Questions handled in by students on 2015-12-07.

Admin - 

1) What are the main big topics covered in the course (list for revision)?

1') What is the best source of questions with answers specific to the course material apart from past papers?

2) Do the lecture notes by Miles contain all examinable material, or do we go by your notes?

3) When it comes to revision for the exam, will revision lectures be available / what resources do you suggest we use online?

4) Will exam questions be of a similar style to the assignments?

Lecture - 

1) In terms of "specialization", what's the order in which we should consider these geometries, i.e. is projective geometry contained in Affine, Euclidean?  What are the relations between them?

2) Why do we not add an element, \infty, to the real line so that we can represent all of projective space with one affine chart?

3) Can we projectivise subspaces other then \RR^n such as S^2 or \HH^2?  What would it look like.

Further - 

1) You said that \LL^3 was like 2 space dimensions and 1 time dimension.  Does this mean that a "space" with 2 time dimensions and 1 space dimension is essentially the same, since you would just get the negative of the Lorentz dot product? (x_1, y_1, z_1) \circ (x_2, y_2, z_2) = -x_1 x_2 - y_1 y_2 + z_1 z_2 (- \circ rearranged)

1') To get S^2 you take \RR^2 and make the points "at infinity" along the axes be the same point.  To get \HH^2 you make them meet a circle.  If you make the axes meet in different ways do you get different geometries which make sense? [many figures] e.g. making the ends of the y-axis meet and the ends of the x-axis meet gives a torus And in higher dimensions?

1'') What property of a space tells you how many types of pencil it has, or is it an intrinsic property?

1''') If you define a projective space over \HH^n does it end up behaving nicely?  Is it at all like S^{n-1}? [figure]

1'''') Does the hyperboloid of 1 sheet have a meaningful geometry?