I have a mathematical program that runs on 64 bit Ubuntu. I need to run it on my Windows 7 workstation, preferably without having to install Ubuntu instead of Windows, because this is a short term need. Is there a way I could run 64-bit Ubuntu as a virtual machine on my Windows 7 box?
You can use VirtualBox and if you really just want Ubuntu for the software i.e. rather than for the whole Unity interface based experience, you might consider Lubuntu instead.
Lubuntu is simply the official LXDE version of Ubuntu and while its spartan blue/grey looks may not win any beauty contests, it does work a lot better than other forms of Ubuntu in VirtualBox.
LXDE is a desktop environment designed for less capable hardware and therefore it is quite suited to virtualisation software, which typically isn't able to use your hardware as efficiently as a bare metal installation.
This really isn't an Ubuntu specific question, but you need to install Cygwin inside your Windows 7 session. Cygwin is a Linux emulator. You can then run your Linux executable from within your Cygwin session.