I am a master student of some engineering (other than EE/CS) in a decent US institution who will apply for a PhD in mathematics in Fall 2014 (possibly in PDE / applied analysis, but not sure). I finished with three A's and five A+'s in eight upper-undergrad / grad math courses, along with four more to come this fall. Since my intention for a math PhD came pretty late, I am only able to look for supervisor this semester. The sad thing, as expected, is that all three potential supervisors declined my request softly. My previous engineering research experiences are mostly experimental, with little to do with math.
So I have roughly one full semester left, and my questions are:
Is it possible to gain any sort of research experience in this semester without a supervisor?
If not, then how can I show my research potential by other ways?
Will the three LORs solely from course lecturers be convincing, even if the lecturers are well-known, and I did pretty well in their courses?
In addition, is abstract algebra a must or at least very important in math PhD application? Since I have to make a choice between Algebra II and Dynamical systems, and I don't have Algebra I.
Thank you for your responses and criticisms.