Phd program
SYLLABI (Elective Courses)
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SYLLABI (Elective Courses)Introductory PhD courses assume the knowledge of the material of fundamental MS level Algebra, Analysis, Probability and Complex function courses, and Intermediate PhD level courses assume the material of the four mandatory PhD courses 1)APPLIED FUNCTIONAL ANALYSIS Course coordinator: Gheorghe Morosanu No. of Credits: 3, and no. of ECTS credits: 6 Level: introductoryPhD Prerequisites: - Brief introduction to the course: Basic concepts and fundamental theorems are presented. Some significant applications are analyzed to illustrate the power of functional analysis. Special attention is paid to linear and nonlinear evolution equations in Banach spaces. The goals of the course: The main goal of the course is to introduce students to some of the most important aspects of functional analysis, including ties with other fields of pure and applied mathematics. The learning outcomes of the course: By the end of the course, students are enabled to do independent study and research in fields touching on the topics of the course, and how to use these methods to solve specific problems. In addition, they develop some special expertise in the topics covered, which they can use efficiently in other mathematical fields, and in applications, as well. They also learn how the topic of the course is interconnected to various other fields in mathematics, and in science, in general. More detailed display of contents (week-by-week): 1. Metric spaces, topological properties, Bolzano-Weierstrass theorem; normed linear spaces, a characterization of finite dimensional normed spaces 2-3. Arzelà-Ascoli theorem. Peano theorem. Banach fixed point theorem. Applications to differential and integral equations 4-5. Linear operators. The dual space. Weak topologies. Hilbert spaces. Projections on closed convex sets. The Riesz representation theorem, Lax-Milgram theorem 6. Orthonormal systems in Hilbert spaces, Fourier series 7. Bochner integral, scalar and vector-valued distributions, Sobolev spaces 8. Eigenvalue problems for linear compact operators. The Hilbert-Schmidt theory 9. Semigroups of linear operators. The Hille-Yosida theorem 10. Linear evolution equations in Banach spaces, applications 11-12. Monotone operators and nonlinear evolution equations, applications References:
2)EVOLUTION EQUATIONS AND APPLICATIONS Course coordinator: Gheorghe Morosanu No. of credits: 3, and no. of ECTS credits: 6 Prerequisites: - Course level: intermediatePhD Brief introduction to the course: After a short historical introduction, some of the most important results will be presented, including the Hille-Yosida theorem as well as the main existence results for linear and nonlinear evolution equations. Some applications will be discussed to illustrate the theoretical results. The goals of the course: The main goal of the course is to introduce students to the main topics and methods of the theory of evolution equations in Banach spaces. The learning outcomes of the course: By the end of the course, students are enabled to do independent study and research in fields touching on the topics of the course, and how to use these methods to solve specific problems. In addition, they develop some special expertise in the topics covered, which they can use efficiently in other mathematical fields, and in applications, as well. They also learn how the topic of the course is interconnected to various other fields in mathematics, and in science, in general. More detailed display of contents (week-by-week): 1. Preliminaries of linear and nonlinear functional analysis 2. Uniformly continuous and strongly continuous semigroups of linear operators. Definition, examples, properties 3. The Hille-Yosida and Lumer-Phillips theorems 4. Solving linear evolution equations by the semigroup approach. Applications to linear partial differential equations 5. Monotone operators. Minty’s theorem on maximality, surjectivity, perturbation results 6. Subdifferentials of convex functions 7. Existence and uniqueness for evolution equations associated with monotone operators 8. Existence theory for the case of evolution equations associated with subdifferentials 9. Stability of solutions. Asymptotic behavior, periodic forcing 10-12. Applications to nonlinear parabolic and hyperbolic partial differential systems.
1. G. Morosanu, Nonlinear Evolution Equations and Applications, Reidel, 1988. 2. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, 1983.
In recent years functional methods have become central to the study of many mathematical problems, in particular of those described by differential equations. Significant progress have been made in different areas of functional analysis, including the theory of accretive and monotone operators (founded by G. Minty, F. Browder, H. Brezis) and the nonlinear semigroup theory (developed by Y. Komura, T. Kato, H. Brezis, M.G. Krandall, A. Pazy, a.o.). As a consequence there has been significant progress in the study of nonlinear differential equations associated with monotone or accretive operators. Our aim here is to emphasize the importance of functional methods in the study of a broad range of boundary value problems. Many applications will be discussed in detail. The goals of the course: The main goal of the course is to introduce students to some important functional methods and to show their applicability to various boundary value problems. We intend to discuss specific models in appropriate functional frameworks. Using functional methods, we sometimes are able to propose new models which are more general than the classical ones and better describe concrete physical phenomena. The learning outcomes of the course: By the end of the course, students are enabled to do independent study and research in fields touching on the topics of the course, and how to use these methods to solve specific problems. In addition, they develop some special expertise in the topics covered, which they can use efficiently in other mathematical fields, and in applications, as well. They also learn how the topic of the course is interconnected to various other fields in mathematics, and in science, in general. More detailed display of contents (week-by-week): Week 1: Function spaces, scalar and vector-valued distributions Weeks 2-3: Monotone operators, convex functions, subdifferentials Weeks 4-5: Operator semigroups, linear and nonlinear evolution equations Week 6: Elliptic boundary value problems (formulation, assumptions, existence results, applications) Weeks 7-8: Parabolic problems with algebraic boundary conditions (formulation, assumptions, existence and uniqueness results, stability, applications) Week 9: Parabolic problems with dynamic boundary conditions (formulation, assumptions, existence and uniqueness results, stability, applications) Weeks 10-12: Hyperbolic problems with algebraic and/or dynamic boundary conditions(formulation, assumptions, existence and uniqueness results, stability, applications) Reference: Gheorghe Morosanu, Functional Methods in Differential Equations, Chapman&Hall/CRC, 2002 and some chapters of other books
Basic principles and methods concerning dynamic control systems are discussed. The main concepts (observability, controllability, stabilizability, optimality conditions, etc.) are addressed, with special emphasis on linear systems and quadratic functionals. Many applications are discussed in detail. The goals of the course: The main goal of the course is to introduce students to the theory of optimal control for differential systems. We also intend to discuss specific problems which arise from real world applications in order to illustrate this remarkable theory. The learning outcomes of the course: By the end of the course, students are enabled to do independent study and research in fields touching on the topics of the course, and how to use these methods to solve specific problems. In addition, they develop some special expertise in the topics covered, which they can use efficiently in other mathematical fields, and in applications, as well. They also learn how the topic of the course is interconnected to various other fields in mathematics, and in science, in general. More detailed display of contents (week-by-week): Weeks 1-2: Linear and nonlinear differential systems (local and global existence of solutions, continuous dependence on data, stability, differential inclusions) Week 3: Observability of linear autonomous systems (definition, observability matrix, necessary an sufficient conditions for observability) Week 4: Observability of linear time varying systems (definition, observability matrix, numerical algorithms for observability) Week 5: Input identification for linear systems (definition, the rank condition in case of autonomous systems, examples) Week 6: Controllability of linear systems (definition, controllability of autonomous systems, controllability matrix, Kalman’s rank condition, the case of time varying systems) Week 7: Controllability of perturbed systems (perturbations of the control matrix, nonlinear autonomous systems, time varying systems) Week 8: Stabilizability (definition, state feedback, output feedback, applications) Week 9: General optimal control theory (Meyer’s problem, Pontryagin’s minimum principle, examples) Weeks 10-11: Linear quadratic regulator theory (introduction, the Riccati equation, perturbed regulators, applications) Week 12: Time optimal control (the general problem, linear systems, bang-bang control, applications)
5)PARTIAL DIFFERENTIAL EQUATIONS Course coordinator: Gheorghe Morosanu No. of Credits: 3, and no. of ECTS credits: 6 Prerequisites: - Course Level: introductoryPhD Brief introduction to the course: After a short introduction into the main typical problems, some of the most important methods and techniques are described, including both classical and modern aspects of the theory of partial differential equations. Some applications are included to illustrate the theoretical results. The goals of the course: The main goal of the course is to introduce students to the main topics and methods of the theory of partial differential equations. The learning outcomes of the course: By the end of the course, students are enabled to do independent study and research in fields touching on the topics of the course, and how to use these methods to solve specific problems. In addition, they develop some special expertise in the topics covered, which they can use efficiently in other mathematical fields, and in applications, as well. They also learn how the topic of the course is interconnected to various other fields in mathematics, and in science, in general. More detailed display of contents (week-by-week):
4-5. Eigenvalues and eigenvectors. Applications to the Laplace operator with different boundary conditions
9-10. The semigroup approach for linear parabolic and hyperbolic PDE’s 11-12. Boundary value problems associated with nonlinear PDE’s References: 1. H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2011. 2. A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, 1969. 3. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, 1983. 6) APPROXIMATION THEORY Course Coordinator: Andras Kroo No. of Credits: 3, and no. of ECTS credits: 6 Prerequisites: - Course Level: introductoryPhD Brief introduction to the course: The main theorems of Approximation Theory are presented among others about approximation by polynomials, and by the application related splines. The goals of the course: The main goal of the course is to introduce students to the main topics and methods of the Approximation Theory. The learning outcomes of the course: By the end of the course, students are enabled to do independent study and research in fields touching on the topics of the course, and how to use these methods to solve specific problems. In addition, they develop some special expertise in the topics covered, which they can use efficiently in other mathematical fields, and in applications, as well. They also learn how the topic of the course is interconnected to various other fields in mathematics, and in science, in general. More detailed display of contents (week-by-week):
Reference: R. DeVore and G. Lorentz, Constructive Approximation, Springer, 1991.
7) NONLINEAR FUNCTIONAL ANALYSIS Course Coordinator: Denes Petz No. of Credits: 3, and no. of ECTS credits: 6 Prerequisites: - Course Level: intermediatePhD Brief introduction to the course: The main theorems of Non Linear Functional Analysis like the von Neumann Minimax theorem on the existence of sadle points, and various applications of the theory are presented. The goals of the course: The main goal of the course is to introduce students to the main topics and methods of the theory of Non Linear Functional Analysis. The learning outcomes of the course: By the end of the course, students are enabled to do independent study and research in fields touching on the topics of the course, and how to use these methods to solve specific problems. In addition, they develop some special expertise in the topics covered, which they can use efficiently in other mathematical fields, and in applications, as well. They also learn how the topic of the course is interconnected to various other fields in mathematics, and in science, in general. More detailed display of contents (week-by-week):
Reference: D. Petz, The algebra of the canonical commutation relation, Leuven University Press, 1990. 8) SPECIAL FUNCTIONS AND RIEMANN SURFACES Course coordinator: KarolyBoroczky No. of Credits: 3, and no. of ECTS credits: 6 Prerequisites: Complex Function Theory Course Level: introductory PhD Brief introduction to the course: Some interesting topics in one complex variable are presented like gamma function, Riemann’s zeta function, analytic continuation, monodromy theorem, Riemann surfaces, universal cover, uniformization theorem The goals of the course: The goal of the course is to acquaint the students with the basic understanding of special functions and Riemann surfaces The learning outcomes of the course: By the end of the course, students areexperts on the topic of the course, and how to use these methods to solve specific problems. In addition, they develop some special expertise in the topics covered, which they can use efficiently in other mathematical fields, and in applications, as well. They also learn how the topic of the course is interconnected to various other fields in mathematics, and in science, in general. More detailed display of contents (week-by-week): Week 1: Analytic continuation, Monodromy Theorem Week 2: Normal families Week 3: Blaschke products, The Mittag-Leffler theorem Week 4: The Weierstrass theorem Week 5: Euler’ Gamma Function Week 6: Riemann’s zeta function Week 7: Riemann surfaces Week 8: Simply connected Riemann surfaces, hyperbolic structure on the disc Week 9: Covering spaces, Universal cover Week 10: Covering the twice punctured plane, Great Picard theorem Week 11: Differential forms on Riemann surfaces Week 12: Overview of uniformization theorem and Riemann-Roch theorem
J. B. Conway: Functions of one complex variable I and II, Springer-Verlag, 1978. 9) COMPLEX MANIFOLDS Course Coordinator: Robert Szoke No. of Credits: 3, and no. of ECTS credits: 6 Prerequisites: - Course Level: intermediatePhD Brief introduction to the course: The main theorems of Complex Manifolds like the Hodge decomposition theorem on compact Kahler manifolds are presented. The goals of the course: The main goal of the course is to introduce students to the main topics and methods of the theory of Complex Manifolds. The learning outcomes of the course: By the end of the course, students are enabled to do independent study and research in fields touching on the topics of the course, and how to use these methods to solve specific problems. In addition, they develop some special expertise in the topics covered, which they can use efficiently in other mathematical fields, and in applications, as well. They also learn how the topic of the course is interconnected to various other fields in mathematics, and in science, in general. More detailed display of contents (week-by-week): Week 1 Basic definitions, examples and constructions. Week 2 Differential forms on manifolds, (p,q) forms. Week 3-4 Tangent bundle, vector bundles, bundle valued forms and Dolbeault cohomology groups, metrics, Hodge * operator. Week 5-6 Sobolev spaces of sections, differential operators between vector bundles and their adjoint, symbol. Week 7 Pseudo-differential operators. Week 8-9 Parametrix for elliptic differential operators, fundamental decomposition theorem for self-adjoint elliptic operators and complexes. Week 10 Harmonic forms, complex Laplacian, Week 11 Kahler manifolds. Week 12 Hodge decomposition theorem on compact Kahler manifolds. References: 1. K. Kodaira: Complex manifolds, Holt, 1971. 2. R.O. Wells: Differential analysis on complex manifolds, Springer, 1979.
The course introduces the students to the theory of unbounded operators, C*-algebras, orthogonal polynomials and mathematical foundations of certain area of quantum theory. The goals of the course: The main goal of the course is to introduce students to the main topics and methods of the theory of CCR algebras. The learning outcomes of the course: By the end of the course, students are enabled to do independent study and research in fields touching on the topics of the course, and how to use these methods to solve specific problems. In addition, they develop some special expertise in the topics covered, which they can use efficiently in other mathematical fields, and in applications, as well. They also learn how the topic of the course is interconnected to various other fields in mathematics, and in science, in general. Yüklə 233,54 Kb. Dostları ilə paylaş:
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