Quantum information science is a new and rapidly developing discipline that studies the general laws governing the transmission, storage, and processing of information in systems that obey the laws of quantum mechanics. It makes extensive use of matrix theory and operator theory on Hilbert spaces to investigate the capabilities of such systems and to develop principles for their rational design. An important consequence of the development of quantum information science is a substantial clarification of the logical structure and foundations of quantum mechanics and of its relation to reality. These lectures are devoted to the foundations of the mathematical theory of quantum systems: states, observables, communication channels, their entropic and information characteristics, and quantum information-processing protocols. The lectures provide a detailed introduction to the statistical structure of quantum theory and the necessary background from classical information theory; hence, knowledge of the basic mathematical subjects—analysis, linear algebra, and probability theory—is sufficient for understanding the material. In the first part of the course, we use the “standard” statistical model of a quantum system, but consider it in comparison with the classical statistical description. A consistent development of this approach leads to a generalized statistical model of quantum mechanics, in which observables (in general, “unsharp”) are described by probability operator-valued measures, while evolutions (in general, irreversible) are described by completely positive maps. Introductory material on these concepts, which underlie mathematical models of quantum information systems in the presence of noise and errors and have numerous applications in quantum information science, is presented in the second part of the course.
This text is based on a course delivered by the author at the Steklov Mathematical Institute of the Russian Academy of Sciences in the fall of 2013. It provides a systematic account of the birational geometry of rational surfaces over perfect fields that are not necessarily algebraically closed. The text focuses on three classical problems: rationality criteria, the L & uuml;roth problem, and the classification of finite subgroups of the plane Cremona group. Modern methods, including Mori theory and the Sarkisov program, are discussed in detail in the surface case.
In modern algebraic geometry, many interesting invariants are associated with algebraic varieties. Some of them, such as the dimension, are merely numbers, while others are more complicated. The derived category of coherent sheaves is one of the richest invariants of an algebraic variety. The properties of this category reflect the underlying geometry, but the categorical point of view often reveals additional algebraic structures that can be useful even for classical questions concerning coherent sheaves and vector bundles. This text provides an introduction to the study of derived categories of coherent sheaves. It is based on the lecture course given by the author at the Steklov Mathematical Institute in the spring of 2024. The text presupposes familiarity with algebraic geometry and basic homological algebra and is suitable for advanced undergraduate and graduate students.
Topological insulators are solid bodies characterized by a broad energy gap that is stable under small deformations. This motivates the use of topological methods in their study. A key role in solid-state theory is played by symmetry groups. Kitaev pointed out a relation between the symmetries of some classes of solids, including topological insulators, and Clifford algebras. According to this observation, the quantization of topological insulators should reduce to the theory of irreducible representations of Clifford algebras. The next important step was made by Kennedy and Zirnbauer, who introduced the notion of pseudosymmetries. While the algebra of observables of a topological insulator is generated by Hamiltonians satisfying commutation relations with symmetry operators, the quantum observables are described by complex structures on the Nambu space that satisfy anticommutation relations with pseudosymmetries. This correspondence determines the quantization of topological insulators.
The article addresses problems concerning bodies of zero resistance and invisible bodies within a model where a solid body interacts with a medium of non-interacting particles that reflect from the body in a perfectly elastic manner. Known results are reviewed, some statements are proved, and several conjectures and open problems are formulated.
