Is analysis an area of maths we understand fully, or are there many open questions?

Question

I apologise for my naivety, and just want to clarify I know very little about analysis, except it involves the fundamental proofs of calculus. That being said, analysis seems like an area that would be interesting to research if I ever got to that stage, so was wondering if there are many open questions left in analysis, and if so, what would be an example?

Many thanks.

Is there an example that you could give, or do they not have simple names? — Jamminermit, Feb 28 '20 at 16:01
See Innocent looking open problems ... AND Are there any series ... AND the paper Real Analysis in Computer Science: A collection of Open Problems AND "A fixed point theorem for differentials?" in this blog post AND this search. — Dave L. Renfro, Feb 28 '20 at 16:09
Did you read the Wikipdia article about unsolved problems in mathematics ? A major problem in analysis that is still quite open is determining whether a given number is rational. Unknown examples include the Euler-Mascheroni-constant, $\ e+\pi\ $ and $\ e\cdot \pi\ $ — Peter, Feb 28 '20 at 16:19
You would probably get a much clearer understanding of your own question after a first year graduate course in analysis which goes deeper into the topic beyond just fundamental proofs in calculus. Also, this would be a good investment of your time in preparation for any topic of mathematical research. That's why a one year course in analysis is required for every mathematics graduate program on the planet. — Lee Mosher, Feb 28 '20 at 16:36
@Peter: The problems you mentioned are not really from analysis, but would probably fall under the purview of number theory, probably MSC2010 classification 11J72 (11: Number Theory; 11Jxx: Diophantine approximation, transcendental number theory; 11J72: Irrationality; linear independence over a field). However, it's a very good bet that a lot of hard-core analysis will be involved in any proofs of these results that might eventually be discovered. — Dave L. Renfro, Feb 28 '20 at 16:59
@DietrichBurde I love most areas of maths, but calculus is something I find the most intuitive to understand I suppose. — Jamminermit, Feb 28 '20 at 17:29

score 1 · Answer 1 · answered Feb 28 '20 at 16:22

If we are dealing with headings as broad as "analysis" then this includes analysis of ordinary and partial differential equations (ODE and PDE) which is a huge, huge part of mathematics. The study of PDE might be the single biggest field of mathematics if there were a way of measuring such a thing.

Is analysis an area of maths we understand fully, or are there many open questions?

1 Answers1