Mathematical analysis studies functions of real and complex variables, functional spaces, and operators on them.
Complex analysis studies properties of analytic functions of one or several complex variables, conformal and quasiconformal mappings. The spaces of analytic functions with reproducing kernel (Paley–Wiener, de Brange, Fock spaces, etc.) and properties of systems of reproducing kernels in these spaces (problems of completeness, basis property, etc.) are mainly studied. Special attention is paid to the connection between the theory of de Brange spaces and the theory of canonical systems.
Operator theory studies properties of linear (not necessarily continuous) maps between normed spaces. Special attention is paid to the functional models of the abstract operators in the spirit Szőkefalvi-Nagy and Foias, for example, to the models of the finite rank perturbations of self-adjoint and unitary operators.
Harmonic analysis deals with the study of various integro-differential operators on functional spaces, including singular integral operators and pseudo-differential operators. One of the main research tools is the Fourier transform and its generalizations.
Probability theory is a part of mathematics that studies random events, forms, and processes. It has extremely rich connections with other domains of mathematical research (especially with mathematical and functional analysis, mathematical physics, etc), and enormous variety of practical applications.
Theory of random processes studies random variables depending on some parameter — time instant, point in the space, and so on. There exist many specific classes of random processes: processes with independent increments, stationary processes, Markov processes, diffusion processes, Gaussian processes and so on. Every kind of processes has a wide field of applications and appropriate investigation tools.
Limit theorems of probability theory usually describe a situation where the interaction of large number of independent or weakly dependent random quantities (summation of large number of independent random variables is a typical example of such interaction) generates a universal probabilistic object. Limit theorems also include investigation of asymptotic behavior of rare events’ probabilities (large and small deviation theories). Limit theorems represent the most important tool of description of fundamental probabilistic laws and have many useful applications in applied fields.
Stochastic geometry studies random objects of geometric nature, for example, random convex sets,random point processes, etc. Nowadays, this direction is actively developing due to various applications in telecommunication networks, statistical physics, molecular biology, astrophysics, stereology, spacial statistics, compressed sensing etc.
Combinatorial objects such as permutations, partitions, graphs etc. have a natural notion of size, and one can study their behavior as the size of objects grows. It turns out that for many of such objects an analogue of the law of large numbers takes place: almost all objects of large size have similar structure in a certain sense. Investigation of this phenomenon, and that of deviations from the limit structure, is a branch of mathematics on the cusp of probability theory, combinatorics and analysis.
Among the people working in this field is Yu.V. Yakubovich.