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AMSI Summer School 2010 Course Materials

Geometry and Group Actions (GGA) All enquiries should be directed to: Grant Cairns E: G.Cairns@latrobe.edu.au

Synopsis

Not only are there many geometries, but there are many approaches to geometry. We will look at an approach that one could arguably say is the mainstream, modern approach. Here, instead of building on a choice of axioms or postulates, one chooses a particular set, equipped with a particular structure, and one studies the resulting geometric properties, that are invariant under the group of automorphisms of the structure. In this subject, we'll start with some classical geometries (Euclidean, inversive, hyperbolic, Minkowskian). We'll then look at group actions, use them to generate some group theoretic notions, and then return to play with geometry again.

Contact Hours

7 hours of lectures per week, with consultation as requested/required. For information on timetabling please visit the timetable web page. My office is Room 217, Physical Sciences 2. Feel free to see me at any time, I'll be in most days.

For Credit (FC) and Not For Credit (NFC)

Are you taking this subject for credit in your home institution? If so, I need to know. I’ll take a roll in the first class. You can change your mind afterwards, but students taking a subject for credit must confirm their intention to do so by the start of the last week of the school.

Prerequisites

The course doesn't assume any background in geometry and the course doesn't use topology. You will need to have done a basic course on linear algebra. When we get to groups we will start with the definition, so technically it is possible to do this subject without having done group theory before. However, the pace is quite fast, so it is preferable that you have already done an introductory course on groups, or algebra in general.  The main thing to note is that the course involves a lot of "proofs", so it is essential that you have done some proof-based mathematical studies, and that you’re comfortable and confident in writing out arguments in a coherent and rigorous manner.

Background

While the course doesn't assume any specific background knowledge in geometry, the geometry part of the course is undoubtedly the most challenging. The group theory part of the course is more straight-forward, and the assignment problems are usually done by doing the (hopefully) obvious thing. The geometry part sometimes requires you to just see how to do it. A good preparation for this course would be to read a little geometry. Classical texts include:

• H.S.M. Coxeter, Introduction to geometry, Wiley Classics Library, 1989.
• Dan Pedoe, Geometry, Dover Publications,1988.
• M.J. Greenberg, Euclidean and non-Euclidean geometries, W. H. Freeman and Co, 2007.

The approach in these texts is not the same as the one adopted in this course, but the geometric flavour is similar.

Assessment

I'm open to negotiation, but the proposal is:

• Three assignments (total: 75%), due in at 2.15pm, by the commencement of the Wednesday lectures on Jan 20, Jan 27, and Feb 3.
• Take-home exam (25%). Handed out at the end of the School and due in by 5pm Friday Feb 12.

Assignments and the exam may be submitted in person, by fax (03 9479 2644) or by email (G.Cairns@latrobe.edu.au).

Late Submission Policy

Because of the short duration of the Summer School, it's important that I mark assignments and get them back to you as soon as possible, so you can get some feedback. So I am very reluctant to allow extensions of deadlines. After all, it is true that is in the nature of Summer Schools that students are in part assessed on their ability to complete a series of demanding assessment tasks in a very short time period. The Late Submission Policy is therefore rather draconian:

• Your mark (whether it be for an assignment and the exam), will be multiplied by 48/(48 + x), where x is the number of hours late.
• No assignments will be accepted after students answers have been returned.

Resources

Lecture notes

Lecture notes (which do get quite opinionated at times) are available for download below. A printed copy of the Lecture notes will be provided for students registered in this subject.

This subject will follow the lecture notes closely; the intention is to cover the four chapters in the four weeks. Chapter 1 is the longest, and will take over a week to go through.