Sage and group cohomology

I’ve been thinking about computing group cohomology by computer
recenty, I thought I’d point out some features of Sage (the leading
free and open-source alternative to Magma, Maple, Mathematics, etc..).

Specifically, with the 4.1.1 release there was added a package for
computing p-group cohomlogy. Below is the link and description.

Also, if you don’t know about Sage, you can download and install it
here: http://www.sagemath.org/

One nice side-effect of installing Sage is that you get a whole bunch
of other useful mathematics programs pre-compiled, configured, and
installed for you, like GAP (for doing group-theory, symbolic linear algebra, and much more).

————————

http://sage.math.washington.edu/home/SimonKing/Cohomology/

New optional package p_group_cohomology version 1.0.2 (Simon A. King,
David J. Green) #6491 — The package p_group_cohomology can compute the
cohomology ring of a group with coefficients in a finite field of
order p. Its features include:

* Compute the cohomology ring with coefficients in
for any finite p-group, in terms of a minimal generating set and a
minimal set of algebraic relations. We use Benson’s criterion to prove
the completeness of the ring structure.
* Compute depth, dimension, Poincare series and a-invariants of
the cohomology rings.
* Compute the nil radical.
* Construct induced homomorphisms.
* The package includes a list of cohomology rings for all groups
of order 64.
* With the package, the cohomology for all groups of order 128 and
for the Sylow 2-subgroup of the third Conway group (order 1024) was
computed for the first time. The result of these and many other
computations (e.g., all but 6 groups of order 243) is accessible in a
repository on sage.math.

B_2, twice

B_2, twice, originally uploaded by brunoj.

… went to New Orleans last weekend, found this ornamental rail around the front plate of a tomb that reminded me of the root system of type B2.

Now that my facebook is dead…

Now that my facebook is dead…

I am updating this blog, my blog, again. Posts to look forward to:

1. The academic job search (and how / why it sucks)
2. A fabulous “labor day” vacation in New Orleans
3. Faster Mustache 2009 (and how my team is going to win it) (incidentally, the last post on this blog was results from Faster Mustache 2008, the greatest urban bike relay in the world).
4. Georgia Cyclocross season 2009 (and how the Loosenuts race team is going to dominate it)
5. Go (and Go-Moku) and why I love it more than chess
6. An awesome poll (polls are a new feature on wordpress)

And since no one actually follows this blog, mostly it will be me talking to myself again, not a healthy thing.

Producing high quality EPS and PDF diagrams for mathematics

I’ve been thinking over the course of the last 6 months or so about the best way to produce EPS and PDF diagrams for inclusion in mathematical documents typeset using LaTeX or pdfLaTeX.

There are many options and many pieces of software, both free and non-free, for doing this. I want to explain the method I settled on as the “best” after trying many different approaches.

I have some requirements: The method must

1. produce high quality vector based EPS or PDF diagrams (not raster images),
2. produce output suitable for inclusion in LaTeX documents which are typeset using pdfLaTeX (so they should be PDF files or EPS files if you include the right package \usepackage{epstopdf} ?). Also this means that the ‘psfrag’ method is ruled out since this doesn’t work under pdfLaTeX,
3. allow for complex diagrams to be produced using a GUI (as opposed to ‘pstricks’ or ‘xy’),
4. allow LaTeX fonts, symbols, macros to be used for labels in the diagram,
5. be cross platform (running under linux and Mac OS X at least).

Here is a partial list of software that I tried at some point in the last 6 months while engaged in a project that required many diagrams to be produced.

1. xfig / jfig
2. Inkscape
3. pstricks package and the graphical editor for it: latexDraw
4. xy package
5. Adobe Illustrator
6. psfrag
7. ‘fragmaster‘ script for getting psfrag functionality in standalone diagrams (see also ‘pdfrack‘)

To cut to the chase, here is the procedure I settled on. It uses GUI drawing program in combination with the ‘fragmaster’ script listed above to get PSFRAG-like functionality in a standalone diagram which can then be converted to PDF and included in a LaTeX document.

1. Using Inkscape or xfig/jfig, produce the diagram. Insert key-letters or keywords where you want LaTeX typeset labels to appear. For instance you could put an ‘a’ where you want to appear. For example:

   

2. Save the diagram as an EPS file using the following pattern for the filename: filename_fm.eps . The ‘_fm.eps’ ending is for the fragmaster script. Make sure not to convert text to paths when saving. This is the default in Inkscape for example when saving as EPS and there is a dialog which asks about this after you hit ’save’.
3. Next we’re going to use the ‘fragmaster’ script, which you can download here: [link]. You need perl installed to run the script, test by running
   perl -v in a terminal. This script runs with no arguments. It expects two files:
   • filename_fm.eps
   • filename_fm

   and leaves as output the file ‘filename.eps’. The ‘filename_fm’ file contains a dictionary which translates the key-letter/keywords from step 1 into LaTeX code. For the example in step 1 I have the following content in ‘filename_fm’:
   
% Just an ordinary comment
%
% A special comment:
% fmopt: width=10cm
%
% Another special comment:
% head:
% \usepackage{amsmath}
% \usepackage{amsfonts}
% end head
\psfrag{beta}[Bc]{$\beta(3)$}
\psfrag{cdb}[Bc]{$\mathbb{D}_{10 \times 8}(\widetilde{\beta(3)})$}

   The commented lines following ’special comment’ are important. They set parameters which LaTeX uses to type set the labels which appear in the ‘\psfrag’ commands. There are three arguments passed to the ‘\psfrag’ command: 1st is the keyword that appears in your EPS diagram that you want to replace with some typeset labels, 2nd is a positioning setting (see the psfrag documentation for more details), and 3rd is the LaTeX code for the label.
   All that remains is to run the script: (assuming the two files above and the fragmaster.pl script are in the same directory)

   ./fragmaster.pl

   Note: You probably need to make fragmaster.pl executable before hand: chmod a+x fragmaster.pl or else run it through perl: perl fragmaster.pl .

4. Now you’ve produced an EPS file called ‘filename.eps’. There is one more (sometimes necessary, sometimes not) step before we have a finished EPS diagram. If your labels occur on the outskirts of the diagram (for instance at the bottom of the diagram in the example of step 1) it can happen that they are partially cut-off by the bounding box of the EPS file. WHen you save your EPS file in the drawing program, the bounding box is set to be just large enough to contain all visible elements. However, when the keyword labels are replaced with LaTeX typeset labels, they may be larger than the keywords and fall outside the original bounding box. One solution to this is the following: enlarge the bounding box by hand. Here’s what I do:
   • edit ‘filename.eps’ using ‘vi’ or whatever text editor you like.
   • at the begining of the file you see lines such as the following:
     

     
%!PS-Adobe-2.0 EPSF-2.0
%%Creator: dvips(k) 5.96 Copyright 2005 Radical Eye Software
%%Title: fm.dvi
%%CreationDate: Thu Jul 12 09:15:33 2007
%%BoundingBox: 0 642 284 792
%%DocumentFonts: CMR12 CMMI12 MSBM10 CMR8 CMSY8 CMEX10
%%EndComments


     The important line is the one starting with %%BoundingBox: . The 4 numbers which follow this define the bounding box for the contents of the EPS file. The first two numbers are (x,y) coordinates for the lower-left corner of the diagram and the second two numbers are coordinates for the upper-right corner. To enlarge the bounding box slightly lets subtract 10 from each of the 1st two numbers and add 10 to each of the last two numbers:

     %%BoundingBox: -10 632 294 802

     Negative values are perfectly fine.

5. Now you should have a beautiful EPS diagram with LaTeX typeset labels:

   
   

   Now, to produce a PDF file for inclusion in a document which is typeset using pdfLaTeX you can simply run the ‘epstopdf’ command: epstopdf filename.eps

6. Sometimes the labels don’t appear in quite the right position after running fragmaster. One option is to go back to the drawing program and adjust the position of the keywords slightly to compensate, save as EPS to ‘filename_fm.eps’, re-run fragmaster, and re-enlarge the bounding box. Another option is to change the positioning arguments to the ‘\psfrag’ commands in the ‘filename_fm’ control file and re-run fragmaster, etc. It usually takes me a couple of runs to get everything perfectly aligned and looking good.

Update 9-22-09: Changed the link to Fragmaster since Vogel’s old homepage no longer exists. The new link points to ctan.org

Ph.D. Defense

During Defense, originally uploaded by brunoj.

I defended my Ph.D. dissertation on Friday. The title of my dissertation is:

On the Singular Chern Classes of Schubert Varieties via Small Resolution 

Above is a photo taken by Sarah at the end of the public talk portion of the defense.

Copies of my thesis

Copies of my thesis, originally uploaded by brunoj.

It is a momentous occasion! I just delivered hard copies of my thesis to the professors on my thesis committee. It is 12:14am and the cleaning crew is rocking out in the hallway as they work. I too am rocking out.

Quote

James Newman (Mathematician and Historian):

The theory of groups is a branch of mathematics in which one does something to something and then compares the results with the result of doing the same thing to something else, or something else to the same thing.

That sounds like what I do.

LaTeX Support

So wordpress is supposed to have good LaTeX support, so I thought I would try it out. Here is a formal expression from my thesis.

Here is the Weyl Character Formula:

Here it is again as a fraction:

Update: I completely screwed up the Weyl Character formula trying to reproduce it from memory. That’s embarrassing.

Sage 2.5 Installed


ben@calamity:~$ /usr/local/share/sage-2.1.4/sage
----------------------------------------------------------------------
| SAGE Version 2.5, Release Date: 2007-05-08 |
| Type notebook() for the GUI, and license() for information. |
----------------------------------------------------------------------
sage: n=next_prime(2^110)*next_prime(2^120)
sage: qsieve(
Display all 1349 possibilities? (y or n)
sage: qsieve(n)
([1298074214633706907132624082305051, 1329227995784915872903807060280345027],
'')
sage: