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I have a genuine question, as someone with an interest in mathematics but alas, no natural ability in the area. Is there some kind of 'map' or list of mathemat

I have a genuine question, as someone with an interest in mathematics but alas, no natural ability in the area. Is there some kind of 'map' or list of mathematics areas, problems, etc. that professional mathematicians consult for ideas when conducting research? Likewise physics? It strikes me that we've collectively had at least 5,000 years of math-related research stretching back to ancient civilisations. Surely most discoveries have been made? How much more math ore is there to mine? When I read Richard Feynman claim (paraphrasing) that the joy of physics, math is the theoretical discovery of new knowledge without applications, I wonder where on earth today's mathematicians are looking for new ideas and discoveries? And how do they get research funding? As a CS researcher, funding is hard to come by if I don't propose a tangible product. When I say, e.g., let's re-examine the relational model and extend it, or do something interesting in AI, the question is always the same: what is the product? Mathematicians on the other hand never create a 'product', so I'm confused who, why and where they get their money? Lots of questions, I know, but I'm missing something here.

miscPerson 6y ago
I only really know about the company-sci touching parts, but — Mathematics didn’t have a strong notion of proof until the late 1800s/early 1900s. Contradictory “proofs” in calculus created a need for a more rigorous system of reasoning. (It took until the 1950s to formalize what an ‘infinitesimal’ is — but those were used in early calculus prior to limits.) That gave us electronic computers, as the effort to reduce proofs to things that could be mechanized bore fruit. Every program is a proof! Since then, the idea of proof and structuring mathematics has been radically redefined — introducing things like category theory and type theory as alternatives to set theory, while studying the impact of certain axioms and developing tools around “reverse mathematics”, which is fitting a set of axioms to the theorems you want to be true. Modern mathematics is working on the study of logic topology, extending our reasoning tools to deal with complexities around proving equality in the hopes we can automatically verify mathematics. These tools overlap heavily with AI research. The applications here are what you might guess: new AI methods, data analysis methods, verification of software, etc. The DOD has paid a fair bit to support that research into software verification, for instance. The NSF and others fund “big data” analysis. I’m less familiar with other parts of mathematics, but — The study of knot theory has impacts for physical sciences, from particle physics (anyons) to fluid dynamics. In particular, there’s some work to be done in higher order knot theory and computational knot theory (ie, how to be efficient). MSFT is building a quantum computer based on this — and has suggested that higher order knots might not require as expensive of hardware. In number theory, we still don’t know much about almost all real numbers. Things like Chaitins constant exist — and such uncomputable, normal numbers form the bulk of the reals — but we don’t really know how to get our hands on them. In less exotic research, elliptic curves are used in cryptography. There’s some work I don’t quite understand in building out a homomorphic encryption — where we can operate on encrypted data. In many areas, we’re still working through the algebra-geometry correspondence, which we got hints of 400 years ago but only formalized once topology and category theory were invented — and still are building tooling around. And there’s lots of areas I have no idea about — but I assume are being similarly productive. Ones that I know of, but can’t comment on: fractals, differential equations, bifurcation theory, and chaos theory.
GaussBonnet 6y ago
Background in pure math here. Generally researchers learn about what problems are fashionable as they talk to others in the field. As a PhD student, your advisor should give you ideas for problems to work on. Solutions to old problems tend to open up new lines of inquiry. It's a potentially infinite process, with the only limit being the capacity of the human mind. The odd perfect number problem has been unsolved for thousands of years: https://en.wikipedia.org/wiki/Perfect_number https://en.wikipedia.org/wiki/Perfect_number Why would anyone want to know the answer to this problem? The truth is that the motivation for most pure math research is purely aesthetic. You could also ask arts or english departments "How can you apply your paintings or novels?!?". Some mathematicians (mostly geometers and topologists) are inspired by problems in physics, but most aren't. In truth you never know what structure or theorem might have some future application; number theory was totally "useless" until modern cryptography made (some of) it useful. For just a tiny taste of one small area of modern math, I dare you to click on any of the links here: https://en.wikipedia.org/wiki/Floer_homology https://en.wikipedia.org/wiki/Floer_homology We are flush with structures to investigate. The idea that "most discoveries have been made" is nonsensical in a domain where the discoveries to be made are literally infinite. To give a hint as to the infinite nature of mathematical inquiry, you probably are familiar with the idea of a function mapping a number to another number. A good deal of modern math is involved with much higher order functions; we can have functions that map functions to numbers, functions to functions, functions to spaces, and so on and so on. And then we can consider functions between those functions (and so on). Category theory is an attempt to give a framework to some of these "meta" relations. It should be obvious there is no limit to these structures and no limit to the number of problems one could pose about them.

miscPerson 6y ago

I only really know about the company-sci touching parts, but — Mathematics didn’t have a strong notion of proof until the late 1800s/early 1900s. Contradictory “proofs” in calculus created a need for a more rigorous system of reasoning. (It took until the 1950s to formalize what an ‘infinitesimal’ is — but those were used in early calculus prior to limits.) That gave us electronic computers, as the effort to reduce proofs to things that could be mechanized bore fruit. Every program is a proof! Since then, the idea of proof and structuring mathematics has been radically redefined — introducing things like category theory and type theory as alternatives to set theory, while studying the impact of certain axioms and developing tools around “reverse mathematics”, which is fitting a set of axioms to the theorems you want to be true. Modern mathematics is working on the study of logic topology, extending our reasoning tools to deal with complexities around proving equality in the hopes we can automatically verify mathematics. These tools overlap heavily with AI research. The applications here are what you might guess: new AI methods, data analysis methods, verification of software, etc. The DOD has paid a fair bit to support that research into software verification, for instance. The NSF and others fund “big data” analysis. I’m less familiar with other parts of mathematics, but — The study of knot theory has impacts for physical sciences, from particle physics (anyons) to fluid dynamics. In particular, there’s some work to be done in higher order knot theory and computational knot theory (ie, how to be efficient). MSFT is building a quantum computer based on this — and has suggested that higher order knots might not require as expensive of hardware. In number theory, we still don’t know much about almost all real numbers. Things like Chaitins constant exist — and such uncomputable, normal numbers form the bulk of the reals — but we don’t really know how to get our hands on them. In less exotic research, elliptic curves are used in cryptography. There’s some work I don’t quite understand in building out a homomorphic encryption — where we can operate on encrypted data. In many areas, we’re still working through the algebra-geometry correspondence, which we got hints of 400 years ago but only formalized once topology and category theory were invented — and still are building tooling around. And there’s lots of areas I have no idea about — but I assume are being similarly productive. Ones that I know of, but can’t comment on: fractals, differential equations, bifurcation theory, and chaos theory.

GaussBonnet 6y ago

Background in pure math here. Generally researchers learn about what problems are fashionable as they talk to others in the field. As a PhD student, your advisor should give you ideas for problems to work on. Solutions to old problems tend to open up new lines of inquiry. It's a potentially infinite process, with the only limit being the capacity of the human mind. The odd perfect number problem has been unsolved for thousands of years: https://en.wikipedia.org/wiki/Perfect_number https://en.wikipedia.org/wiki/Perfect_number Why would anyone want to know the answer to this problem? The truth is that the motivation for most pure math research is purely aesthetic. You could also ask arts or english departments "How can you apply your paintings or novels?!?". Some mathematicians (mostly geometers and topologists) are inspired by problems in physics, but most aren't. In truth you never know what structure or theorem might have some future application; number theory was totally "useless" until modern cryptography made (some of) it useful. For just a tiny taste of one small area of modern math, I dare you to click on any of the links here: https://en.wikipedia.org/wiki/Floer_homology https://en.wikipedia.org/wiki/Floer_homology We are flush with structures to investigate. The idea that "most discoveries have been made" is nonsensical in a domain where the discoveries to be made are literally infinite. To give a hint as to the infinite nature of mathematical inquiry, you probably are familiar with the idea of a function mapping a number to another number. A good deal of modern math is involved with much higher order functions; we can have functions that map functions to numbers, functions to functions, functions to spaces, and so on and so on. And then we can consider functions between those functions (and so on). Category theory is an attempt to give a framework to some of these "meta" relations. It should be obvious there is no limit to these structures and no limit to the number of problems one could pose about them.