# Analysis, Geometry and Topology

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T

he Department of Analysis, Geometry and Topology (AGT) was created in March 2011 by merging the former departments *Complex Analysis, Geometry and Topology,** and** Real and Functional Analysis.*

The Department of Complex Analysis was one of the first departments within the Institute of Mathematics and Informatics at the Bulgarian Academy of Sciences with clearly identified research areas and well-developed scientific potential. It was one of the departments originating from the former Department of Advanced Analysis, directed by Academician Lyubomir Tchakalov until 1962. In 1962 it was divided into three separate departments:* Complex Analysis*, *Real and Functional Analysis**,* and *Differential Equations**. *Previous chairs of the department of Complex Analysis have been Academician Lyubomir Iliev (1962–1988), and Corresponding Member Ivan Dimovski (1988–2004). Since 2004 chair of the department is Corresponding Member Oleg Mushkarov. Initially, research in the department was focused on topics from traditional areas of classical function theory like geometric function theory, distribution of zeroes of entire and meromorphic functions. Gradually, modern areas like multi-dimensional complex analysis, complex geometry, special functions, integral transforms, fractional and operational calculi were added. In the course of time, three main directions of scientific research were formed, which by the year 2011 were grouped into the following thematic projects: functions of a complex variable, transform methods and special functions, several complex variables, and complex geometry.

The Department of Geometry and Topology was established in 1993 by merging the former Department of Geometry and Department of Topology. Until March 2011 the department was headed by Assoc. Prof. Georgi Ganchev. The Department of Geometry was one of the first departments at the Institute of Mathematics. Throughout the years chairs of the department have been Academician B. Petkanchin and Prof. G. Stanilov. The Department of Topology was founded in 1971 and has been headed by Prof. D. Doichinov and Prof. S. Nedev. The main areas of research in the Department of Geometry and Topology have been differential geometry of manifolds and general topology.

The Department of *Real and Functional Analysis was founded in 1958 and **has been headed by** Prof. Y. Tagamlitzki, Prof. S. Troyansky**,** a**n**d Prof. K. Kirtchev. *The main areasof research in the department have been the geometry of Banach spaces and spectral theory of linear and non-linear operators.

Since its formation in 2011 until 2020 the Department of Analysis, Geometry and Topology has been headed by Corresponding Member Oleg Mushkarov. At present, research in the department has been concentrated in two basic research projects with the participation of 23 scientists (including PhD students).

From July 2020 head of the department is Corresponding Member Nikolai Nikolov.

## Research Projects

**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.

*Research team:*

Acad. Oleg Mushkarov, Corr. Member Nikolai Nikolov, Corr. Member Stefan Ivanov, Prof. DSc Johann Davidov, Prof. DSc Ludmil Katsarkov, Prof. PhD Velichka Milousheva (Coordinator of the Project), Assoc. Prof. PhD Vestislav Apostolov, Assoc. Prof. PhD Stoyu Barov, Chief Assist. Prof. PhD Petar Petrov, Chief Assist. Prof. PhD Antoni Rangachev.

**Mathematical Analysis and Applications**

*Annotation:*

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.

*Research team:*

Acad. Stanimir Troyanski, 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. DSc Emilia Bazhlekova.