What does mathematics look like in the 21st century? I’m in no position to make any declarations, what with not being an expert on math history and all, but I’d like to offer up a couple of brief observations to think on.
If I had to name one candidate for the overall flavor of 21st century mathematics, I’d say complex adaptive systems. Why? Because it encapsulates the transition we’re seeing from the rigid, linear, and static to the complex, nonlinear, and dynamic. I think a lot of this in particular has been motivated by a couple of things: 1. our great and ever-increasing numbers as humans, and 2. the increasing complexity of the technology we use to accomplish our tasks. Given an exponential increase in population coupled with an exponential increase in the complexity of the technology being used virtually every second of every day, some new mathematics were due to emerge. Among the more interesting examples you have things like fractal geometry, cellular automata, and ‘system of systems’. New variations of these and other approaches are appearing daily in academic journals, and some make it to market.
The pace is increasing, to the degree that the landscape is changing faster than anybody can keep up with. That’s technology as a whole. For mathematics, an entirely new era is on its way in, motivated by society and the thirst for new technology. The reigning paradigms of this century will likely be vastly complex networked systems, and how to describe them accurately. This includes anything from social networks to artificial intelligence, transportation systems (including space traffic), economics, neuroscience, biology – almost anything you can think of. What’s becoming apparent is that everything that was off limits to traditional mathematics is becoming accessible through new frameworks.
These are exciting times! The future is bright, and there is surely no end to the amount of adventure a motivated person can have this century.