- #1
- 12,000
- 2,307
future direction of mathematics
In regard to the question of where math is going and what is the important recent work and most important work needed, I am of course not qualified to say, not having a global enough grasp of what is being done. that being said, i still have an opinion.
for a more qualified view, see the lecture from 1900 by hilbert at
http://aleph0.clarku.edu/~djoyce/hilbert/problems.html
and the commemorative one attempting to keep things going in 2000 at UCLA.
I will make the obvious remark that since the "riemann hypothesis" occurs as problem 8 on hilbert's list, and remnains unsoved today, it is still very important. one may also consult the list of Clay institute problems, including this and the so called "hodge conjecture", for other questions that at least could be worth a lot of money to the solver.
In my own opinion, one of the most important phenomena has been the reconnecting of mathematics wihth physics, with the consequent energizing of both subjects. Witten and the people doing quantum gravity have made big advances possible in enumerative algebraic geometry at least.
string theory also benefits in reverse from knowledge about riemann surfaces. this interaction between physics and analysis was key to riemann's confidence in the correctness of the results he ahd insufficient proofs for. Physical insight still stands today as as confidence booster for results that the mathematicians only achieve a foundation for sometimes much later.
I would also say the work unleashed by wiles, and the outstanding problems generalizing his results on modularity of representations is important. i am even less qualified here.
my own work is in a specialized area begun by riemann, of understanding the relationships between curves and abelian varieties, and their moduli spaces. as such it keeps me interested in the developments surrounding moduli spaces coming from quantum phenomena however, e.g. cohomology of moduli spaces, as well as singularity theory especially non isolated singularities, and vanishing cycles. Higgs bundles in higher dimensions also seem to play a role in problems of interest to me but are hard to compute with effectively when one gets away from curves.
anyone else?
In regard to the question of where math is going and what is the important recent work and most important work needed, I am of course not qualified to say, not having a global enough grasp of what is being done. that being said, i still have an opinion.
for a more qualified view, see the lecture from 1900 by hilbert at
http://aleph0.clarku.edu/~djoyce/hilbert/problems.html
and the commemorative one attempting to keep things going in 2000 at UCLA.
I will make the obvious remark that since the "riemann hypothesis" occurs as problem 8 on hilbert's list, and remnains unsoved today, it is still very important. one may also consult the list of Clay institute problems, including this and the so called "hodge conjecture", for other questions that at least could be worth a lot of money to the solver.
In my own opinion, one of the most important phenomena has been the reconnecting of mathematics wihth physics, with the consequent energizing of both subjects. Witten and the people doing quantum gravity have made big advances possible in enumerative algebraic geometry at least.
string theory also benefits in reverse from knowledge about riemann surfaces. this interaction between physics and analysis was key to riemann's confidence in the correctness of the results he ahd insufficient proofs for. Physical insight still stands today as as confidence booster for results that the mathematicians only achieve a foundation for sometimes much later.
I would also say the work unleashed by wiles, and the outstanding problems generalizing his results on modularity of representations is important. i am even less qualified here.
my own work is in a specialized area begun by riemann, of understanding the relationships between curves and abelian varieties, and their moduli spaces. as such it keeps me interested in the developments surrounding moduli spaces coming from quantum phenomena however, e.g. cohomology of moduli spaces, as well as singularity theory especially non isolated singularities, and vanishing cycles. Higgs bundles in higher dimensions also seem to play a role in problems of interest to me but are hard to compute with effectively when one gets away from curves.
anyone else?
Last edited: