Algebra, operator theory and financial mathematics

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.

Achieved results will be important for the development of the mathematical sciences. Since we will be studying problems that have been raised in the international mathematical community, we expect that their solutions will attract a considerable amount of attention. The results will be very important for the development of algebra, operator theory and their applications to mathematical finance. New results will shed light on the structure of operators and families of operators. They will also be important in the study of the invariant subspace problem, real algebraic geometry, abstract group theory, and elsewhere. Many of our results have already attracted a considerable amount of attention. We expect that the same will be true going forward. We will intensively pursue and advance scientific collaboration with leading mathematicians around the world. We will publish our results in prestigious international scientific journals, present them at the international scientific meetings and at invited lectures at foreign universities.

We transfer the newest scientific results to our students. In this way we contribute to social and economic development. Our results are, we believe, an important part of Slovenian mathematical research, which is fundamental for many other sciences. Research in the field of financial mathematics will contribute to the transfer of knowledge to the students of the new study program of Financial Mathematics at the University of Ljubljana. In addition, this part of our research is directly applicable to the financial sector of the economy (insurance companies, banks, other financial institutions, etc.). We have already encountered a considerable interest from the Slovenian financial industry, the central bank, and others.