I don't know if this is an acceptable question to make here, but I wanted to know what kinds of things are being researched that can lead to major breakthroughs in math. I'm not talking about usual research but something that can be really big and promising.
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There are essentially two different ways to approach the question that might lead you to different conclusions.
But before that, I think it would be nice to share a quote about the current state of knowledge in mathematics, and where we are headed. This is paraphrased from an anatomist who was remarking on the state of his field near the turn of the century.
As our level of mathematical knowledge increases over time, it becomes correspondingly difficult to comprehend, let alone answer, the questions that remain. Some of them fall into the category of easy questions with incredibly complicated answers (like Fermat's last theorem), while some are seemingly vague and nebulous to all but the specialists that engage with them. Questions like P-NP are considered to be very deep with incredibly complicated answers, yet it would take quite a while to explain to someone without extensive background what the questions surrounding P-NP are, and why they need to be answered.
The first approach to the question would be "what is the current trend in applied mathematics?" I don't have very much experience with the subject, but there are very important open questions in computer science and number theory, such as the aforementioned P-NP questions. As stated before, integer factorization is highly relevant to cryptography, and a "fast" algorithm to factorize large integers would be considered quite groundbreaking.
The second approach would be to ask about the trends in "pure" mathematics, or just mathematics for mathematics' sake. Current research is highly specialized. It is difficult to prioritize one field over another. Even highly abstract fields like topology have important applications to other theoretical fields. Theoretical physics has borrowed heavily from group theory and topology. Mathematicians like to ask questions about things without regard to practical application. We like to classify manifolds because it's fun (the classification of manifolds is not complete, however, but it is a highly studied field). It just so happens that manifolds are enormously useful to frame problems in physics and other sciences.
I am not sure this is a 100% match, but these are of interest to a wide audience: http://www.claymath.org/millennium/
I just wanted to add that sometimes results from "the usual research" happen to "hit on" something "phenomenal", or, as is often the case, the implications of some result from "usual research" has far-reaching consequences that one could not have anticipated.
Theory often "trickles down" to applications, in ways that one cannot necessarily know, and in turn, "needs in the real world" often motivate breakthroughs in theoretical research.
So I wouldn't dismiss, outright, "the usual research."
I simply felt obliged to point out that the foundations of "breakthrough" research are years in the making, and such work, behind the scenes, often goes unnoticed. I didn't take your question to be dismissive; it's just that those credited for breakthroughs wouldn't be in the spotlight (historically, or in the future) were it not for the "usual" work of his/her teachers, predecessors and/or contemporaries.
I think, P vs NP is currently the problem with the most important implications to society. It concerns the complexity of algorithms, which can be described as the asymptotical growth rate of the number of calculations needed by a computer to solve a problem. P denotes the class of problems like solving systems of linear equations and calculating the determinant of a matrix which can be solved comparatively easily (in a number of steps proportionaly to a P olynomial in the number of digits of the input, hence the name). NP is a class of problems like the factorization or discrete logarithm problem which may not be easy to solve but whose solution can be easily checked if one is given the correct solution. Clearly $P \subseteq NP$. Most mathematicians believe that $P \ne NP$, meaning that there exist problems belonging to NP, but not to P.
The most currently used encryption algorithms rely on the difficulty to solve certain NP-Problems. For example the RSA cipher could be deciphered if one had a fast algorithm that given the product $pq$ of two large primes $p$ and $q$ (somewhat like 50 or 100 digits each), returns those two primes.
In, other words, our whole encryption system could break down if someone published an efficient algorithm for solving NP-complete problems (which are "maximally hard" NP problems, i. e. problems which enable us to solve every NP problem efficiently if we can solve them efficiently; the factorization problem is presumably not NP-complete and so it could be possible that you could factor an integer quickly but you are not able to find a Hamilton circle in a graph in polynomial time).
On the other hand, if someone proved $P \ne NP$ we could be very optimistic that our current encryption standards are not effectively breakable (althou.
Joseph Teran at UCLA works on things like computer graphics and virtual surgery, and has a lot of cool videos (showing output from his algorithms) on his web page. Ron Fedkiw at Stanford also does cool computer graphics stuff.
The "deep learning" algorithms for machine learning being developed by Geoffrey Hinton, Andrew Ng, etc. seem very exciting. This area is making a lot of progress right now.