I'm an undergraduate-level self-taught mathematics student. I'm estimating I might be around the second year of the degree on mathematics in terms of general knowledge. While I love the subject, i'm a little puzzled by the state of the art today:
I've been wondering if i've got enough knowledge to read some actual academic publications; but, to my surprise, even with a basic knowledge of fields like analysis, topology, abstract algebra, number theory, set theory, linear algebra, field theory and functional analysis, I couldn't even assess the topic of the publications by reading the title. I don't think the problem is poor titling tendencies, so i'm left with two options:
- Academic mathematical journals (like AMS) are far too advanced publications for anyone with anything less than a PhD.
- The topics are indeed intractable for anyone but very specialised experts in some specific domain of mathematics.
The first option seems ridiculous to me for a reason: PhDs and masters are supposed to refine the knowledge a student already has (possibly in an specific branch of math), not teaching whole new branches of mathematics with hundreds of sophisticated standard definitions. While PhDs are supposed to have a significantly higher understanding than ordinary graduates, they should at least be able to speak the same language, so to say.
The second option makes way more sense: math is profound, so how sophisticated your research topic may be depends on your level of expertise in some domain...but that makes me wonder: why would all experts prefer to publish results about specific, particular, almost cryptic subjects for a narrow audience instead of encompassing, insightful results for a more general audience? Is this some kind of institutional requirement? How is that tendency beneficial for mathematics in the long run? Are mathematical journals generally readable for graduated students?