18.786 Computing Notes

Part of the goal of this course is to get you used to performing numerical experiments, since that is and has always been how many results in number theory were first discovered. There are a number of software packages that can be used for this, each of which has various strengths and weaknesses.

For this course, I am using a brand-new system called SAGE (Software for Algebra and Geometry Exploration), which I find superior to other systems because the authors (William Stein and his collaborators) did not simply make up a poorly thought-out programming and interaction environment. Instead, SAGE is built on top of the programming language Python, which is one of the most popular scripting languages in use today mainly because it is well-conceived: its code is easy to read and write, and it facilitates object-oriented programming. The mathematical internals of SAGE are modeled on Magma, a system which is justly renowned for facilitating the manipulation of complex mathematical objects (rings, modules, groups, schemes...) but which is proprietary and unavailable on Athena.

You can access SAGE in several ways.

The best way to access SAGE for reasonably-sized calculations is to install it on your own computer, either by downloading a binary (for Windows, MacOS, Linux) or compiling it yourself (SAGE is totally open-source). If you need help doing this, some of your classmates may be more helpful than I can be.
On the math department network, you'll find SAGE on the network in the directory /home/r1/kedlaya/sage-complete/. I don't yet have it set up to run on all computers on the network: I recommend running on either hypatia (shared server, extremely fast, but sometimes high load) or dwork (my office computer, also very fast, low load). I think all of the interfaces work, but I haven't tried them all yet.
On Athena, you can add the course locker 18.786 and then run SAGE ("add 18.786; sage"). This should work both under Linux and Solaris, but particularly the Solaris build is a little iffy (and takes a very very long time to load); let me know if there are problems. Note: to use some external interfaces (Maple, Mathematica, Maxima, maybe Singular, and maybe others) you have to add the corresponding Athena lockers before running SAGE (add (maple|mathematica|maxima|singular); I hope to fix that at some point.
For short calculations, you can use William Stein's SAGE calculator from any web browser.
If all else fails, I may be able to ask William for a guest account on his very fast computer, which will always have the latest version of SAGE installed. Let me know if you want me to do this.

The SAGE home page is where you will find binaries, source code, documentation, and other helpful stuff; there is also a sourceforge page.

If you have never seen Python before, you might also want to check out the home page, which includes lots of documentation, including a tutorial. Even if you don't really know anything about programming, it's relatively easy to get started with Python. (Since this is a number theory course and not a programming course, the programming will tend to be pretty light.)