The IPâ€™s goal is to offer a new innovative more open approach to teaching based on use of modern computer applications and presentation of first line scientific research. As an example, computer methods will be applied in mathematical models applicable both in physical and mathematical analyses. The focus of this course in terms of its content and methods applied is on mathematics, physics, computer use and computer science. Students will learn about - numerical analysis and computer-assisted proofs for photonic crystals (Maxwellâ€™s equations, photonic band gaps),
- calculus of variations, applications to elasticity and optimization
- functional equations and inequalities
- convexity and generalized convexity.
Students will use in (1) and (3) the numerical toolbox and computer algebra system Matlab for searching solutions of functional equations and inequalities and will write new computer applications. Some of the newest scientific achievements will be presented directly by their authors. Students will approach a new theory by attending lectures and tutorials, where problems are discussed and computer experiments are carried out. Students will also learn how to write a computer application in computer algebra system in order to solve functional equations and inequalities. Similarly, students will gain experience in using programming languages for the numerical solution of Maxwell's equations and in using programming tools for establishing rigorous enclosures in the area of computer-assisted proofs. After the course students will be able to understand how the basics concepts from convexity and from the calculus of variations can be used to describe the deformation of elastic materials. They will experience how analysis, functional analysis and mechanics are combined. At the same time they will realize that similar notions of convexity are the key to solving problems in finite dimensional optimization theory. Students will be also able to use computer methods to solve problems related to the above mentioned issues. Students will be able to understand how to approach and elaborate mathematical models for various practical problems arising from the industrial and economic environment. They will know how the basics concepts from convexity and from the calculus of variations can be used to describe the deformation of elastic materials. They will experience how analysis, functional analysis and mechanics are combined. Simultaneously they will realize that similar notions of convexity are the key to solving problems in finite dimensional optimization theory. Students will be also able to use computer methods to solve problems related to the above mentioned issues. Students will obtain new convexity concepts from a generic way of thinking, to identify those properties of the convex objects that are necessary in solving optimization problems, to use some main convex programming techniques. |