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Исследователи направления занимаются алгебраической геометрией (включая как\ классическую, так и производную, и некоммутативную геометрию), арифметической\ геометрией, алгеброй. В частности, проводятся исследования в области зеркальной\ симметрии, теории моделей Ландау–Гинзбурга, бирациональной геометрии и\ программы минимальных моделей, теории автоморфизмов алгебраических многообразий\ и алгебраических групп, теории арифметических решеток, изучение дзета-функции\ алгебраических многообразий, теории многомерных аделей, алгебраической K-теории и\ теории алгебраических циклов.
\ \Области исследования: алгебраическая коммутативная и некоммутативная\ геометрия, производные и триангулированные категории, (квази)когерентные пучки,\ мотивы, K-теория, зеркальная симметрия, бирациональная геометрия, программа\ минимальных моделей, многообразия Фано, алгебраические группы, арифметические\ многообразия, многомерные адели.
", eng:"Our researchers study algebraic geometry, both classical and derived\ noncommutative, arithmetic geometry, and algebra. The subjects of their interest\ include Mirror Symmetry, the theory of Landau-Ginzburg models, birational\ geometry and the Minimal Model Program, automorphisms of algebraic varieties\ and algebraic groups, the theory of arithmetic lattices, the study of zeta-functions\ of algebraic varieties, the theory of higher-dimensional adeles, algebraic K-theory,\ and the theory of algebraic cycles.
\ \Research areas: algebraic commutative and noncommutative geometry,\ derived and triangulated categories, (quasi-)coherent sheaves, motives, K-theory,\ Mirror Symmetry, birational geometry, Minimal Model Program, Fano varieties,\ algebraic groups, arithmetic varieties, higher-dimensional adeles.
", }, 2:{rus:"Наши исследования относятся к широкому спектру областей в геометрии,\ топологии, математической и теоретической физике, таких как алгебраическая и\ геометрическая топология, гиперболическая геометрия, дискретная геометрия\ с приложениями в кристаллографии, теория особенностей, теория узлов,\ математическое и физическое моделирование, квантовая теория поля, нелинейные\ солитонные уравнения, точно решаемые модели статистической механики, квантовые\ алгебры, пуассонова геометрия интегрируемых систем. В частности, мы изучаем\ структуры на многообразиях и теории кобордизмов, эквивариантную и торическую\ топологию, волновые фронты гиперболических операторов, алгоритмические проблемы\ теории узлов, пространства модулей и группы классов отображений, вложения\ многообразий и полиэдров, выпуклые многогранники, голографическую дуальность и\ смежные вопросы квантовой теории поля, сигма-модели, технику анзаца Бете,\ интегрируемые иерархии и их алгеброгеометрические решения, уравнения Пенлеве и\ изомонодромные деформации, представления квантовых алгебр, дуальности\ в интегрируемых системах.
\ \Области исследования: алгебраическая топология, геометрическая топология,\ торическая топология, теория особенностей, теория узлов, дискретная геометрия,\ квантовая теория поля, нелинейные солитонные уравнения, точно решаемые модели\ статистической механики, квантовые алгебры, пуассонова геометрия, голографическая\ дуальность, сигма-модели.
\ ", eng:"Our research includes a wide range of topics in geometry, topology, mathematical\ and theoretical physics such as algebraic and geometric topology, hyperbolic\ geometry, discrete geometry and applications to crystallography, singularity\ theory, knot theory, mathematical and physical simulation, quantum field theory,\ nonlinear soliton equations, exactly-solvable models of statistical mechanics,\ quantum algebras, Poisson geometry of integrable systems. In particular, we study\ structures on manifolds and cobordism theories, equivariant and toric topology,\ wave fronts of hyperbolic operators, algorithmic problems in knot theory, moduli\ spaces and mapping class groups, embeddings of manifolds and polyhedra, convex\ polytopes, holographic duality and related topics in quantum field theory, sigma-\ models, Bethe ansatz techniques, integrable hierarchies and their algebro-\ geometric solutions, Painleve equations and monodromy preserving equations,\ representations of quantum algebras, dualities in integrable systems.
\ \Research areas: algebraic topology, geometric topology, toric topology,\ singularity theory, knot theory, discrete geometry, quantum field theory, nonlinear\ soliton equations, exactly-solvable models of statistical mechanics, quantum\ algebras, Poisson geometry, holographic duality, sigma-models.
" }, 3:{rus:"Исследования по направлению "Анализ и аналитическая теория чисел"\ посвящены актуальным проблемам теории функций, аналитической теории чисел и\ комплексного анализа.
\ \Одной из основных рассматриваемых проблем является изучение теории пространств\ Соболева на нерегулярных областях конечномерных евклидовых пространств. Также\ разрабатываются новые приложения теории интерполяции линейных операторов в теории\ ортогональных рядов. Одно из направлений исследований – изучение\ свойств коэффициентов Фурье непрерывных функций относительно произвольных\ полных ортонормированных систем. Изучаются различные $L$-функции, связанные\ с голоморфными формами, исследуется поведение дзета-функции Римана и ее\ аналогов на критической линии. Проводится исследование различных задач\ суммирования в конечных и бесконечных полях. Изучается кэлерова геометрия\ трех классов бесконечномерных комплексных многообразий, включая пространства\ петель компактных групп Ли, бесконечномерные грассмановы многообразия и\ универсальное пространство Тейхмюллера. Полученные результаты применяются\ в теории струн. Изучается и обосновывается адиабатическая предельная\ конструкция для уравнений Гинзбурга–Ландау, Зайберга–Виттена и\ Янга–Миллса в различных пространствах, включая компактные римановы\ поверхности. Разработывается теория конструктивных рациональных приближений\ аналитических функций. Проводится исследование свойств полиортогональных\ многочленов.
\ \Области исследования: тригонометрические ряды, ортонормированные системы,\ пространства Соболева, пространства функций положительной гладкости, теория сумм\ произведений, $L$-функции, явление универсальности, пространство Тейхмюллера,\ комплексные многообразия и римановы поверхности, рациональные аппроксимации.
\ ", eng:"Research in the direction of "Analysis and Analytic Number Theory"\ is devoted to topical problems in the theory of functions, analytical number theory,\ and complex analysis.
\One of the main problems under consideration is the study of the theory of Sobolev\ spaces on irregular domains of finite-dimensional Euclidean spaces. We also\ develop new applications of the theory of interpolation of linear operators in the\ theory of orthogonal series. The study of the properties of the Fourier coefficients\ of continuous functions with respect to arbitrary complete orthonormal systems is\ in our focus. We study different $L$-functions associated with holomorphic\ forms and investigate the behavior of the Riemann zeta function and its analogs on\ the critical line. Research on various summation problems in finite and infinite\ fields is performed. We study the Kähler geometry of three classes of\ infinite-dimensional complex manifolds, including loop spaces of compact Lie\ groups, infinite-dimensional Grassmannian manifolds, and the universal\ Teichmüeller space. The results are applied to string theory. We study and\ justify the adiabatic limit construction for Ginzburg-Landau, Seiberg-Witten, and\ Yang-Mills equations in various spaces, including compact Riemann surfaces. The\ theory of constructive rational approximations of analytic functions is developed.\ The study of the properties of multiple orthogonal polynomials is performed.
\ \Research areas: trigonometric series, orthonormal systems, Sobolev spaces,\ spaces of functions of positive smoothness, sum-product theory, $L$-\ functions, universality phenomenon, Teichmüeller space, complex\ manifolds and Riemann surfaces, rational approximations.
" }, 4:{rus:"\ Мы проводим исследования в трех направлениях. Первое направление включает \ в себя различные аспекты теории динамических систем. Особое внимание здесь мы \ уделяем системам классической и статистической механики. В частности, мы \ развиваем теорию гамильтоновых систем и их случайных возмущений. Второе \ направление наших исследований – теория оптимального управления, \ включая задачи, связанные с экономическим моделированием. Третье направление \ посвящено развитию математических методов в различных областях механики \ сплошных сред, таких как гидродинамика, теории упругости и пластичности, теория \ детонации, и др.
\ \Области исследования: теория оптимального управления, экономическое \ моделирование, динамические системы, включая системы классической механики, гамильтоновы системы и их случайные возмущения, неравновесная статистическая \ механика, вариационные методы, математические методы механики сплошных сред.
\ ", eng:"In our section, we pursue research in three directions. The first is the study of\ various aspects of the dynamical systems theory, mostly (but not only) related to\ the systems of classical and statistical mechanics; in particular, to Hamiltonian\ systems and their stochastic perturbations. The second direction of our research is\ optimal control theory, including problems related to economic modeling. The\ third direction is devoted to the development of mathematical methods in diverse\ areas of continuum mechanics, such as hydrodynamics, elastic media, dispersive\ media, detonation theory, etc.
\ \Research areas: optimal control theory, economic modeling, dynamical\ systems in classical mechanics, Hamiltonian systems and their stochastic\ perturbations, variational methods, non-equilibrium statistical mechanics,\ mathematical methods in continuum mechanics.
" }, 5:{rus:"Данное направление исследований связано с классическими и современными \
вопросами теории вероятностей, математической статистики, квантовой теории \
информации и математической физики. Предметом изучения математической физики \
являются математические модели в физике и других естественных науках. Наши \
исследования в этой области включают дифференциальные уравнения \
в частных производных, уравнения общей теории относительности, квантовую \
теорию поля и теорию струн, математические основания статистической физики. \
Современная математическая физика использует различные геометрические, \
алгебраические и вероятностные методы, функциональный и
Области исследования: уравнения математической физики, бесконечномерный \ анализ, математические задачи общей теории относительности, математическая теория \ открытых квантовых систем, квантовые каналы, квантовая криптография, квантовая \ теория информации, теория мартингалов, проверка гипотез, стохастические процессы, \ стохастический анализ, случайные матрицы.
\ ", eng:"Research in this direction focuses in classical and modern issues in probability\ theory, statistics, quantum information theory and mathematical physics.\ Mathematical physics studies mathematical models in physics and other natural\ sciences. Our research in this field includes partial differential equations, equations\ of general relativity, quantum field theory, and string theory, mathematical\ foundations of statistical physics. Modern mathematical physics uses various\ geometric, algebraic, and probabilistic methods, functional and $p$-adic\ analysis, etc. Infinite-dimensional analysis as a widely used tool in both probability\ theory and mathematical physics is also within the scope of our research. Research\ topics in probability and statistics include such actively developing areas as\ hypothesis testing, sequential analysis, martingale theory, stochastic analysis, and\ random matrix theory. An important field of research is quantum information\ theory investigating capacities and other characteristics of quantum\ communication channels, quantum key distribution, quantum error-correcting\ codes, and dynamics of open quantum systems.
\ \Research areas: equations of mathematical physics, infinite-dimensional\ analysis, mathematical problems of general relativity, mathematical theory of open\ quantum systems, quantum channels, quantum cryptography, quantum information\ theory, martingale theory, hypothesis testing, stochastic processes, stochastic\ analysis, random matrices.
" }, 6:{rus:"Дискретная математика и математическая логика являются важными разделами \ математики. С одной стороны, эти разделы изучают фундаментальные вопросы \ математики, такие как проблемы существования и эффективности алгоритмов или \ доказуемости и недоказуемости математических утверждений. С другой стороны, \ эти разделы особенно важны для приложений математических computer science и data \ science. Сотрудниками центра исследуются такие направления как асимптотическая \ комбинаторика, комбинаторная и геометрическая теория групп, теория сложности \ вычислений, теория доказательств, неклассические логики, теория доказуемости и \ формальная арифметика, ветвящиеся процессы, случайные блуждания и многие другие.
\ \Области исследования: асимптотическая комбинаторика, комбинаторная и \ геометрическая теория групп, теория сложности вычислений, теория доказательств, \ неклассические логики, теория доказуемости и формальная арифметика, ветвящиеся \ процессы, случайные блуждания.
\ ", eng:"Discrete mathematics and mathematical logic are important areas of mathematics.\ On the one hand, these areas address foundational aspects of mathematics, such as\ the problems of existence and efficiency of algorithms, or provability or non-\ provability of mathematical statements. On the other hand, they are especially\ relevant for applications of mathematical methods in computer science and data\ science. At SIMC, among other topics, we study asymptotic combinatorics,\ combinatorial and geometric group theory, computational complexity theory, proof\ theory, non-classical logics, provability logic and formal arithmetic, branching\ processes, random walks.
\ \Research areas: asymptotic combinatorics, combinatorial and geometric\ group theory, computational complexity theory, structural proof theory, non-\ classical logics, provability logic and formal arithmetic, branching processes,\ random walks.
" }, 7:{rus:"\Данное направление посвящено решению математических задач и разработке \ теоретических методов, необходимых в активно развивающейся области квантовых \ технологий. Квантовые технологии используют особые свойства индивидуальных \ квантовых систем, таких как фотоны, атомы, молекулы, для различных приложений, \ начиная от лазерной химии до квантовой криптографии. Направления исследований \ включают методы квантового управления, теорию открытых квантовых систем, \ квантовую информацию, квантовую криптографию, квантовую сложность, квантовую \ томографию, квантовую теорию многих частиц, неравновесную квантовую теорию \ и др. Сотрудники направления также активно участвуют в образовательной \ деятельности в области квантовых технологий, курируют магистерскую программу \ "Методы квантовых технологий и математической физики" на кафедре \ Московского физико-технического института на базе Математического института \ им. В. А. Стеклова, разработку и чтение новых учебных курсов по \ квантовым технологиям для студентов, аспирантов и молодых ученых научно- \ образовательного центра Математического института им. В. А. Стеклова \ и других университетов и институтов, организацию конференций, \ популяризацию квантовых технологий.
\ \Области исследования: методы квантового управления, теория открытых \ квантовых систем, квантовая информация, квантовая криптография, квантовая сложность, \ квантовая томография, неравновесная квантовая теория.
\ ", eng:"The research direction focuses on solving mathematical problems and developing\ theoretical methods necessary for the now actively growing field of quantum\ technologies. Quantum technologies exploit specific properties of individual\ quantum systems such as photons, atoms, molecules for a variety of existing and\ potential applications ranging from laser chemistry to quantum cryptography.\ Problems that are investigated by the researchers of the direction include quantum\ control, the theory of open quantum systems, quantum information, quantum\ cryptography, quantum complexity, quantum tomography, many-body quantum\ systems, non-equilibrium quantum dynamics, etc. Members of the direction are\ also actively involved in educational activities in the area of quantum technologies,\ they are in charge of the master degree program "Methods of Quantum\ Technologies and Mathematical Physics" in the (based at Steklov\ Mathematical Institute) chair of Moscow Institute of Physics and Technology,\ developing and reading of novel training courses on quantum technologies for\ students, graduate students, and young scientists, Research and Education Center\ of Steklov Mathematical Institute and other Universities and Institutes,\ organization of workshops, popularization of quantum technologies.
\ \Research areas: quantum control, open quantum systems, quantum\ cryptography, Markovian dynamics, non-Markovian dynamics, quantum\ tomography, quantum complexity, non-equilibrium quantum dynamics.
" }, }; const research_results_arr={ 1:{ eng:"\\ Alexander Efimov solved several fundamental hard problems in non-commutative geometry: he proved a conjecture of M. Kontsevich on homotopy finiteness of derived categories of coherent sheaves on separated schemes of finite type over a field of characteristic zero [1], he found counterexamples to conjectures of Kontsevich about generalized Hodge-to-de Rham degeneration for dg-categories [2], and, in particular, he obtained a negative answer to a question of Bertrand Toen, a world known top expert in this field.\
\ Sergey Gorchinskiy and Denis Osipov completed a local theory of the higher-dimensional Contou-Carrere symbol, in particular, characterizing the symbol by a universal property. One-dimensional theory has been developed by a number of authors, including P. Deligne, A. Beilinson, S. Bloch, H. Esnault. Two-dimensional theory was partially treated by Osipov and Xinwen Zhu, but already in the two-dimensional case, one needs completely different methods, which were finally developed by Gorchinskiy and Osipov with the help of a novel geometric approach to iterated Laurent series.\
\ Sergei Novikov discovered a mathematical relationship between two important regimes in physical systems: the high-speed rotation in the Clebsch case of the rigid body motion in an ideal fluid and the regime of a two-dimensional particle drift in a spatially inhomogeneous strong magnetic field. The former regime is interesting because of its applications to gyroscopes, while the latter is related to Hannes Alfven's problem in the particle accelerator physics. In the 1980s, Sergey Novikov proved that the Clebsch case is mathematically equivalent to the charged particle motion on a two-sphere in the (unquantized) Dirac monopole field. Nevertheless, the relationship between the two above mentioned regimes has not been noticed before.\
\ Leonid Chekhov obtained important results on relationship between two approaches to Poisson structures on moduli spaces of Riemann surfaces. For decades, two competing descriptions of geometric and Poisson structures on moduli spaces of Riemann surfaces uniformized by Poincare existed in the literature: the first Fenchel-Nielsen description in terms of coordinates of lengths and twists, and the second, more modern, description by Thurston, Penner and Fock in terms of shear coordinates based on ideal-triangle decompositions of Riemann surfaces with holes. Both these descriptions generate Goldman Poisson bracket on the set of geodesic functions, but one crucial brick in the construction was missed: how to construct canonical variables of lengths and twists out of Thurston’s shear coordinates? A technical problem was that Poisson relations depend on normalization of twist coordinates, so it was important to find a normalization ensuring commutativity of the twist coordinates. It was shown that the proper normalization is the one proposed several years ago by Nekrasov, Rosly and Shatashvili. Leonid Chekhov has proved the canonicity of thus constructed coordinates and generalized these coordinates to the case of Riemann surface with marked points on boundaries. This opens perspectives of constructing the so-called Yang-Yang functional on moduli spaces of Riemann surfaces in a consistent way.\
\ Stefan Nemirovski and Vladimir Chernov introduced a natural interval topology on groups of contactomorphisms and Legendrian isotopy classes and explored its relations with the Alexandrov topology in Lorentz geometry.\
\ Armen Sergeev has described Kähler geometry of three classes of infinite-dimensional complex manifolds, including loop spaces of compact Lie groups, Hilbert-Schmidt Grassmann manifolds and universal Teichmueller space. Using the obtained results it is constructed the quantization of the theory of half-differentiable strings which was not done before.\
\ Viktor Buslaev extended the Schur criterion for formal power series to the case of formal Newton series with nodes in the sequence of points of the unit disk. Necessary and sufficient conditions for the existence of a function that is holomorphic in the unit disk, takes values from the closed unit disk and coincides with every partial sum of the Newton series at the given nodes were found.\
\ Ilya Shkredov, Misha Rudnev, and Sophie Stevens, using a new technology, proved a sum-product result in the form of Balog and Wooley with the best possible exponent. New estimates of additive and multiplicative energies in average were obtained. The exponent in the famous sum-product problem for real numbers was improved.\
\ Alexander Tyulenev and Sergey Vodopyanov solved a Whitney-type problem for the first-order Sobolev spaces under some assumptions on the corresponding exponent of integrability and on the regularity of the corresponding Hausdorff contents.\
\ Lev Lokutsievskiy and Mikhail Zelikin solved Newton’s aerodynamic problem in the subclass of convex bodies with vertical plane of symmetry and smooth side boundary. In 1993, it was found that Newton's axially symmetric solution is not optimal in the class of all convex bodies. The exact optimal shape remains unknown. The shape found by Lokutsievskiy and Zelikin is optimal in the described subclass and it is very close to the optimal one in the entire class for large heights due to Wachsmuth's numerical experiments.\
\ Andrey Kulikovskii, Andrej Il'ichev, Anna Chugainova, and Vladimir Shargatov first proposed to consider the structure of a spontaneously emitting shock wave and established that the structure is stable. Stability of the spontaneously emitting shock wave is an important problem that has been studied by many authors for more than half a century.\
\ In a series of works by Alexander Holevo [1]-[3] devoted to multimode Gaussian measurement channels, the fundamental property of the \"Gaussian maximizer\" was established, on the basis of which the most important information characteristics of such channels were calculated. The accessible information was obtained for a gauge-invariant Gaussian ensemble of states, which closed the question posed in the early 1970s. The classical capacity of a channel satisfying the \"threshold condition\" that covers the gauge-invariant case was calculated. The entropy reduction was found and based on it, the entanglement-assisted capacity for a general-type quantum Gaussian measurement channel was determined.\
\ Mikhail Katanaev and Daniel Afanasev have finished classification of all global solutions of general relativity with an electromagnetic field and a cosmological constant assuming that four-dimensional space-time is the warped product of two surfaces. There are three cases: the direct product of two constant curvature surfaces, the warped product of Lorentzian surface on constant curvature Euclidean surface, and the warped product of constant curvature Lorentzian surface on Euclidean surface. Each case has several subcases. Totally, there are 56 topologically different solutions. In particular, they describe black and white holes, cosmic strings, wormholes, and other physically interesting solutions.\
\ Vladimir Podolskii and Alexander Kozachinsky proved an open conjecture in circuit complexity theory concerning the depth of monotone boolean circuits computing the MAJORITY function of $n$ boolean variables. They explicitly showed how to compute this function by a circuit of logarithmic depth consisting of MAJORITY functions of only three variables.\
\
Stepan Kuznetsov solved a problem by Buszkowski (2007) on the Lambek calculus extended with iteration, or Kleene star, axiomatised by means of
\
Fedor Pakhomov and James Walsh have isolated a natural well-founded partial order structure on the boolean algebra of sentences of second-order arithmetic defined in terms of
\ Alexander Pechen and Denys Bondar proved uncomputability, in the Turing sense, of quantum control problems for open and closed systems. The proof was done by establishing a relation between discretized quantum control problems and Diophantine equations and using the negative answer to the 10th Hilbert problem. In addition, this technique allowed them to construct quantum control problems belonging to different complexity classes. In particular, an example of the control problem involving a two-mode coherent field was shown to be NP-hard.\
\ Dmitry Ageev, Andrey Bagrov, and Askar Iliasov have studied the Coleman–Weinberg potential of $p$-adic field theory. One-loop quantum corrections to the effective potential were computed. In contrast to the limit $p \\to 1$, in the $p \\to \\infty$ limit the theory was shown to exhibits very peculiar behavior with emerging logarithmic terms in the effective potential, which has no analogue in real theories.\