You definitely don't want to just pull the plug this can cause problems with the SD Card and filesystem. 

From the command line or a terminal window you can enter the following to do a clean shutdown:

sudo shutdown -h now


or from the LXDE GUI

click the shutdown button (red power button) in the menu bar at the bottom right corner of the screen. 
then click shutdown.

Once it says "system halted" you can remove the power cord if needed

How do I turn off my Raspberry Pi?
I understand that it's possible, thankfully, to have your dissertation in a different topic from your master's thesis, but I seem to have a problem (see below). Do you know any other problems might I run into?

Background on me:

I am finishing up my master's in mathematical finance and now plan to pursue a PhD with my dissertation being in stochastic analysis/calculus, a branch of mathematics used in finance and physics.

My background on stochastic analysis:

I have had 3 classes related to stochastic analysis and 2 classes applying the concepts.

My thesis, being on the P-side of mathematical finance, has extremely little relevance to stochastic analysis, which is more used in the Q-side of mathematical finance.

The ONLY time stochastic analysis ever came up was encountering something in a paper that gives probabilities in the Q-side, and I thought of using a theorem to convert them to the P-side but later found out the paper was unnecessary or inapplicable (I forgot; I think I figured out another solution).

One problem I seem to have:

Some PhD applications seem to require a PhD proposal including a hypothesis, literature review and methodology.

I am not quite sure how mathematical research is conducted. Most of my research has been mainly in finance and statistics. What might I be able to do about this?

I am considering taking a second master's if needed but hopefully it need not come to that.
