Several Complex Variables, Differential Geometry and Topology
Goals and tasks:
Investigations in the theory of holomorphic functions of several complex variables and the geometry of complex and almost complex manifolds. Investigations in the field of differential geometry of Riemannian manifolds, almost Hermitian manifolds and almost contact metric manifolds, differential geometry of Riemannian and Kaehlerian manifolds with a distribution. Investigations in the field of topology and tomography related to convex geometry and infinite dimensional topology. Education and supervision of MSc and PhD students in the fields of the research project.
Main fields of research:
- Invariant (pseudo-) metrics on domains in n-dimensional complex space.
- Geometric and analytic properties and characteristics of complex convex domains.
- Geometry of four-dimensional manifolds.
- Complex surfaces endowed with additional geometric structures.
- Hermitian geometry of twistor spaces of four-dimensional Riemannian manifolds.
- Contact geometry of twistor spaces of odd-dimensional Riemannian manifolds.
- Differential geometry of surfaces and hypersurfaces in Euclidean and Minkowski spaces.
- Local theory of surfaces in pseudo-Euclidean spaces with neutral metric.
- Convex projections of sets in Banach spaces and infinite dimesional topology.
Acad. Oleg Mushkarov, Corr. Member Nikolai Nikolov, Corr. Member Stefan Ivanov, Prof. DSc Johann Davidov, Prof. PhD Velichka Milousheva (Coordinator of the Project); Assoc. Prof. PhD Georgi Ganchev, Assoc. Prof. PhD Vestislav Apostolov, Assist. Prof. PhD Strashimir Popvasilev, PhD Petar Petrov, PhD Stoyu Barov, PhD Antoni Rangachev.
Mathematical Analysis and Applications
Mathematical Analysis (Calculus) is one of the oldest and best developed mathematical disciplines, both in world and national scales. Traditionally, under the notion “Mathematical Analysis” it is meant a group of trends occupying in the AMS mathematical subject classification the positions from 26 to 49, that is more than ¼ of all mathematical disciplines. These lie in the base of mathematical models from mathematical physics and many other natural and engineering sciences. The applicable aspects of Calculus (Classical and Fractional) have a great importance and are closely related with development of theory of differential equations, of numerical analysis, software applications, computer algebra systems (Mathematica, Maple, MATLAB), and contemporary information technologies. Mathematical Analysis’ applications are permanently growing in modeling in chemistry, biology, biomedicine, financial mathematics, control theory, signals recognition, etc.
Goals and tasks:
Studies on applied mathematical analysis of one and many variables in real and complex domain, in functional analysis, spectral analysis for differential equations; publication of books/ book chapters, surveys, articles, educational handbooks; publication of two international mathematical journals, indexed in Web of Science/ resp. Scopus: Fractional Calculus and Applied Analysis, specialized on Project’s subjects; and International Journal of Applied Mathematics, on Maths’ applications in wider sense; organization of specialized international conferences and forums, among which the traditional ones: Transform Methods and Special Functions, Complex Analysis and Applications, Geomеtric Function Theory and Applications, etc.; editing of conference proceedings and of volumes with selected works of known Bulgarian mathematicians; education of MSc and PhD students on the subject “Mathematical Analysis” under one of department’s doctoral programs.
Main fields of research:
- Special functions, entire functions, orthogonal polynomials, series in them;
- Fractional calculus (integration and differentiation of fractional order): theory, generalizations, applications to mathematical models of systems of fractional order;
- Integral transforms and transmutation operators;
Operational and convolutional calculi of local and nonlocal boundary value problems for differential operators of integer and fractional order, generalized Duhamel principle;
- Analytical and numerical analysis for differential and integral equations of fractional and higher integer order and their solutions, maximum principle, subordination principle;
- Geometric theory of functions of one complex variable: investigation of special classes of univalent functions and relationships between them, coefficients estimates, distortion theorems;
- Distribution and geometry of zeros of classes of entire functions and polynomials;
- Approximation theory by rational functions in the complex plane: Pade approximations, best rational approximations in Chebyshev and Lp-metrics, rational Chebyshev approximations of real-valued functions, measure theory and orthogonal polynomials;
- Multivariable complex analysis, theory of complex manifolds with additional geometrical structures;
- Applications of variational and topological methods and nonlinear analysis for differential and difference equations – critical points, existence, number and multiplicity of solutions;
- Stability and well-posedness of problems for partial differential equations, spectral theory, inverse scattering, traveling waves, applications;
- Theory of Banach spaces, approximations of norms, quasi-additive subspaces.
Acad. Stanimir Troyanski, Corr. Member Ivan Dimovski, Prof. DSc Peter Rusev, Prof. DSc Jordanka Paneva-Konovska (Coordinator of the Project), Prof. DSc Virginia Kiryakova, Prof. DSc Sevdzhan Hakkaev, Prof. DSc Ralitza Kovacheva, Prof. DSc Stepan Terzian, Assoc. Prof. PhD Emilia Bazhlekova.