One of the highest authorities in modern mathematical sciences. Through his pioneering and monumental work in mathematical sciences, especially in functional analysis-which has experienced tremendous development this century and not only affected other areas of mathematics but has also provided indispensable mathematical tools for the physics of elementary particles and quantum mechanics-he has brought up and inspired many prominent mathematicians in the course of his creative career. He has provided key ideas and deep insights to the whole of mathematical sciences and made outstanding contributions to the advancement of the field.
Abstract Functions and Linear Operators
On Normed Rings
Irreducible Unitary Representations of Locally Compact Groups (With D. A. Rajkov)
Generalized Functions 1-6 (With G. E. Shilov and others)
Izrail M. Gelfand Collected Papers, Three Volumes.
Through pioneering and monumental works especially in functional analysis, Dr. Gelfand made outstanding contributions to the advancement of mathematical sciences. In the course of his creative career, he also brought up and inspired many prominent mathematicians through collaboration, who, in turn, continue to play leading roles in mathematical sciences.
Functional analysis, which studies infinite dimensional spaces of functions by analytic and algebraic means, is one of the areas in mathematics which experienced tremendous development in this century. Its developments not only affected other areas of mathematics, but provided indispensable mathematical tools in the physics of elementary particles and quantum mechanics as well. Dr. Gelfand played crucial roles in these developments.
His doctoral thesis in 1938 on commutative normed rings and his work on the non-commutative rings of linear operators on Hilbert spaces provided decisive steps for subsequent developments of functional analysis, and deeply affected such areas as algebraic geometry and physics.
Groups, which describe symmetries in nature, are basic objects of study in mathematics and physics. Dr. Gelfand obtained numerous fundamental results on unitary representations of locally compact groups, semi-simple Lie groups and even infinite dimensional groups. His works not only brought about revolutionary changes in representation theory and physics, but had enormous impact on number theory and geometry as well.
The six volumes on generalized functions, published in collaboration with several co-authors, also contain accounts on such diverse topics as differential equations, representations, homogeneous spaces, integral geometry, automorphic functions, and stochastic processes, and continue to be rich sources of inspiration.
The extraordinary collection of over 460 research papers and monographs by Dr. Gelfand also contain remarkable results on the inverse spectral problem, topological invariance of the index of elliptic operators, geodesic flows on homogeneous spaces, co-homology of infinite dimensional Lie algebras, integral geometry, and numerical analysis. The breadth and depth of his achievements cause astonishment.
Throughout these works, Dr. Gelfand often sees unexpected deep connections between previously unrelated matters and obtains results which lead to new lines of further development.
For over 40 years, Dr. Gelfand has been conducting an extremely creative and inspiring seminar, at Moscow University, which turned out many prominent mathematicians.
Dr. Gelfand is still at the forefront of creative activity in research on hypergeometric functions. Key ideas and deep insights provided by Dr. Gelfand are expected to have lasting influence on the development of mathematical sciences.
1. This lecture is based on my experience with my work in pure and applied mathematics, neurophysiology, cell biology and cybernetics.
The first observation I have made it that classical mathematics, which is extremely useful in many domains of science, statistics, and technology turns out to be useless in many other domains such as biology, medicine, psychology, etc. While thinking over these matters, I found out that there are profound reasons for these things and, consequently, we must look at these matters from a much broader picture. The final step took more than three months of hard work and I am very grateful to the Inamori Foundation which gave me the opportunity and the pretext to do it. I am also happy that I have come to the conclusions which from my point of view fit the principles of the Inamori Foundation.
2. In this lecture I will explain that there are two archetypes which have been built into Man from the very beginning, and that there is a duality which is caused by the contradictions between these two archetypes.
In the second archetype Man is a part of all living nature and cannot separate himself from it. Even if he could, it would only be temporarily and then only with the understanding of the limits of such a separation. Perhaps this is the point which constitutes the difference between cleverness and wisdom. We know so little about living systems that it is hopeless to understand the whole picture from our knowledge of small isolated parts even though they are very remarkable, as for example, in the case of the genetic code.
At a first glance the dualism is not universal and mathematics, for example, is connected only with the first archetype. But I think that mathematics clearly belongs to the second as well.
3. It seems to me that one of the main features of the modern world is its extreme globalization and the worldwide dissemination of the problems caused by this globalization. All the modern means of communication (cars, airplanes, telecommunications) practically turn the world into a united system with strongly interactive parts. But we cannot say that this is true for the spiritual aspects of humanity. Therefore there is a strong imbalance between the logical and the technological side (first archetype) and the spiritual side of life (second archetype).
In my lecture I will note some of the very serious problems for mankind caused by this imbalance.
4. There are two essential reasons for the necessity of creating adequate languages.
One of them is that globalization causes the necessity of interacting (communicating) with many different parts of the world in which there are different traditions, cultures, and so on. And if there is no what I call Oadequate languageO, the misunderstandings which arise are dangerous.
The other reason is that this, let us say, contradiction exists not only between different parts of the world, but also between the two archetypes themselves. If the language is not adequate, the second archetype will be suppressed, because the first archetype has many more capabilities.
Of course, no adequate language can unify both of these archetypes which are two sides of Man, but at least it gives them the possibility of interacting. In my lecture I will try to explain a bit about the notion of adequate language.
5. For the mathematics of the 20th century the notion of structure is very important. I think that this notion may be useful for the creation of adequate languages. The elementary components of the structure I will call structural units. These structural units must satisfy three conditions:
1) the inner structure of the structural unit is much more complicated than the way in which it interacts with the outside world;
2) a part of a structural unit is not a structural unit;
3a) the principle of reduction: the parts of the structural units which do not function are eliminated, as for example in the process of evolution.
3b) the principle of abundance; the non-functioning parts of the structural units manage to find a job within the structural unit.
There are many interesting examples of types 1, 2, 3a and 1, 2, 3b in biology, sociology and so on.
6. In the last part of the lecture there will be some discussion of the mathematicians’ responsibility toward society and afterwards some remarks about some new directions in mathematics.