4

(Sorry, last soft question!) Borwien, in The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike, (probably quite rightly) says of the Riemann Hypothesis, that

No layman has ever been able to understand it and no mathematician has ever proved it.

Aside from being one of the most famous problems in mathematics, is this the most "difficult" to understand (by either layman of mathematician!) of all mathematical problems, or are others more "complicated" conceptually?

I realise that this is a highly subjective question (and perhaps one that is not worthy of this site), but I am interested in other problems that are perceived as being "difficult". I realise, of course, that the Millenium problems (themselves, based on the Hilbert problems, I believe) are all strong contenders. But are these the most " difficult" to understand?

For me, the axiom of choice, any problem associated with Cantor, or any definition involved in higher dimensions, such as Hilbert spaces, quaternions, etc. present considerable conceptual barriers, to mention just a few!

martin
  • 8,998
  • 4
    Even more intriguing are the questions that are straightforward to understand yet shockingly difficult to solve, e.g., finding general solutions to the Navier-Stokes equations. These are the problems you know are really difficult because you've ruled out the possibility that it's just your lack of understanding of the question that's preventing you from arriving at an answer. – David H Jun 12 '14 at 22:13
  • The striving for smoothness is a disease that seems to plague most mathematicians! – martin Jun 12 '14 at 22:19
  • RH is definitely easier to understand than say Hodge conjecture since to understand the former you just need basic complex analysis (undergrad material), while the latter requires differential and algebraic topology (grad level material in most places). – Moishe Kohan Jun 12 '14 at 22:20
  • Does educational structure mirror ability to learn/understand? (I don't know the answer - just playing Devil's Advocate!) – martin Jun 12 '14 at 22:22
  • In turn, the easiest equivalent formulation of RH is due to Nicolas and deserves to be better known, see http://math.stackexchange.com/questions/323144/eulers-phi-function-worst-case – Will Jagy Jun 12 '14 at 22:25
  • Robin, I think, no? – martin Jun 12 '14 at 22:26
  • Due to in turn to Ramanujan... – martin Jun 12 '14 at 22:27
  • Robin was a student of Nicolas. The thing called Robin's Criterion is a consequence of the criterion of Nicolas. Having fiddled with both, I can tell you that the Nicolas version is much easier to work with. – Will Jagy Jun 13 '14 at 00:47
  • 1
    Dear Martin, Note that there is a big difference between "difficult to understand" and "difficult to solve". But not always --- e.g. the general conjectures of the Langlands program are difficult to understand (more difficult than the Millennium Problems, I would say), and seem to be difficult to solve! Regards, – Matt E Jun 13 '14 at 01:15

1 Answers1

2

Here is the list of Millenium prize problems listed in order the difficulty in understanding their statements. Here a "understand X" does not include "appreciate significance" or "be able to read research papers papers discussing X". Some of the ordering is a bit random (problems 1 through 3 can be listed in any order.)

  1. RH is far from the "most difficult to understand" mathematical problems. If you took, say, undergraduate complex analysis, you can understand the statement of RH simply by reading the Wikipedia article. This, in part, explain why it attracts so many mathematical cranks. Will's comments will tell you how to get a reformulation of RH without even undergraduate complex analysis, undergraduate calculus would suffice here.

  2. Similarly, in order to understand the problem about NS equation, all you need is an undergraduate class in partial differential equations (and reading Wikipedia article). (Strangely, cranks seem to ignore this one, I guess they just do not like PDEs.)

  3. Same goes for P vs NP, you just need good undergraduate-level CS courses to get this one.

  4. To understand 3-d Poincare conjecture, you need to take a good upper division topology class (which gets you to the definitions of manifolds and the fundamental groups; many top-tier universities teach such classes). If not, then one semester of a standard graduate algebraic topology class will suffice here.

  5. YM-existence and mass gap problem would require you to take a graduate course in differential geometry or topology (bundles, connections, etc).

  6. Hodge conjecture can be understood after taking graduate courses in complex analysis, differential topology and algebraic topology (no algebraic geometry is really needed here even though the problem comes from algebraic geometry). OK, I am assuming that your courses in differential topology and complex analysis cover Hodge theorem and definition of (anti)holomorphic forms in several complex variables. It is a bit of a stretch, but you get it in better places.

  7. Birch and Swinnerton-Dyer (BSD) Conjecture would be the hardest. Only a handful of math departments would offer graduate number theory courses covering L-functions.

However, anybody who got through the first few years of graduate school (in pure math) and is willing to spend few hours (say, less than 10) reading some unfamiliar material, can understand (the statements of) all the Millenium Problems. Does it mean that, say, BSD problem is the "most difficult to understand in the entire mathematics"? Not at all. There are many (less famous) problems in a variety of area of mathematics whose understanding would require much greater time investment (on the scale of years, not hours, unless you happen to specialize in that particular area, of course). For instance, just take a look at this list of open problems in algebraic topology. Or here for problems in model theory.

Moishe Kohan
  • 97,719
  • Dear studious, The Hodge Conjecture asserts the existence of algebraic cycles. So it seems a bit hard to understand without knowing some algebraic geometry. (Maybe you will tell me that you're going to use Chow's theorem to work entirely in the complex analytic world. Okay ... .) Certainly I was able to understand BDS well before I could understand the Hodge Conjecture, and I don't think I'm alone. After all, the definition of the $L$-function of an elliptic curve is pretty easy (just make a Dirichlet series using the number of solutions modulo various primes $p$). Regards, – Matt E Jun 13 '14 at 01:09
  • @Matt E: yes, one can define algebraic cycles as complex analytic subvarieties. – Moishe Kohan Jun 13 '14 at 03:08