## Table of Contents

The PDF version of this page can be downloaded by replacing `html`

in the its address by
`pdf`

.
For example `/html/sheaf-cohomology.html`

should become `/pdf/sheaf-cohomology.pdf`

.

## Log term in energy of \( f: \overline{\mathbb{H}^2} \longrightarrow \overline{\mathbb{H}^{n+1}} \).

Let \(\mathbb{H}^2 \) be coordinated by \( (t,s) \) with \( t>0,s\in \mathbb{R} \) and \( \mathbb{H}^{n+1} \) be cordinated by \( x> 0, y^1,\dots, y^n\in \mathbb{R} \).

Define \( f: \overline{\mathbb{H}^2} \longrightarrow \overline{\mathbb{H}^{n+1}} \) by \[ (t,s) \mapsto \left(t+t^2, \gamma^i(s)\right) \] whose image is the vertical cylinder over the curve \( s \mapsto (\gamma^i(s)) \).

If \( n=2 \), one can take \( (y^1,y^2) := \left(\frac{2s}{s^2+1},\frac{s^2-1}{s^2+1} \right) \). Its image is the unit circle in \( \{x=0\} \), only filled once as \( s \) varies in \( \mathbb{R} \).

Now that \[ dx = (1+2t)dt,\quad dy^i = \dot \gamma^i ds, \] the pullback by \( f \) of Poincaré metric \( g = \frac{1}{x^2}(dx^2 + \sum d{y^i}^2) \) on \( \mathbb{H}^{n+1} \) is \[ f^* g = \frac{1}{t^2 (1+t)^2} \left( (1+2t)^2 dt^2 + \sum \dot {\gamma^i}^2(s) ds^2 \right) \] Its trace w.r.t the Euclidean metric on the half 2-plane is \[ \tr f^* g = \frac{(1+2t)^2}{t^2(1+t)^2} + \frac{\varphi(s)}{t^2(1+t)^2} \] where \( \varphi \) only depends on \( s \) and can be replaced by \( a^2\varphi \) for any constant \( a>0 \) when one modifies the boundary curve \( \gamma \to a\gamma \)).

Integrate by \( s\in \mathbb{R} \) and \( t \) in \( (t_{\min},1) \) (we can suppose \(t\leq 1 \) since we are only interested in the energy near boundary, \( t_{\min} = \frac{-1 + \sqrt{1+4\epsilon}}{2} = \epsilon + O(\epsilon^2) \)):

\[ \int_t \frac{1}{t^2} + \frac{1}{(1+t)^2} + \frac{2}{t(1+t)} + \const(\varphi) \left( \frac{1}{t^2} - \frac{2}{t(1+t)} + \frac{1}{(1+t)^2} \right) \] where \( \const(\varphi) \) can be scaled up \( a^2 \) times when one replaces \( \gamma \) by \( a\gamma \). Choose \( a \) so that the term \( \frac{2}{t(1+t)} \) remains, which integrates to \( \log t_{\min} = \log(\epsilon) + O(\epsilon) \).

It is clear that one needs a certain regularity of \( f \) at boundary of the initial space so that the \( \log\epsilon \) term does not appear. The "Poincaré half-plane to Poincaré disk" transform is only smooth and diffeomorphic in the interior. The boundary map \( (y^1,y^2) := \left(\frac{2s}{s^2+1},\frac{s^2-1}{s^2+1} \right) \) in the example above (\( n=2 \)) is the restriction on boundary of the half-plane-to-disk transform.

## Commutative diagram revisited

### xypic and xyJax

To my knowledge, the only way to type commutative diagram on the web is to use `xyJax`

, a
third-party extension of `MathJax`

that render diagram using `xypic`

. This is also how
Stacks Project was set up.

The syntax of `xypic`

is almost the same as `tikz-cd`

, to a basic user the only difference
is how arrows are typed. I learn `xypic`

syntax by rewriting a GUI editor for `tikzcd`

by
Yichuan Shen to output `xypic`

code. Here is the `xypic`

version of the editor. I also
host a copy of the `tikz-cd`

editor here.

The configuration of `xyJax`

or any third-party extension MathJax was not very easy for
me, since it seems that MathJax CDN no longer hosts third-party extensions. So I have to
host my own copy of `xyJax`

and tell MathJax CDN to use it, as indicated in its
documentation. Also, one also has to reconfig a path in `xyJax`

.

To config MathJax in org-mode, apropos `org-html-mathjax-template`

.

## LaTeX in Inkscape: Incompatibility between `ghostscript`

and `pstoedit`

Since Ghostscript 9.22, certain flags (e.g. `dREALLYDEALYBIND`

) are deprecated and `pstoedit`

,
a piece of software that Inkscape uses to render LaTeX, remains unchanged since 2005. This
makes Inkscape users unable use LaTeX. Textext developers claims that they will switch to
`pdf2svg`

frontend in their version 0.8 to avoid this trouble with `pstoedit`

. Meanwhile,
the only solution seems to be downgrade ghostscript to 9.21 each time one uses Inkscape.
In ArchLinux, one can downgrade a package using `pacman`

:

ls /var/cache/pacman/pkg/ | grep ghostscript pacman -U /var/cache/pacman/pkg/package-old_version.pkg.tar.xz

or a AUR package called `downgrade`

downgrade ghostscript

I had good experience with both of them (`pacman`

to downgrade `php`

to be compatible
with `nextcloud`

, and `downgrade`

for ghostscript)

## LaTeX indentation in org-mode

I was told, in accordance with my experience, that we visually process text better if each line in a paragraph is approximately below 80 characters. This fact is also omnipresent on the internet, standardized tests, books, etc.. being among the fundamentals of web design (except wikipedia, that's why I use Wikiwand).

In Emacs, the key binding `M-q`

will execute the `fill-paragraph`

function that
automatically shrinks text and insert "soft" newlines to shorten line below a certain
threshold. This function however does not respect the LaTeX structure, e.g. it inserts
line break inside inline equation and merge display equation.
format paragraph.

This is a long long long long long long long long long paragraph with equation \( 1+1 = 2+0=3-1=4-2 \) with an equation in display \[ 1+ 1 = 2\].

The problem has been noticed around the Internet here and there.

Meanwhile AucTeX does not have this annoying problem. It turns out that AUCTeX maps the
`M-q`

key to a different function, called `LaTeX-fill-paragraph`

. So the temporary fix is
to load `latex.el`

in org-mode and maps `M-q`

to `LaTeX-fill-paragraph`

(load "latex.el") (global-set-key "\M-q" 'LaTeX-fill-paragraph)

## A (decent) map of mathematics

I saw this map of mathematics somewhere on YouTube and then again on the door of someone's office at the I2M. Although this map may help popularise mathematics, it should raise an eyebrow of a serious math student (or even an attentive highschool student, as for example 1 was listed among prime numbers in an earlier version of this map).

For a decent map of Mathematics, made by a mathematician, see this.

## My 2016-2017 internship

I have just recently finished my Stage3A at Institut de Mathématiques de Marseille. It is also in this period that I start this blog. The memoire can be founded here [PDF]