Suppose Mr A. is a graduate student who, for some reason, does not want to go into teaching or academic research. Mr A. did a Doctoral dissertation, but it is in a narrow area. This area is not relevant to several industries which hire mathematicians. Is it prudent for Mr. A to apply to these jobs anyway?
In other words: if one wants to work in industry does the dissertation topic matter?
I know several people who got jobs in industry, doing work entirely unrelated to the topic of their Ph.D. theses (and to anything else they learned as students). The people who hired them seemed to have the attitude that they want to hire smart people who can learn the relevant background information reasonably quickly, can understand the problems they should work on, can make a real contribution to solving them, and can communicate effectively with other employees (not just with other mathematicians). The Ph.D. degree in mathematics strongly suggests (though it doesn't strictly imply) that one has at least some of these qualities. Letters of recommendation and interviews add more of the desired information (or of its negation).
From my experience interviewing people for engineering jobs: not really. What matters more is the quality of one's research, and a general willingness to learn about new kinds of problems and new methods.
The results of pure research are inapplicable as they are anyway... :)
My usual gripe: strictly speaking, the answer is "it depends" and anyone who says otherwise is making assumptions that are unwarranted given how little we know about Mr A (Dr A?) and his situation. The type of industry matters, of course (some have a more established history of employing mathematicians than others), as well as the country that Mr A lives in. So as much as I can appreciate the encouraging words that we can give to Mr A, I want to insist that any answer must by definition have a localized domain of validity that may or may not overlap Mr A's situation.
So I don't know if the topic of Mr A's dissertation will matter, but it seems to me that it shouldn't matter to Mr A: according to the question, Mr A has essentially wrapped up the dissertation and so, to paraphrase Rumsfeld, will have to look for a job with the topic he has rather than the topic he wishes he had.
So if I can offer Mr A some advice, it would be rather to do his homework about the type of industry he wants to join. The fact that the dissertation topic was relevant to the industry may or may not matter, but what will matter for sure is your ability to make the case that you (not your dissertation) have something to offer to your prospective employer. I think scaaahu made a good point in pointing out that some employers/colleagues may have misconceptions about what mathematicians do. But more generally, you should research what the industry does and in what way you can be of help.
(For the record: I know a few mathematicians who have successful careers in various industries. As far as I can tell, all of them were very diligent in preparing for the industry in question, and they targeted one specific type of activity.)
With rare exceptions, nobody reads dissertations except the author and his or her committee. But people might read papers that come out of dissertations.
Credibility radiates outward from one's research topic. If you write a dissertation on Peruvian tree frogs, then you're an expert on Peruvian tree frogs. But you probably know something about Peru and something about other kinds of tree frogs. And presumably you know basic biology pretty well.
A PhD in any math topic implies credibility in math. However, you may want to help the reader see that. The dissertation or paper title may be so full of jargon that a non-expert may not be able to tell what general area of math it's in, or even that it's in math.
For example, my dissertation was something like "Microstructure models with secondary flux". What's that? Physics? Engineering? Math? What kind of math? So I might explain that I looked at partial differential equation models applied to fluid flow.
I am a retired software engineer from a well-known company in U.S. Allow me to answer this question based on my personal experience and observation.
If you decide to find a job in industry, please don't limit yourself to only those companies which you believe would hire math PhD's. You never know. You should send your resume to the companies who offer jobs you are interested in.
However, you need to be aware of the expectations from your boss and co-workers after you start your job. Most people outside the math world have misconceptions about math. For example, they think a math PhD can do calculation very fast. So, if your performance does not meet thier expectations, you'll be in trouble.
Now, to answer your question, does your dissertation topic matter? No, particularly if it's in a narrow area your employer has no use. Is your math PhD useful to them? Yes. because they can use the general math knowledge and problem solving skills you bring to them. Your challenge is how to go from academia to industry and from math proof to real life applications. Once you overcome them, everything would be fine.
I think this pretty much case by case. And in most cases it will not give big advantage, however there are other cases. In our company they do NOT want to take people without PhD (RnD placed in Moscow, Russia of some international company).
Also in some RnD centers of big companies they sometimes need people who will not ask - "what to do?", but people who will invent new tasks. I guess for such position PhD would be preferable.
Also there are different situations when people are hired, one is when they precisely know what kind of person they need e.g. must know "Haskell" or any other specific and quite narrow expert - in this case I guess PhD is not big advantage, unless it is precisely on the field.
However other situations happen - company reorganize or build a new RnD center - so they need to hire in short term (say 1/2 year) dozens experts - there might be lack of such experts on the market - so they can decide to take people without experience and train them. In this case PhD would be an advantage - showing you are smart enough, and hopefully will manage to learn things which are necessary.
PS
If the question is "apply or not apply" - of course - apply. I know examples where positions seemed to be not relevant - but nevertheless people were hired, and actually I am the one of such examples :)
Except in very unusual circumstances, the dissertation is mostly irrelevant in industrial employment. Far more relevant are willingness to learn, ability to communicate with people in other disciplines (learning to speak the language they speak), and a decent breadth in science and mathematics.
I am a software engineer, working in an industry (approximation algorithms, computational lithography, semiconductor manufacturing) completely unrelated to my Math PhD (low-dimensional topology, moduli spaces). Sometimes I wish I majored in Applied Math instead. But then I remember why I majored in Pure Math to begin with and regret nothing.
I don't know about others, but for me going into Pure Math was not about developing a career but about following my intellectual curiosity. That doesn't make much practical sense, but neither does education in Music or Art History. People do that because the subjects are interesting to them, not because their education will be useful to somebody else. Similarly, the topic one one's dissertation should be something of interest to himself, regardless of practical applicability.
That said, education in Mathematics has this odd quality of being useful. Most of the things I do as an engineer involve Calculus, Statistics, Linear Algebra, and a lot of Numerical Analysis and Algorithms, and a little bit of Functional Analysis. Most of the above is a part of any graduate education in Mathematics. The rest could be picked up by an average mathematician in a relatively short time. Computer programming skills (not college level, but industrial quality) would be required too, and, alas, that's not taught sufficiently at Math departments.
Anyway, here's a nice slide that's completely contrary to what I just said:
