I believe that any non-trivial idea will sooner or later find application in real life. However "sooner" is better than "later":)

If we look at famous open problems - e.g. Millennium Prize problems - there is no clear practical outcome from their solution (imho). (Well, P vs NP of course will be of greatest importance if one will produce quick algrorithm to solve all NP problems - but it seems it is not plausible).

**Question** what are open problems which solution will have some "real life" outcome in **visible future** ?

My answer. It might be not perfect from various sides, but still...

**Problem** Construction of **short** and effective error-correcting codes.

**Background**
Recent advances in error-corrrecting codes are LDPC and turbo-codes - invented in 90-ies
(more recent polar codes)
produced codes which are highly effective for quite a **long** length of the codewords
(thousands bits or may be even dozens thousands). These codes are widely used in practive,
e.g. 3G-smart-phones use turbo-codes.

http://en.wikipedia.org/wiki/Low-density_parity-check_code

http://en.wikipedia.org/wiki/Turbo_code

However such codes are not effective for short sequences of bits (dozens bits). That is why a problem.

**Practical importance**
In cell networks not only data is transmitted, but various auxilliary information
- e.g. acknowledge receipts (data has been received or not), various channel quality indicators (measuring how bad are propogation conditions).
Amount of such data is not big - dozens of bits - so we need short codes.
Up to recent times this was not considered as important issue, because it might be just 1% of data transmission. But with the advance of smartphones and "always on-line"
feature this appears to be critical problem.
Since "always on-line" means sending very often and very short amounts of data, but for each data transmission one needs to send also auxilliary information - so
the amount of auxilliary information becomes comparable with data transmission.

**Literature**
This problem is mentioned in influential article in telecommunication community
"Is the PHY layer dead?" http://repositori.upf.edu/handle/10230/13026

("PHY(=physical) layer" - is a part where most mathematical-consuming algorithms were concentrated (e.g. error-correcting codes, statstical algorithms for signal estimation etc.)). So the title has a flavor "math is not useful for wireless telecom anymore?" (However now RM (Resource managament) layer is consuming math (from game theory to optimization) so we may not worry :).

theoreticalproblems with practical applications. Come to think of it, if this is an accurate description of what you are after, "practical applications for theoretical open problems" might be a better title than what you have now. – Charles Staats Feb 7 '12 at 23:04However "sooner" is better than "later"... Does this opinion apply to $E=mc^2$ ? – Denis Serre Feb 8 '12 at 6:25