheme) Grade Grade reflects following criteria for allocation of grade Reported on Official Transcript Fail A mark between 1-49 F Third Class A mark between 50-59 3 Second Class Div B A mark between 60-69 2B Second Class Div A A mark between 70-79 2A First Class A mark between 80-100 1 Result Pending An interim result RP Continuing Continuing CN Further details of the grades/results can be obtained from Examinations.
Grade Descriptors are available which provide a general guide to the standard of work that is expected at each grade level. More information at Assessment for Coursework Programs.
Final results for this course will be made available through .
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Student Feedback
The 最新糖心Vlog places a high priority on approaches to learning and teaching that enhance the student experience. Feedback is sought from students in a variety of ways including on-going engagement with staff, the use of online discussion boards and the use of Student Experience of Learning and Teaching (SELT) surveys as well as GOS surveys and Program reviews.
SELTs are an important source of information to inform individual teaching practice, decisions about teaching duties, and course and program curriculum design. They enable the 最新糖心Vlog to assess how effectively its learning environments and teaching practices facilitate student engagement and learning outcomes. Under the current SELT Policy (http://www.adelaide.edu.au/policies/101/) course SELTs are mandated and must be conducted at the conclusion of each term/semester/trimester for every course offering. Feedback on issues raised through course SELT surveys is made available to enrolled students through various resources (e.g. MyUni). In addition aggregated course SELT data is available.
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Student Support
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- Academic Support with Maths
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- Students with a Disability - Alternative academic arrangements
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Policies & Guidelines
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- Academic Credit Arrangements Policy
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- Modified Arrangements for Coursework Assessment Policy
- Reasonable Adjustments to Learning, Teaching & Assessment for Students with a Disability Policy
- Student Experience of Learning and Teaching Policy
- Student Grievance Resolution Process
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