The course objectives are:

To learn about research at the beginning of the PhD program, in order to
(a) critically read published research,
(b) start thinking about a thesis topic,
(c) make an informed decision on a PhD thesis supervisor,
(d) hopefully shorten the time to get a PhD degree and research publications

To work with faculty members of your choice (one per month) whose research you might have interest in, based on reading their research web page. Also keep in mind, that faculty members who have many current PhD students might be less likely to take on an additional PhD student this academic year.

To read research papers and write mathematics. The latter includes getting familiar with LaTeX (or one of its variants such as Scientific Word, lyx), which is the standard software for mathematical writing.

To get comfortable with scientific computing. Most thesis research involves scientific computing since original research means developing new methodology and implementing them. Depending on the research, scientific computing might be implemented in Splus/R, matlab, SAS/IML, C/C++, Fortran, and/or perl/python etc.