Certainly, one of the main discoveries of Shannon was that the amount of information can be measured without reference to the meaning of the information. Therefore it is perhaps not too wrong to call Shannon Theory that field, all those fields, where those information measures play an important role. In this sense coding theorems for sources, for channels and multiuser networks certainly fall into this field, but so do applications of information theory outside communication like MDL or maximum entropy techniques in statis- tics, or applications to large deviation theory in probability theory, or to the theory of Gibbs fields.

Some people might find this definition too broad, but at least that is a possible point of view. Of course I do not think it is a very basic question what kind of terminology we adopt. It is more important what kind of problems we are doing. Now the next part of the question would be what are the problems which we consider important and what failures do we see. Again for me this means something else than probably for the majority. I am not very familiar with the applied field. I was not aware that now we are able to achieve channel capacity by algorithmic methods in some very real situations. But I am happy to learn this fact.

From the purely mathematical point of view, there are many problems, very basic problems, which could not be solved so far. I do not think that is a failure, it is probably due to the very difficult nature of those problems. For example, a fundamental problem of coding theory is to determine the maximum number of codewords with a prescribed minimum distance, at least asymptotically. This very basic problem appears extremely hard to solve, even though, perhaps, the algebraic geometry codes provide a tool to get closer to the solution. There are basic problems about zero error capacity. Nobody has any means of computing the zero error capacity of a channel even in principle, except for special cases, although zero error capacity is a more basic concept than ordinary capacity.

Clearly, one would be more interested in transmitting information without any errors than with a small probability of error. In certain devices it is impossible, but when it is possible to transmit without errors, one would like to know what rates are achievable. Many people have worked on this problem and the answers are still far away. It often helps to give new formulations and design new problems which perhaps look more general and more difficult. Sometimes the more general problem will be less difficult to solve. I think it is important to work even on such questions which may look hopeless to solve completely. Small steps forwards may also be valuable.

For example, I just mention one problem on which I have worked recently, that is the problem of channel capacity for a specified decoding metric. There is no hope or very little hope to solve this problem completely because this would include the solution of the zero error capacity problem. That would be a great achievement, but even if this cannot be done, any partial results may bring us closer to understand the difficulties or the workings of such mathematical problems. Of course there are many, many difficult open problems in multiuser information theory. This morning we have heard a talk in which the solution of one of those problems was announced, and I would be very happy if this solution would turn out to be correct. There are plenty of problems in statistics, in combinatorics which have an information theoretical flavor. An appropriate formulation often leads to good answers. Very often we cannot answer problems because we are not able to formulate them properly.

I would like to emphasize, I do not consider information theory a part of communication theory. It is much broader. Communication is just one field of applications. But there are other fields, statistics, physics, biology, genetics (as we saw today) where information measures can turn out to be useful. So, I think it is important to view the field in this generality.