# How to properly address an unsolved problem that is very hard to solve in mathematical thesis?

I'm preparing my thesis presentation and I'm trying to address a situation in which the general case has no known solution and is an open, very hard to solve, problem. It involves non-isolated singularities, so, if you're a mathematician, you could have an idea regarding this kind of difficulty.

How can I spell this out in a more appropriate way? I don't want to say "one should notice that solving this problem in the non-isolated singularities case is very hard". It seems out of place and subjective (I mean, what do I mean by "very hard"? That the case I solved in my thesis was easy?).

• Is there previous work? If so, the difficulty of the problem should be evident through your explanations (and possible conjectures) towards why previous works haven't yielded a complete solution. Commented Nov 16, 2017 at 14:49
• Is it provably hard in some sense? For example, in computer science, an optimization problem may be difficult because it has been proven to be NP-complete (e.g. traveling salesman), or its complexity itself may still unknown (e.g. factorization of composite integers). Commented Nov 16, 2017 at 15:44
• Do you mean probably as in "there is a probability that it's hard" or in "I can proof that the mathematical knowledge so far is not enough to solve this problem" or "I can proof that this is not easy - somehow"? It is simply because this singular set might have topological properties for which some techniques will not work. Commented Nov 16, 2017 at 15:48
• A problem with no known solution isn't quite the same as a very hard problem. In fact, one is rather the opposite of the other. Commented Nov 17, 2017 at 10:47
• Shouldn't the title be "How to properly address an unsolved problem subjectively as 'very hard to solve' correctly in a mathematical thesis?" Commented Nov 17, 2017 at 15:34

I am assuming you're talking about a presentation to be given at the defence of a PhD thesis.

Let me address a few points you raise.

• It is perfectly appropriate to include subjective judgements about the difficulty or significance of certain results in a mathematical thesis (or paper). Mathematicians make these judgements all the time in deciding what to work on and in evaluating other people's work. What matters is that you have a solid rationale for the judgements you make. In your situation, the rationale for considering the general case "very hard" is that it involves non-isolated singularities. That seems perfectly reasonable to me.

• If you're worried about presenting subjective judgements as if they were statements of fact, there are various standard forms of wording you could try. Instead of writing "solving this problem... is very hard" you could write "solving this problem... appears to be very hard" or "solving this problem...is generally considered to be very hard" (assuming the latter is true). If there are published works affirming that the general case is hard, you can cite them in support of your point.

• Finally, asserting that the general case of a problem is very hard does not imply (in any sense of that word) that a special case is easy, only that it is easier. But that is completely fine: experienced mathematicians are well used to people restricting themselves to more tractable special cases. I don't have any examples to hand, but I have read many papers in which the author does exactly this. This is not interpreted as an "admission" that the special case is "too easy". (In fact, often when someone does prove something that really is too easy, their rhetoric is the opposite: they try to puff it up to make it sound more difficult than it really is.)

• Totally agreeing on the third point. Once you have devoted a whole thesis to a topic, it seems relatively easy to you because you are now an expert on it. But objectively, this problem might still be hard. On the other hand, people telling me in talks that the problem they solved was oh so hard immediately seem suspicious.
– Dirk
Commented Nov 16, 2017 at 12:27

I don't think you have anything to worry about. Presumably you and your advisor have agreed that the problem you did solve is hard enough to justify a thesis, else you would not be at the presentation stage. That's what your audience will judge. A small improvement in understanding the zeroes of the zeta function could make a fine thesis; no need to apologize for not settling the Riemann Hypothesis. Just put your work in context. Keep the audience interested in what you have done.

Finally, almost all the time the formal thesis defense is just a formality. You want to do it well, but need not fear failure.

Can you say something along the lines of "The general case is a long-standing open problem..."

So, perhaps explain that the general case is unsolved, but, with the following assumptions, the following solution is relevant for this particular case or cases...