Category Archives: basic mathematics

Grothendieck

https://minhyongkim.files.wordpress.com/2014/12/grothendieck2.pdf

Proofs

I thought I would post a short email exchange, just in case other people are confused by this issue as well.

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Email:

hi, this is where the confusion arose as you mention some proofs specifically but at the bottom where it says “Needless to say, it is assumed that you will have a full
understanding of the material surrounding the results listed above, especially the definitions and the examples.” does this include the proofs of the theorems?

>
> For the answer to this, look at the course summary on the course webpage.
>
> Best,
>
> MK
>

>> do we need to know the proofs that are in the online notes but not in the
>> class notes?

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The answer to the last question is: Yes, of course you need to know the proofs. The paragraph quoted was put in to emphasize that you need to know *more* than the proofs. The obvious intention was to discourage people from pure memorization, which has been a topic of discussion since I taught the course last year. Let me once again refer people to the the posts on the blog surrounding last year’s exam. The exchange reproduced above seems to illustrate once again a tendency to indulge in a `minimal reading’ of requirements. Be careful. to spell it out once more, what is meant is:

`You need to know the proofs and understand them fully. This means you need *also* to understand the material surrounding the theorems and their proofs.’

Geometry of complex numbers

Dear professor,

How has your Easter been?

I was doing some revision and I came across a question I would like to ask you. Please enlighten me on it.

Find the least value of |z| if |z - i| = |z - 2|.

Thank you for your time.

Hope you had a good holiday.

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Reply:

The Easter break went by quite quickly. I was mostly working at the Institut Des Hautes Etudes Scientifiques in France, but my family got to roam around Paris a bit and attended some Easter events at the Notre Dame Cathedral .

Regarding your question, note that |z-w| for any two complex numbers z and w is just the distance between them regarded as points in the plane. In particular, you should be able to draw the line (why?) defined by the condition

|z - i| = |z - 2|,

after which the answer should be easy.