Advice for choosing an MSc or PhD. I have recently graduated from the University of Warwick with a 1st Class in my Master of Mathematics(MMath) degree and would really like to continue my education in mathematics by doing a PhD.
The area of maths that I am interested in is Differential Geometry but unfortunately during my final year at university I was not able to do a research project on it. In fact I did not do a research project at all but rather what Warwick University calls a Maths-in-Action Project. This project was very basic and it did not require doing any research at all.
I was just wondering then considering that I do not have any research experience and that I will have to take a year out of education whether it is advisable to do a 1 year MSc degree before going on to do a PhD. I know for a MSc degree you have to do a research project so this will give me valuable experience in research. I am also aware that a MSc degree has to be self-funded but this is not a problem.
Advise on this will be very much appreciated!
 A: I am currently purusing a PhD degree in some second-tier university in US. I do not think you should waste any time in getting a master degree because there is no substantial difference between courses in two levels (some Cambridge Part III courses are definitely research level). The difference is during the first 1-2 years of PhD you decide your field and your advisor, in 3-4 year you work on your theis problem to graduate. While in a master program you are often not required to finish a thesis, and often has no formal advisor, usually only required to finish a certain number of courses. 
However, to be realistic if you want to increase your chances of getting into a better PhD program and be better prepared to do research, then a one year master program is not a bad choice. At least it is better than going to a bad PhD program and work in relative isolation. Most PhD students learn material from seminars and books, lecture notes, etc, and often the advisor do not have time to guide students in every step during their research. Therefore it is necessary to know some basics in your field before you started research, and a good master program with reasonable difficulty courses might be helpful.  Maybe you can try to write a paper in your field(may not be publishable, but still good exercise)? I would guess you will not have trouble to get a decent grade, references, etc after your master program, and you may get into a better PhD program afterwards. 
Also an extra year may give you some elbowroom to decide what you are really interested; I came to my university without realizing there is no professor specializing in my field of interest, and I either need to shift focus or transfer out. So if you are applying for PhD, you should 'know' if your target school's professors' research is compatible with your personal taste. And if you are undecided(a lot of people are only interested in some abstract idea or mathematical structure), then imagine yourself is already a graduate student - who you want to be? what you are interested? with whom you want to work with? You may just apply and decide when the offer comes and school year begins, but I think it make sense to think about this when you started the application process. 

I did not notice that you are planning to learn differential geometry. If yes, then you should check with the top geometry PhD schools in US, UK and France (maybe Russia as well) to see what their programs are like. Differential geometry does not require a lot of foundational reading like algebraic geometry, but it relies heavily on PDE. So you may need to study some PDE seriously. Most of what I wrote earlier still applies to this case (that is, a good master program preparation helps your chance to get into a top PhD program). But if you are convinced to study differential geometry, and believed you could not get a decent research experience in a typical master program, then you should apply for PhD. 
A: If you are interested in doing mathematical research, then you will want to get the PhD. A masters in math usually prepares someone for a career doing analyst-type problems that wouldn't necessarily be interesting for someone who has a passion for mathematical research. 
In the US I don't think there is any need to get the Master's before doing a PhD - most PhD programs only assume you have a Bachelor's and will award you a Master's during the process (usually after you've finished the required classes, but before you're finished your dissertation), but I'm not sure where exactly you're looking to do your graduate studies.
Doing a masters before your PhD may not require any research work (probably just a bunch of classes) so it may not help you with your goal of acquiring research experience (btw, you should research your school of choice first. I could be wrong here). If you want research experience before PhD, you may consider applying for some kind of undergraduate research opportunity (or something analogous for master's students if you choose to do the master's first). 
A: I guess the answer depends on the institution (and more generally the country) you plan to apply to. Just a rough and incomplete guide.


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*In US the entry requirements for most PhD programs is a Bachelor's Degree. In fact most Master's programs are cut off versions of PhD programs (I think).

*In UK MSc is regarded higher than MMath, but is not strictly necessary for PhD

*In Germany a written thesis is an entry requirement for PhD. So MMath won't quite cut it.

