Inverse problems are natural for many applications. For many centuries people are searching for hiding places by tapping walls and analysing echo. This is a particular case of an inverse problem. Generally, inverse problems are those of finding some characteristics of medium from knowledge of some fields interacting with the medium. These fields (or some of their characteristics) are usually measured outside the medium, for instance, on its boundary.

Mathematically, a problem may be formulating in the following way: Assume the behaviour of a field is described by a differential equation Pu = f where f is a source and coefficients of P reflect properties of the medium. Assume we are able to measure u outside M, where M is a region occupied by the medium. What can we say about coefficients of P (and sometimes f)?

The name ``inverse" has come from the fact that it is traditional for mathematics to consider the problem of finding u in the case when P and f are given. Then the problem of determination of P from u may be regarded as ``inverse" to the one described above.

Although inverse problems are quite usual for applications, interest towards them in mathematics was provided mostly by the development of quantum mechanics. It was Heisenberg who conjectured that quantum interaction was totally characterized by its scattering matrix which was a particular case of inverse data at infinity.

I study mostly inverse boundary problems in several variables. In the one-dimensional case the corresponding problems were solved in the fifties in classical works by Borg, Gelfand and Levitan, Krein and others. Although since that time much effort was made to generalise one-dimensional results to several dimensions, many difficulties are still not overcome. In my last works, I deal with the multidimensional Gelfand inverse problem. This is the problem of finding an elliptic operator from the knowledge of its spectral data on the boundary of M. And although the problem itself sounds very abstract (dealing for example with operators on differentiable manifolds), it models a number of important practical problems arising in medical imaging, geophysical prospecting, etc.

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