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A new map of Spec Z[x]

A little over two years ago I started my Atlas of \mathop{\mathrm{Spec}}\mathbb{Z}[x]. Today I added another map, from a (forever draft?) book of David Mumford and Tadao Oda.

I urge you to check out the website of David Mumford, who, despite having switched from algebraic geometry to vision in the 80s maintains an excellent overview of all his work in geometry. In particular, he recently posted this draft version of a book which is over 40 years old, which contains a TeX’ified version of his map of the affine line over the integers on page 120.

Announcement for a seminar

Yes, I am still alive. Today I would like to announce a new series of lectures in our graduate student seminar. We are going to talk about Hodge-to-de Rham degeneration, and the proof of Deligne–Illusie of this. Feel free to join in if you are interested, and somewhere in the neighbourhood of Antwerp.

Brute forcing first species counterpoint

The last post in the music category dates from almost 3 years ago. And even longer ago I started a not so long-lived series on computational composition using Strasheela. Time to change this situation.

Last week as a programming finger exercise I decided to implement a brute-force first species counterpoint solver in Python, enumerating all legal solutions and then scoring them. The first part is finished now, and can be seen at GitHub.

There are several directions in which I would like to take this little project. First of all it needs to be faster, which might be feasible by multi-threading (something I’d like to try just for getting some experience with multi-threading in Python). Observe that the code certainly hasn’t been written with speed in mind, I went for conceptually clear code not some super-fast C-type implementation. Then of course, I want to implement the scoring system. But before that, there are some rules that aren’t implemented yet (e.g. hidden fifths and octaves come to mind) and some of the rules have an ugly implementation.

Going to second species will be a significantly harder task: one has to interpret the function of a note correctly.

Feel free to chime in with suggestions! I have discovered a similar project and I should definitely try to implement something like melodypy.com but for now you’ll have to look at plaintext and hear the notes in your head.

Examples of spectral sequences in algebra and algebraic geometry

I just gave a lecture on examples of spectral sequences in algebra and algebraic geometry for the ANAGRAMS seminar and without further ado, here are the notes. Nothing new or insightful in there, just some facts and examples taken together to help me during the lecture.

Nitpicking on Čech cohomology notation

A long time ago I wrote about how to produce Čech cohomology groups in LaTeX, while in the previous post there is the Čech-to-derived spectral sequence making an appearance, and when I took the first screenshot in that blogpost I realised there was something wrong with the positioning of the exponent. Compare:
Positioning of the exponent in Cech cohomology: bad
to
Positioning of the exponent in Cech cohomology: good
The first one is the naive approach, obtained by using

\DeclareMathOperator\cHH{\check{\HH}}
\DeclareMathOperator\HH{H}

The problem with it is that the accent raises the height of the box, thereby raising the exponent too. But visually speaking the q and p should be on the same height. This is done by using the \smash macro (which already exists in Plain TeX, and is not part of mathtools which I mistakingly thought up to today) which basically annihilates the height from its argument in all layout computations. It can create a horrendous mess, but here it is the solution.

To make sure that our macro is as generally applicable as possible we do insert the height of the argument which will then act as the total height of the result using the \vphantom macro:

\newcommand\cechit[1]{\smash{\check{#1}}\vphantom{#1}}

\DeclareMathOperator\cHH{\cechit{\HH}}
\DeclareMathOperator\HH{H}

which results in the second screenshot.

The irony is that if I wouldn’t be using macros, there wouldn’t be a problem, as TeX will not treat it as one high box if you hardcode everything. Oh well…

Adding an equation to the table of contents

Next week I’ll be giving a lecture on spectral sequences in algebraic geometry, and in the notes I’m preparing I wanted to put the spectral sequence below the title of the corresponding section in the table of contents. The \addcontentsline command was known to me, but \addtocontents wasn’t, and there is one caveat in using it: you need to make sure you put \par at the end of the what you are adding (if it is actual text), forcing the next item to get its own paragraph.

So the macro

\newcommand\tocequation[1]{\addtocontents{toc}{\vspace{.7em}#1\par}}

does the trick (change the vertical spacing to your liking), and using it just requires to say \tocequation{$a^2+b^2=c^2$} after the corresponding \section. A small excerpt of the result is

Table of contents containing an equation

The final results will be posted somewhere next week. I’ll be posting a follow-up blogpost on some nitpickery regarding the notation for Čech cohomology later.

What to do if a proof is psychologically uncomfortable?

Robert Thomason suggests the following in his article The classification of triangulated subcategories:

The reader may find the indirectness of the proof of this useful corollary psychologically uncomfortable. If so, rather than dosing himself with a benzodiazepine, he may find relief in deducing Corollary 2.3 from the following criterion for equality of two classes in \mathrm{K}_0(\mathcal{D}), whose proof is very similar to that of Lemma 2.2. […]

I never checked whether we have diazepines next to the pens in the office supplies closet in the department.

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