The question in brackets in the title is my main mathematical question, but does not reflect my initial motivation for writing this post. It is in fact above all for personal reasons that I'm affording to call on your experience and your advice.
I will be starting a PhD next year. I don't know yet with whom, or exactly what I want to get into. I have long been interested in the topology of manifolds, and seduced by the algebraic approach to problems of a topological nature. I like algebra, but questions of purely algebraic nature have always interested me less: for example I have never been concerned with arithmetics or technical commutative algebra.
It has been clear in my head for a long time that I will be doing algebraic topology as a research subject. I really enjoyed all of the introductory courses, then came an introductory course on model categories. For the first time, I completely forgot why I was there, what I was doing. Now I have a lot more perspective, I know how to appreciate the benefits of this theory and I understand its motivations. However, I am still as repelled by the "obscure formal arguments" side that this theory can have. Then came a course on infinity categories/operads, which I decided to abandon before the end, telling myself that the day I need it, I will take it up calmly, having clear ideas and motivations. The reason I am describing that it is to emphasize my view of mathematics as I like them: having concrete motivations for studying something, if possible coming from topological situations, and use formal and technical theories if it is appropriate. I love abstract theories but only if I can see why they are useful.
I am currently working on a master's thesis, and I will have to find a PhD before the end of the year. I am currently doing stable homotopy theory, I really like it for now despite the profusion of algebra. In particular, I'm studying Thom spectra, and I was initially very seduced by their link with the classification of manifolds up to cobordism. Also, I had for a while for project to discover this theory (stable homotopy) because of its power to prove concrete results ("external to the theory"): which spheres have multiplicative structures, how to find a good framework to compute various cobordism groups, how to properly study loop spaces, understanding the homotopy groups of spheres. I must admit that I am a little confused by the subject on which I am offered to work: is $H \mathbb Z / p^k$ an $E_n$-Thom spectrum for $ n> 2 $. It's interesting for sure, but I have no idea of the motivation to study this question, if not a motivation that I would call a "formal motivation": it is formally natural to ask this question. But I have no idea of the geometric issues that may be behind it. We are gradually moving away from Mahowald's theorem which is directly linked to the cobordism of manifolds. What's the point ?...
If I decided to put everything on the table on mathoverflow, it is because the people who supervise me do not seem to understand my disarray. I'm very interested in stable homotopy theory, but when I'm working on something, I want to know "why". I want to know the concrete motivation. So far, I have let myself go and have worked on the subjects that have been proposed to me, because I have not met anyone who has really been able to advise me. Today I can no longer afford that, I have to find a PhD and if I choose the wrong person or the wrong subject (in a general sense) it will have too much of an impact on my future.
Here are some mathematical questions: are some people working on stable homotopy theory from a purely geometric prospective ? How much has stable homotopy theory moved from its initial geometric goals ? Is it necessary to do algebraic geometry to do research in stable homotopy theory ? I know that people here like really concrete questions and I am sorry to be that confused and clumsy. I just hope that, first, it is ok to write this kind of post here; that maybe some of you will identify with my experience and will be of good advice, or that experienced researchers in homotopy theory will kindly accept to help me get on the right path. I really feel lost right now.