Plaksa Sergii

Zhitomir Pedagogical Institute, Faculty of Physics and Mathematics, 1979–1984, Master of science, diploma with honors;

Post-graduate studies at the Institute of Mathematics of the Ukrainian Academy of Sciences under the guidance of Professor P.M. Tamrazov, 1984–1989

Ph.D. (in Physics and Mathematics, mathematical analysis), Institute of Mathematics of

the Ukrainian Academy of Sciences, 1989. Theses: Riemann boundary problem and singular integral equations;

Doctor of Sciences (in Physics and Mathematics, mathematical analysis), Institute of Mathematics of the Ukrainian Academy of Sciences, 2006. Dissertation Title: Monogenic functions in boundary problems for elliptic type equations with a degeneration on an axis

Research topics

Complex and hypercomplex analysis;

Analytic function theory of complex variable;

Monogenic function theory in Banach algebras;

Boundary problems theory for monogenic functions;

Boundary problems of mathematical physics;

Singular integral equations and operators;

Perturbation theory of Noetherian and semi-Noetherian operators



Principal scientific results

1. Problem posed by M.A. Lavrentyev: to develop methods for investigation of spatial potential solenoid fields analogous to analytic function methods in the complex plane for plane potential fields.

We constructed analytic functions of vector variable with values in an infinite-dimensional commutative Banach algebra and proved that components of these functions generate axial-symmetric potential functions and Stokes flow functions. In such a way new integral expressions for these functions were obtained and new method for investigation of spatial axial-symmetric potential solenoid fields was found. The suggested method is analogous to analytic function method in the complex plane and is the solution of the Lavrentyev problem for spatial axial-symmetric potential fields. Using new integral expressions for axial-symmetric potential functions and Stokes flow functions, we developed a functional-analytic method for effective solving boundary problems for axial-symmetric potential solenoid fields.

2. We developed an algebraic-analytic approach to equations of mathematical physics. An idea of such an approach means a finding of commutative Banach algebras such that monogenic functions defined on them form an algebra and have components satisfying to beforehand given equations with partial derivatives.

We obtained constructive descriptions of monogenic functions taking values in commutative algebras associated with the two-dimensional biharmonic equation and the three-dimensional Laplace equation by means of analytic functions of the complex variable. For the mentioned monogenic functions we established basic properties analogous to properties of analytic functions of the complex variable: the Cauchy integral theorem and integral formula, the Morera theorem, the uniqueness theorem, the Taylor and Laurent expansions.

3. The classical theorems on stability of the Noetherian properties of operators in complete spaces are well-known. However, attempts to eliminate the requirement of completeness of spaces from theorems on stability of the Noetherian property faced essential difficulties of topological nature.

To overcome these difficulties we have developed algebraic methods for proving theorems on stability of Noetherian and semi-Noetherian properties of operators in incomplete vector spaces. These algebraic methods are indifferent with respect to topological properties of the given spaces and operators.

Noether operators in incomplete spaces appear in the theory of singular integral equations on curves in the complex plane for extended classes of equation coefficients and curves. We studied the Noetherian property for singular integral Cauchy operators in incomplete spaces of quickly oscillating functions on closed Jordan regular rectifiable curves by the mentioned methods of algebraic nature.

4. Solvability of boundary problems for analytic functions theory in domains with boundary not being piece-wise smooth rectifiable depends on combined properties both of the given functions and the boundary. Difficulties increase if the index of a boundary problem is infinite. To overcome these difficulties we developed methods for constructing asymptotic expansions for the Cauchy-type integrals on non-regular rectifiable curves (in particular, on spiral-form curves). Using these asymptotic expansions, we have solved explicitly some boundary problems for analytic functions with infinite index in domains with both regular and non-regular rectifiable boundary. We have also solved explicitly some Riemann boundary problems with quickly oscillating coefficients on closed Jordan rectifiable curves.

Institute of Mathematics of the National Academy of Sciences of Ukraine, Kiev,

Junior Research Fellow, 1989 – 1992

Research Fellow, 1992 – 1999

Senior Research Fellow, 1999 – 2006

Leading Research Fellow, 2006 – present

2nd European Congress of Mathematics (Budapest, Hungary, 1996);

Conference on Differential Equations and their Applications (Brno, Czech Republic, 1997);

XII International Conference on Analytic Functions (Lublin, Poland, 1998);

7-th International Colloquium on Finite or Infinite Dimensional Complex Analysis (Fukuoka, Japan, 1999);

Second International ISAAC Congress (Fukuoka, Japan, 1999);

International Conference dedicated to M.A. Lavrentyev on the occasion of his birthday centenary (Kiev, Ukraine, 2000), plenary lecture ”On the Lavrentyev’s problem: description of axial-symmetric potential fields by means of analytic functions”;

International Conference on Complex Analysis and Potential Theory (Kiev, Ukraine, 2001);

Ukrainian Congress of Mathematics (Kiev, Ukraine, 2001);

International Conference on Factorization, Singular Operators and Related Problems in Honour of Professor Georgii Litvinchuk (Funchal, Madeira, 2002);

International Workshop on Potential Flows and Complex Analysis (Kiev, Ukraine, 2002);

International Conference ”Complex Analysis and its Applications”, (Lviv, Ukraine, 2003);

International Workshop ”Potential Theory and Free Boundary Flows” (Kiev, Ukraine, 2003);

International Conference ”Analytic Methods of Analysis and Differential Equations” (Minsk, Belarus, 2003);

5-th International ISAAC Congress (Catania, Italy, 2005);

International Workshop ”Free boundary flows and related problem of analysis” (Kiev, Ukraine, 2005);

International Conference on Complex Analysis and Potential Theory (Gebze, Turkey, 2006);

6-th International ISAAC Congress (Ankara, Turkey, 2007);

International Conference ”Complex analysis and wave processes in mechanics” within the framework of Bogolyubov Readings (Zhitomir, Ukraine, 2007);

3-rd Schools on Abstract Differential Equations and Ordinary Differential Equations (Mostaganem, Algeria, 2008);

4-th International Conference on Complex Analysis and Dynamical Systems (Nahariya, Israel, 2009);

International Workshop ”Applied Nonlinear Analysis” (Holon, Israel, 2009);

International Conference ”Analytic methods of mechanics and complex analysis” dedicated to N.A. Kilchevskii and V.A. Zmorovich on the occasion of their birthday centenary (Kiev, Ukraine, 2009);

2nd Ukrainian Congress of Mathematics (Kiev, Ukraine, 2009);

3-rd International Workshop on Contemporary Problems of Mathematics and Mechanics (Minsk, Belarus, 2010);

6-th International Conference ”Finsler Extensions of Relativity Theory” (Moscow, Russia, 2010)

Honors and Grants:

1994 – 1995 ISF Grant UB 4000;

1995 – 1996 ISF Grant UB 4200;

1995 – 1998 INTAS Grant 94-1474;

1997 – 1998 Ukrainian – Polish Grant 2M/1401-97;

1999 ISAAC Award for outstanding research achievements in mathematics;

2000 – 2003 INTAS Grant 99-00089;

2008 Grant of London Mathematical Society