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This application is very important for me because completion a PhD degree in Electrical and Computer Engineering is the best chance for a unifying my interest in information technologies and my natural inclination to mathematics. I am confident that when I become a serious and mature researcher my desire to make a contribution to science and to our understanding of this world will remain my main driving force.
It took a long time until personal computers became widely available in this country and until then my only single interest was mathematics. At the age of 14, during my school holidays, when my classmates had a rest, I willingly got up at about 7 in the morning and till the evening was absorbed by mathematics. At that time these were mostly Olympiad problems, but two years later, still being a High School student, I attended a course on inequalities in Kiev State University and conducted my first research work. Essentially it was a proof of Karamat inequality that utilized properties of convex functions and Murhead inequality. It was unforgettable, how main idea of my central proof dawned upon me. Although, relatively simple, it was something innovative, something that no one had ever done before with methods of elementary mathematics. That evening I went to sleep the happiest boy in the world. Later I refined the proof and being a freshman at the Moscow Institute of Physics and Technology(MIPT) presented it at the 52nd MIPT Scientific Conference where it was honored the first prize.
In the third-year at MIPT I started working at the Institute of System Programming (ISP), which is a part of Russian Academy of Sciences. Lectures on computer science at the ISP expanded my knowledge in many important fields such as complexity of algorithms, parallel computing, compiler technology, software engineering and so on. Especially I enjoyed studying new Autonomous Adaptive Control (AAC) method under the guidance of Prof. X. The essence of the AAC method is in simultaneous solution of such problems as pattern recognition, knowledge obtaining, presentation and decision-making. During my work in the ISP I had rare opportunity to interact closely with scientists who stood at the origin of computer science in Russia: Prof. X, Dr. Y and Prof. Z.
In my fourth year I joined Moscow office of NetCracker Corp. that selected several students for training. NetCracker produces software for modeling, analysis and interpretation of complex telecommunication networks. My work in NetСracker gave me solid practical knowledge of many technologies and concepts such as Java, Oracle, XML, Object-Oriented Design and Programming. But much more important is that the project required extensive knowledge of both mathematics and computer science. I personally wrote some modules to NetCracker where actively utilized graph theory, parallel computing and discrete mathematics. Moreover sometimes generalization of classical algorithms was required. For example, I extended Dejkstra algorithm to allow multiple search of optimal paths.
Along with my study of computer science I am involved in research activity in fundamental mathematics. Under supervision of Professor X I conducted research in convex analysis, wrote my thesis ”Integration of Multivalued Mappings“ and defended it with Honors. Essentially it consisted in a study of necessary and sufficient conditions under which there exists Riemann integral of multivalued maps. During this work I acquired a broad range of research experience and strong background necessary for further research. My current research work is devoted to differentiation of multivalued mappings and differential inclusions and is mostly concerned with nonconvex case. One of the most challenging tasks in the work is to obtain Pontryagin maximum principle in Hamiltonian form from Lagrange form (in terms of tangent cones).
My research topic is closely related with and often serves as a background to theory of optimal control and theory of decision making that are widely used in network optimization and graph algorithms – areas that are of great interest to me. Moreover my work on network performance in NetCracker adds up to this framework. Investigations in these fields, in discrete mathematics and in the theory of compilers are widespread at the department and it makes admission to your University particularly desirable. I have strongest incentive to advance as far as I can in this field and to discover something that has not been known before. I am certain that application to Carnegie Mellon is the best possible step to accomplish it and I would regard my admission not only as a great honor but also as a great responsibility and an obligation to work hard.
