Here is a list of open source software that you may find useful. Some, I use almost every day, some I have not yet used, and some may be so ubiquitous that you have even forgotten that it is software.
1. XPP/XPPAUT. Bard Ermentrout wrote XPP in the 1980’s as a dynamical systems tool for himself. It’s now the de facto tool for the Snowbird community. I still find it to be the easiest and fastest way to simulate and visualize differential equations. It includes the equally excellent bifurcation continuation software tool AUTO originally written by Eusebius Doedel with contributions from a who’s who list of mathematicians. XPP is also available as an iPad and iPhone App.
2. Julia. I only learned about Julia this spring and now I use it for basically anything I used to use Matlab for. It’s syntax is very similar to Matlab and it’s very fast. I think it is quickly gaining a large following and may be as comprehensive as Python some day.
3. Python often seems more like a way of life than a software tool. I would probably be using Python if it were not for Julia and the fact that Julia is faster. Python has packages for everything. There is SciPy and NumPy for scientific computing, Pandas for statistics, Matplotlib for making graphs, and many more that I don’t yet know about. I must confess that I still don’t know my way around Python but my fellows all use it.
4. R. For statistics, look no further than R, which is what academic statisticians use. It’s big in Big Data. So big that I heard that Microsoft is planning to write a wrapper for it. I also heard that billionaire mathematician James Simons’s hedge fund Renaissance Technologies uses it. For Bayesian inference there is now Stan, which implements Hamilton Monte Carlo. We tried using it for one of our projects and had trouble getting it to work but it’s improving very fast.
5. AMS-Latex. The great computer scientist Donald Knuth wrote the typesetting language TeX in 1978 and he changed scientific publication forever. If you have ever had to struggle putting equations into MS Word, you’ll realize what a genius Knuth is. Still TeX was somewhat technical and thus LaTeX was invented as a simplified interface for TeX with built-in environments that are commonly used. AMS-Latex is a form of LaTeX that includes commands for any mathematical symbol you’ll ever need. It also has very nice equation and matrix alignment tools.
6. Maxima. Before Mathematica and Maple there was Macsyma. It was a symbolic mathematics system developed over many years at MIT starting in the 60’s. It was written in the programming language Lisp (another great open source tool but I have never used it) and was licensed by MIT to a company called Symbolics that made dedicated Lisp machines that ran Macsyma. My Thesis advisor at MIT bought one of these machines (I think it cost him something like 20 thousand dollars, which was a lot of money back then) and I used it for my thesis. I really loved Macysma and got quite adept at it. However, as you can imagine the Symbolics business plan really didn’t pan out and Macysma kind of languished after the company failed. However, after many trials and tribulations, Macsyma was reborn as the open source software tool Maxima and it’s great. I’ve been running wmMaxima and it can do everything that I ever needed Mathematica for with the bonus that I don’t have to find and re-enter my license number every few months.
7. OpenOffice. I find it reprehensible that scientific journals force me to submit my papers in Microsoft Word. But MS Office is a monopoly and all my collaborators use it. Data always comes to me in Excel and talks are in PowerPoint. For my talks, I use Apple Keynote, which is not open source. However, Apple likes to completely overhaul their software so my old talks are not even compatible with the most recent version. I also dislike the current version. The reason I went to Keynote is because I could embed PDFs of equations made in LaTeXiT (donation ware). However, the new version makes this less convenient. PDFs looked terrible in PowerPoint a decade ago. I have no idea if this has changed or not. I have flirted with using OpenOffice for many years but it was never quite 100% compatible with MS Office so I could never fully dispense with Word. However, in my push to open source, I may just write my next talk in OpenOffice.
8. Plink The standard GWAS analysis tool is Plink, originally written by Shaun Purcell. It’s nice but kind of slow for some computations and was not being actively updated. It also couldn’t do some of the calculations we wanted. So in steps my collaborator Chris Chang who took it upon himself to write a software tool that could do all the calculations we needed. His code was so fast and good that we started to ask him to add more and more to it. Eventually, it did almost everything that Plink and gcta (tool for estimating heritability) could do and thus he asked Purcell if he could just call it Plink. It’s currently called Plink 1.9.
10. Inkscape is a very nice drawing program, an open source Adobe Illustrator if you will.
11. GNU Project. Computer scientist Richard Stallman kind of invented the concept of open software. He started the free software foundation and the GNU Project, which includes GNU/Linux, the editor emacs, gnuplot among many other things.
Probably the software tools you use most that are currently free (but may not be forever) are the browser and email. People forget how much these two ubiquitous things have completely changed our lives. When was the last time you went to the library or wrote a letter in ink?