Question: What would future historians see as the general directions of mathematics in the period of 1970s -- 2020s?
What I look for in an answer: a pointer to a review paper on this topic would be enough, something titled like "Major Directions of Contemporary Mathematical Research, 1970--2000". If such a paper doesn't exist, I hope for an answer that can read like an abstract to such a paper.
Background:
Yesterday at dinner, Cloudy Quartz (my mom) asked me what modern mathematicians are doing, and I was at a loss. I could see the entire building of modern mathematics, but most of it seems to be completed before the 1970s. There are many unsolved questions, but I don't know where something truly new can grow.
And if mathematics does not have some new field to grow, what is left other than polishing the already mature fields, or solving specific applied problems using old tools?
In the end, I gave a small lecture on the history of mathematics. It ends with certain modern achievements in solving particular hard questions (Fermat's Last Theorem, Poincare conjecture), and solving day-to-day questions from science (numerical solvers, analysis of neural networks, quantum path integrals) but no new fields that I could point to.
After some more reflections, I thought of a few trends in contemporary mathematics:
- Formalizing mathematics, teaching computers mathematics, designing AI systems that can do mathematics
- Whatever modern algebraic geometers and topologists are up to, in their vast abstractions.
- Theoretical computer science: the P=NP problem, quantum computing algorithms, cryptography, etc.
- Justifying methods that are found useful in theoretical physics, such as path integrals.
- Solving all the messy problems in applied fields, as mathematicians has always done.
So my question is basically, in the eyes of a mathematical historian from the future, what might be the historical trends of mathematics in the 1970-2020 period?
Appendix: a list of the major fields of mathematics and their periods of major activities.
- Euclidean Geometry: Ancient Greece, then 19th century.
- Noneuclidean geometry: 19th century.
- Projective geometry: a little bit of it in the 15th century, then greatly popular in the 19th century.
- Topology: 19th century (Analysis situs) to around 1970s (algebraic geometry).
- Calculus: 17th century to 19th century (Cauchy's formalization of calculus).
- Abstract algebra: 19th century (Galois, Abel) to about 1960s (Algebra (MacLane and Birkhoff, 1961) already contains all the main areas of modern abstract algebra).
- Algebraic geometry: 17th century (Newton's classification of cubic curves) to 19th century to about 1970s (Grothendieck).
- Probability: 16th century to 1930s (Kolmogorov) to 1970s (stochastic calculus).
- Logic: ancient Greece, medieval Europe, 1870s to 1970s (Godel, Turing, Paul Cohen).