Vladimir G. Boltyanski, Doctor of Sciences, Professor, Lenin Prize winner, Beruni Prize Winner, Corresponding Member of the Russian Academy, Head of Combinatorial Geometry Group of Steklov Math. Institute (MIAN) of the Russian Academy of Sciences, Researcher Titular D of the Center of Investigations in Mathematics (CIMAT) in the are of Basic Mathematics. 19431945:
Soldier in the Soviet Army. Scientific Degrees: 1951:
Ph. D in Phys.Math. Sciences 19511978: Scientific worker, then Senior research fellow of the Dept. of Diff. Equations of the Steklov Math. Institute (MIAN). Doctor thesis (in the Russian sense, in Topology) prepared and defended in 1955. Prof.
of Geometry and Topology of Mechanical Mathematical Dept. of MSU. 19791993:
Head of the Laboratory of Optimization in Research Institute of System
Analysis (VNIISI) of the Russian Academy of Sciences. Member
of the Editorial boards of the journals: I
was the assessor and director of two doctorate theses in CIMAT (Efren
Morales Amaya and Francisco Sánchez Sanchez) and one master thesis
in CIMAT (Hernan Gonzalez Aguilar). FIELDS OF SCIENTIFIC ACTIVITY The first field of scientific interests was set topology, where I had constructed a two dimensional compact metric space with three dimensional square (now this is a classical result). This example allowed me to give a complete solution for the problem of dimensional fullness of compact metric spaces. The next region of interest was the theory of results in Mathematical Theory of Optimal Control. The main result here is the discovery of correct statement and proof of the Maximum Principle, which is the central achievement of modern, nonclassical variational calculus. Among some other results obtained in this region, there are: sufficient conditions of optimality, the theory of the regular synthesis for optimal trajectories, justification of Bellman´s dynamical programming and other results. Now I have obtained (together with Dr. A. Poznyak) a new version of the Maximum Principle in infinitedimensional Banach spaces and several versions of the Robust Maximum Principle. Optimization theory led me to the discovery of the Tent Method that is the general modern tool for solution of different extremal problems (mathematical programming, optimization, minimax problems, etc). Tent Method allowed me to obtain some new, important applications to Combinatorial Geometry. In particular, I obtained the algebraic solution of the SzokefalviNagy problem (formulated in 1995) and some other results in this direction. I have obtained a lot of theorems in Combinatorial Geometry. Among them: the solution of the illumination problem (formulated in 1960) for zonoids and belt bodies, the solution of minimal fixing systems problem (formulated in 1962) for arbitrary convex bodies, the solution of the problem minimal hindering systems (formulated in 1971) for arbitrary convex bodies, the complete classification of the planar, compact, convex figures with respect to the maximal cardinalities of their fixing system This result is obtained very recently together with my pupil, H. Gonzalez.
Recently I obtained some fundamental results on the Illumination problem (formulated in 19571960). My results consist of a solution of the problem for compact, convex bodies with mdM=2, complete solution for threedimensional bodies, for zonoids, for belt bodies, etc. Furthermore, I have some results in the theory of equivalent and equidecomposable figures, which is connected with the Third Hilbert´s problem. Recently, I worked out an alternative version of the General Relativity theory with an explanation of some phonomena in the galaxy, which were not explained in Einstein´s relativity theory. In the region of mathematical didactics, I worked out the mathematical visualization theory in school education, a theory of intellectual computer games with applications for school education etc. I have published many popular books, written many didactical papers and textbooks in Mathematical Education, the last three textbooks were published in the year 2000 and conducted some psychological research in this region. COURSES OF LECTURES. In the International Psychological College, I was responsible for a general Course of Mathematics with applications to mathematical education and the psychology of decision making. The main focus of the course was the main ideas of mathematical notions (not the tools of calculations) with their applications to logical thinking, finding of ways of solutions, and understanding the role of Mathematics in different branches of activity. In 1990, I worked in Washington University (Seattle) during the spring quarter, where I gave a course of Combinatorial Geometry . After that, I visited and gave lectures at the University in Colorado Springs (one month) and the University of Buffalo (one month). During my stay in the U.S.A, I also visited Mexico (one month), where I gave many lectures on the didactics of mathematics in Hermosillo, Guadalajara and other Mexican cities. From 19931999, I visited Germany several times (all together 14 months, including a sabatical for three months in 1999), where I gave several courses of lectures. In Germany I have written (jointly with Prof. H. Martini and Prof. P. Soltan) a fundamental monograph "Excursions into Combinatorial Geometry". The book was published by SpringerVerlag in 1997. Moreover, together with Dr. H. Martini, we have written 8 articles on Combinatorial Geometry which are published in different scientific Journals. From July 1994, I have worked in CIMAT (Centro de Investigacion en Matematicas), Guanajuato, Mexico. I gave six long courses for students and scientific researchers of CIMAT (in Combinatorial Geometry, Linear Optimization, Third Hilbert´s Problem, Convexity and Nonlinear Optimization). The first course was devoted to problems of Combinatorial Geometry. It was a modern course which included methods and results developed recently by scientists from Russia, U.S.A., Germany and other countries. The course was connected with a monograph written jointly by Prof. H. Martini, Prof. P. Soltan, and myself (published by SpringerVerlag in 1997). This course gave some intuitive representation in Euclidean and Banach spaces and was intended for the development of possible applications of convex sets theory in Analysis and other branches of Mathematics. It included the classical Helly Theorem with some its generalizations, partition problem for convex sets into smaller parts (with respect to diameters or other characteristics), different illumination problems for compact, convex bodies, the famous Borsuk problem (with its negative solutions obtained recently ) and other problems. Geometry is now, in a sense, the ideology of modern mathematicians. Before writing some formulas and integrals, a mathematician has to construct a bright geometrical picture of the investigated problem. The aim of the course consisted of collecting very recent results of combinatorial geometry and in giving listeners some representation on modern methods of geometry and their role in applied mathematics. The second course and seminar was devoted to linear optimal control theory. In 19941995 a complete survey of the linear part of the theory was given with detailed proofs and developments of new, modern methods of optimal control. In particular in the course, theory of convex sets and convex cones were widely applied. This was a good illustration for using geometrical ideas in applied mathematics. With the help of these methods, a proof for the Maximum principle was given a general solution of the synthesising problem was sketched with detailed picture of synthesis for controlled objects of second order. There were a lot of other applications. Besides that, some computing methods were developed for solving the linear optimization problem with the help of analog and digital computers. These methods were developed in a geometric style, and they were reduced to a convenient algorithm for finding optimal controls by a quickly convergent sequence of iterations. In the course, many exercises and research problems were proposed as a basis for preparing postgraduate papers and thesis . Some papers were written as a result of the course. That is especially important, since this region of applied mathematics was not developed in Mexico earlier.
The third course was devoted to Third Hilbert´s Problems. The modern state of this region of Geometry was explained for the listeners. The listeners received the detailed lecturesnotes with many exercises. The fourth course was devoted to nonlinear Optimal control theory. In 1997 a survey of the nonlinear part of the theory was given with detailed proofs and development of new, modern methods of optimal control. The complete account of Tent theory was given, where all proofs were conducted by geometrical and topological tools. With the help of the Tent method, a proof of the Maximum principle was given. In fact modern Maximum principle is a collection of Theorems of analogous contents , which differ by statements of optimization problems (Mayer´s problem, Lagrange´s problem, Bolza´s problem, robust optimization problem, etc). The course also contains solution of synthesis problem for second order nonlinear controlled objects (of oscillatory and nonoscillatory types) with detailed pictures of synthesis. As in my other courses, many exercises and research problem are proposed as a basis for preparing postgraduate papers and thesis. Some papers were written as a result of the course. That is especially important, since this region of applied mathematics was not developed in Mexico earlier. The fifth course was devoted to the theory of convex bodies. In the course I gave main theorems of the theory, explained the modern state of the convex bodies theory and formulated several unsolved problems. I am preparing a new fundamental monograph on convexity using materials of this course. Finally, my sixth course (given in 2001) is devoted to the ordinary linear differential equations in Banach spaces. I have explained the Tent Method in Banach space and obtained a new version of the Maximum Principle for linear controlled objects in Banach space. During my stay in Mexico, I prepared lecture notes of my linearoptimization course (about 240 pages in LATEXformat) which contain thoretical text with complete proofs, many drawings, and more than 300 exercises. The
lecturenotes on my three optimization courses will be a basis for a big
modern monograph on optimal control (the part devoted to nonclassical
variational calculus is already published in the monograph "Geometric
Methods and Optimization Problems" published in 1999.

