PURE MTH 4066 - Pure Mathematics Topic E - Honours
North Terrace Campus - Semester 2 - 2018
General Course Information
Course Code PURE MTH 4066 Course Pure Mathematics Topic E - Honours Coordinating Unit School of Mathematical Sciences Term Semester 2 Level Undergraduate Location/s North Terrace Campus Units 3 Available for Study Abroad and Exchange Course Description Please contact the School of Mathematical Sciences for further details, or view course information on the School of Mathematical Sciences web site at http://www.maths.adelaide.edu.au
Course Coordinator: Dr Guo Chuan Thiang
The full timetable of all activities for this course can be accessed from Course Planner.
Course Learning OutcomesIn 2018, the topic of this course is Introduction to topological K-theory.
The letter 'K' in K-theory comes from the German word Klasse, which means "class". Everyone already knows at least one K-theory group. Natural numbers classify vector spaces via the dimension, while an integer, being a difference of two natural numbers, measures the difference between two vector spaces. This is an example of Grothendieck's construction applied to obtain the K-theory of a point.
In the 1960s, Atiyah and Hirzebruch introduced topological K-theory, to classify bundles of vector spaces parametrised by an arbitrary topological space. Their theory turned out to be extremely rich, having deep connections to many areas of modern mathematics such as index theory, differential geometry, operator algebras, and representation theory. In noncommutative geometry, one does not even require a topological space to define and utilise K-theory! Because of its multi-facetedness, K-theory has also found profound applications in string theory, fundamental particle physics, and condensed matter physics.
The aim of this course is to provide a gentle first introduction to the vast and interdisciplinary subject of K-theory, as an invitation to further study and more abstract treatments of the subject. The target audience is final-year undergraduates and beginning graduate students. The approach will be quite concrete, with only the simplest version of K-theory to be discussed.
- Linear algebra (inner product spaces, adjoints, traces, determinants), abstract algebra (groups and rings), basic point-set topology (compactness, connectedness, metric spaces, quotient and subspace topplogies) and analysis. A review will be done at the beginning of the course.
- No knowledge of smooth manifolds, algebraic topology, differential forms or functional analysis is assumed, but it will be helpful to know its rudiments.
On successful completion of this course, students will be able to:
1) Demonstrate an understanding of the basic theory of vector bundles
2) Demonstrate familiarity with a range of examples within this theory
3) Demonstrate an understanding of the definition of the K-groups for a topological space
4) Apply Bott periodicity to compute some K-theory groups
5) Formulate and prove an index theorem for Toeplitz operators
6) Apply the theory in the course to solve a variety of problems at an appropriate level of difficulty
University Graduate Attributes
This course will provide students with an opportunity to develop the Graduate Attribute(s) specified below:
University 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
Recommended ResourcesThere is no textbook for the course, but the following books at beginning graduate level are useful.
• E. Park, Complex topological K-theory
• A. Hatcher, Vector bundles and K-theory
• N.E. Wegge-Olsen, K-theory and C*-algebras: A Friendly Approach
• M.F. Atiyah, K-theory
For differential forms and algebraic topology,
• I.E. Madsen and J. Tornehave, From calculus to cohomology
• A. Hatcher, Algebraic topology
• J.W. Milnor, Topology from the differentiable viewpoint
• R. Bott and L.W. Tu, Differential forms in algebraic topology
Learning & Teaching Activities
Learning & Teaching ModesThe 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.
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- What is K-theory about? (1 lecture)
- Preliminaries on algebra and topology (2 lectures)
- Vector bundles, basic homotopy theory, exact sequences and basic category theory (4 lectures)
- Definition of the K-theory functors (8 lectures)
- Bott periodicty and the cyclic exact sequence of K-theory (6 lectures)
- Hilbert spaces, Fredholm operators, Toeplitz index theorem (6 lectures)
- Further topics and applications (3 lectures)
The University'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 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 RequirementsAn aggregate score of 50% is required to pass the course.
Assessment DetailThere 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.
SubmissionHomework 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.
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 Access Adelaide.
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