Last Update 2022/07/29
Some non-mathematicians tend to view mathematics merely as a collection of technical tools to use whenever these tools are applicable to their work. Some mathematicians have a similar point of view regarding homological algebra and algebraic geometry, one of the most technical areas of mathematics. This restrictive approach often fails for both mathematics and these particular areas of it. As experience shows, one has to use the very ideology of the theory and not just concrete results in order to get real insight when applying homological algebra and algebraic geometry to his/her research.
Nowadays, these areas, which are my primary subjects of research, have an enormous influence on various areas in mathematics, mathematical physics, and string theory. I have developed an approach to noncommutative geometry by means of derived categories, an advanced technique in homological algebra. Derived categories, which first emerged as a very abstract mathematical tool, are now considered to be the most powerful method for describing D-branes of topological field theories in physics. In addition to applications, one has to continually develop the basics of homological algebra and its applications to algebraic geometry, in order to mathematically understand various proposals by physicists aimed at describing the structural evidence of our world. Among my results is the discovery of homological relation between geometry of some algebraic varieties and representation theory, reconstruction of algebraic varieties from their derived categories, a homological interpretation of the Minimal Model Program in birational geometry, introduction of Serre functor and invariants of derived categories based on it, construction of enhancements of triangulated categories, the discovery of the action of the braid group on the set of exceptional collections - particular kind of bases in triangulated categories.
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