Head of program P1-0222: prof. dr. Igor Klep

Duration: 01/01/2022 - 12/31/2027

Abstract: The research program will focus on research in algebra and operator theory, and will explore their applications in financial mathematics. The main fields of research include group theory, real algebraic geometry, the theory of operators on Banach spaces and lattices. In financial mathematics we will conduct research on stochastic analysis. In real algebraic geometry, we will investigate positive noncommutative functions and matrix convex sets. We will also explore the application of our advances to the theory of linear control systems, optimization and quantum information. In group theory, we will develop homological methods for studying the problem of Emmy Noether and its applications in algebraic geometry and K-theory. At the same time, we will also tackle modern combinatoric group theory through Babai's conjecture about the diameter of Cayley's graphs of finite simple groups. In linear algebra and algebraic geometry, we will research the classical problems of simultaneous similarity of tuples of matrices and the variety of commuting matrix tuples. We will study the properties of operators and one-parameter operator semigroups, where we will be the first to systematically go beyond Banach spaces. To this aim we will investigate other known notions of convergence and topology, especially the unbounded ones, on ordered spaces, vector lattices or Banach lattices. We will also be interested in the spectral theory of operators and related operator inequalities, where we intend to settle the 30 year old open problem of Huijmans and de Pagter. Further, we shall continue with the development of tropical methods for the studies of nonlinear operator problems. We will also strive to apply our results in financial mathematics, e.g. in the area of random processes arising from stochastic partial differential equations. With the rising importance of precise and imprecise probability in practical applications, especially in the field of statistics and finance, there is a greater than ever need for a deeper investigation and understanding of existing mathematical models for imprecise probability and for the development of new alternative models. The central role when modeling the dependence of random variables here is played by copulas. We will therefore research copulas, quasi-copulas, multivariate distributions, and the related Sklar’s theorem.

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