Alexander S. Mishchenko - Böcker

In the last few years the use of geometrie methods has permeated many more branehes of mathematies and the seiences. Briefly its role may be eharaeterized as folIows. Whereas methods of mathematieal analysis deseribe phenomena 'in the sm all " geometrie methods eontribute to giving the picture 'in the large'. A seeond no less important property of geometrie methods is the eonvenienee of using its language to deseribe and give qualitative explanations for diverse mathematieal phenomena and patterns. From this point of view, the theory of veetor bundles together with mathematieal analysis on manifolds (global anal- ysis and differential geometry) has provided a major stimulus. Its language turned out to be extremely fruitful: connections on prineipal veetor bundles (in terms of whieh various field theories are deseribed), transformation groups including the various symmetry groups that arise in eonneetion with physieal problems, in asymptotie methods of partial differential equations with small parameter, in elliptie operator theory, in mathematieal methods of classieal meehanies and in mathematieal methods in eeonomies.There are other eur- rently less signifieant applieations in other fields. Over a similar period, uni- versity edueation has ehanged eonsiderably with the appearanee of new courses on differential geometry and topology. New textbooks have been published but 'geometry and topology' has not, in our opinion, been wen eovered from a prae- tieal applieations point of view.

In the last few years the use of geometrie methods has permeated many more branehes of mathematies and the seiences. Briefly its role may be eharaeterized as folIows. Whereas methods of mathematieal analysis deseribe phenomena 'in the sm all " geometrie methods eontribute to giving the picture 'in the large'. A seeond no less important property of geometrie methods is the eonvenienee of using its language to deseribe and give qualitative explanations for diverse mathematieal phenomena and patterns. From this point of view, the theory of veetor bundles together with mathematieal analysis on manifolds (global anal- ysis and differential geometry) has provided a major stimulus. Its language turned out to be extremely fruitful: connections on prineipal veetor bundles (in terms of whieh various field theories are deseribed), transformation groups including the various symmetry groups that arise in eonneetion with physieal problems, in asymptotie methods of partial differential equations with small parameter, in elliptie operator theory, in mathematieal methods of classieal meehanies and in mathematieal methods in eeonomies.There are other eur- rently less signifieant applieations in other fields. Over a similar period, uni- versity edueation has ehanged eonsiderably with the appearanee of new courses on differential geometry and topology. New textbooks have been published but 'geometry and topology' has not, in our opinion, been wen eovered from a prae- tieal applieations point of view.

This is a volume originating from the Conference on Partial Differential Equations and Applications, which was held in Moscow in November 2018 in memory of professor Boris Sternin and attracted more than a hundred participants from eighteen countries.

This is a volume originating from the Conference on Partial Differential Equations and Applications, which was held in Moscow in November 2018 in memory of professor Boris Sternin and attracted more than a hundred participants from eighteen countries.

This book consists of reviewed original research papers and expository articles in index theory (especially on singular manifolds), topology of manifolds, operator and equivariant K-theory, Hopf cyclic cohomology, geometry of foliations, residue theory, Fredholm pairs and others, and applications in mathematical physics. The wide spectrum of subjects reflects the diverse directions of research for which the starting point was the Atiyah-Singer index theorem.