Math 102A: Project# 5: Due December 5, 2019 In-Class Presentations on December 3 and 5

Late projects will not be accepted.

Project Rules. This project is to be a collaborative e?ort between three or four people. You may select your collaborators (they do not have to be from your table.)

Share the work equally. There is to be one write-up per group.

The projects pose questions for you to investigate. There is no unique answer–

just do you own thinking. You are not allowed to consult texts, other people, or the internet. You can talk to the instructor, but nobody else outside of your group. If you consult outside sources you will receive a score of zero on the project.

There are a number of possible projects and di?erent groups will receive di?er-

ent problems. If you have seen this project before, request another project.

Your Question. In class we introduced the study of arrangements of lines. We didn’t prove many details, but we did look at the diagrams in the Grunbaum article

where he displays the di?erent possible of arrangements of small numbers of lines.

In this project you are asked to work out some details and then generalize the

problem to that of arrangements of lines and conics.

Your first problem is to work out the details that show Gunbaum’s figures give

all possible arrangements of three, four or five lines. Provide careful proofs. This

is a good warm up for the next problem which is a more open-ended investigation.

Your second problem is to consider the question of arrangements of lines and

conics. What are the possible ways to arrange one line and one conic, up to projec-

tive equivalence. What about two lines and one conic, or one line and two conics?

What about two lines and two conics, and so forth. If this question is a bit vague,

it is supposed to be–you need to define the parameters. You will note that if the

horizon line does not intersect any of the conics, then they all will appear as el-

lipses. Can you always simplify to this case using projective changes of the plane?

A second issue is what we might mean by cell structure, because now some edges

will be lines and others may be conics.

There are many issues that will come up. Some have multiple resolutions

depending upon how you devise the definitions. Get started and see how far you

can go. Ask your own questions—you are free to take this anywhere that makes

sense. Talk to Prof. Jacob if you need more direction.

Project Write-up. Your project write up should have three parts.

The first section should discuss your process. Include discussion of your con-

jectures and whether they ended up being true or false. Don’t be shy to share

ideas that ended up wrong. This is part of mathematical learning. Also include

where you got stuck and how you overcame being stuck. The writing here should

be informal (but clear) and your ideas can be described intuitively. You might pay

attention to how the precision of your thinking increased with time.

The second part should contain your results. This should consist of carefully

formulated definitions and proofs based upon those definitions. If you cannot prove

a result, but instead give a “plausibility argument”, that is fine, but be sure to

indicate that this is what you are doing. In some projects, you may not have the

tools to prove every result you need. Discuss this issue with the instructor if you

need to. In some projects you may give several di?erent definitions of a term, andDue 11/20 11pm California timezone