8

I am a first year PhD student in mathematics (complex differential and Kahler Geometry) and the other day I had a thought about the difficulty of starting to do research. This is my first post here but I have some in Math Stack-exchange.

First of all I hope this is the right place to post such a question, if it isn't let me know and I will move the post immediately.

Being in the first year of my PhD I am in an intensive study phase in order to get to the limits of research and start working on a project with my supervisor. I am noticing, however, that in order to be able to truly understand a problem (which is within the reach of a doctoral student, so excluding complicated problems that have been unsolved for many years e.g. the existence of a complex structure on the six-sphere) it really takes a lot of study and above all a good knowledge of the basics in other nearby fields of mathematics. Just think of Kahler geometry in which you must have good bases of analysis and geometry (sometimes even algebraic if you want to focus on the various notions of stability), or for example also in the low-dimensional differential topology in which you need to have good basics of algebraic topology and differential geometry. Obviously the more one goes on with the years the more this phenomenon, in my opinion, becomes more and more accentuated. On the other hand, however, thanks to technology we have new tools from which to draw/spread the culture and knowledge in pure mathematics (just think of this site), tools that the researchers of 30 years ago did not have.

At this point, thinking in this light, I asked myself a question: how difficult will it be for a master's degree student to start a PhD in 2050, say? The problems could be two: the first that obviously the research from today to 30 years has gone on and we have gone further and further with knowledge; the second is that university exam programs, to try to cope with the first problematic, are weighed down more and more. The most obvious consequence is that only a few people (the most determined and brilliant) will be able to go ahead and start a research program, or that they will dedicate themselves exclusively to a super specialized topic ignoring everything else (even if I don't see how it is possible in a subject like mathematics, moreover it would be a real shame).

This talk I am giving is exclusive to pure mathematics research, thus excluding the use of a computer as a means to be able to do research in the strict sense of the term (as for example a particle physicist or an applied mathematician can use it to do numerical simulations).

Having said that, I wanted to know what you thought about it and above all what researchers and/or university professors who have been in this field for many years think about it. I hope I have not asked a question that may be trivial, but rather a food for thought. Thanks for reading.

John117
  • 395
  • the notion of writing a dissertation for a doctor's degree goes back more than two centuries; surely the changes over that time have been more substantial than those we can expect in the next decades; so why would you expect a disruptive change? – Carlo Beenakker Jan 29 '21 at 18:09
  • 6
    Imo the biggest factor determining how easy/hard it will be to start doing math research in 2050 will not be the state of our collective math knowledge but the broader state of the world. – Sam Hopkins Jan 29 '21 at 18:10
  • Things will get hyper specialized. – Piyush Grover Jan 29 '21 at 18:12
  • 3
    Your reasoning applies well to old subjects like number theory, where all the low-hanging fruit is picked and remaining morsels require either climbing extremely far out on a branch (specializing) or climbing extremely high up the tree (determined/brilliant). But there are always new, younger subjects emerging with plenty of ripe jewels hanging low — the one that comes immediately to mind for me is graph theory, but I’m sure there are even younger ones. – Alec Rhea Jan 29 '21 at 18:13
  • 1
    Well, unless there is some very big revolution (such as a pervasive use of AI in the proof verification) I guess that a PhD student will work in 20 or 30 years just as today: one starts by learning some subject, and then improves their understanding by "stretching tentacles" in nearby areas. In a first step, a complete understanding of all proofs is not always required, and gaps will be filled later. Of course, the role of supervisor will be an essential one during all the process. I also think that MO or is successors will be important. – Francesco Polizzi Jan 29 '21 at 18:32
  • 5
    @CarloBeenakker On the scale of centuries, one aspect of mathematical research has gotten easier: We no longer need to write our paper in Latin. – Andreas Blass Jan 29 '21 at 18:53
  • 6
    The increase of mathematical knowledge over time can be partly compensated (from the student's point of view) by better systematization and careful foundations for existing knowledge. 18th-century analysis and early 20th-century algebraic geometry relied heavily on intuition to avoid falling into the fallacies that imprecise methods facilitate; nowadays, the corresponding (correct) results can be proved without that hand-waving. – Andreas Blass Jan 29 '21 at 18:58
  • @FrancescoPolizzi ... in 2050 will AI write the proofs? Will AI write the novels (assuming people still read novels)? Will AI write the music? (Something like Facebook's algorithm will accurately suggest music you will like, or even write custom music you will like.) But this is not the place to discuss this. – Gerald Edgar Jan 29 '21 at 19:01
  • 1
    I think there will always be relatively elementary doors opening up into worlds of advanced mathematics now undreamt of. – Lubin Jan 29 '21 at 19:15
  • Possibly helpful is an answer I gave to a related question, When is one 'ready' to make original contributions to mathematics? – Timothy Chow Jan 29 '21 at 21:47
  • 3
    In newer fields there are going to be interesting problems in the foothills that experts will bypass as they go for the higher peaks. – JP McCarthy Jan 29 '21 at 23:06
  • You have to bear in mind that although in the time of Newton there was less mathematical knowledge to learn, Newton also had to learn and practice a bunch of stuff that we do not use any more. There are also new emerging subjects in mathematics which have lots of ''low hanging fruit'' which isn't that hard to pick, the intersection of mirror symmetry/toric geometry might be an area with this type of fruit. – Hollis Williams Sep 29 '22 at 11:55

0 Answers0