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From Sommerfeld diffraction problems to operator factorisation: Lecture 2

Presented by: 
Frank Speck
Thursday 8th August 2019 - 09:00 to 10:15
INI Seminar Room 1


This lecture series is devoted to the interplay between diffraction and operator theory, particularly between the so-called canonical diffraction problems (exemplified by half-plane problems) on one hand and operator factorisation theory on the other hand. It is shown how operator factorisation concepts appear naturally from applications and how they can help to find solutions rigorously in case of well-posed problems as well as for ill-posed problems after an adequate normalisation.  
The operator theoretical approach has the advantage of a compact presentation of results simultaneously for wide classes of diffraction problems and space settings and gives a different and deeper understanding of the solution procedures.  

The main objective is to demonstrate how diffraction problems guide us to operator factorisation concepts and how useful those are to develop and to simplify the reasoning in the applications.  

In eight widely independent sections we shall address the following questions:  
How can we consider the classical Wiener-Hopf procedure as an operator factorisation (OF) and what is the profit of that interpretation?        
What are the characteristics of Wiener-Hopf operators occurring in Sommerfeld half-plane problems and their features in terms of functional analysis?
What are the most relevant methods of constructive matrix factorisation in Sommerfeld problems?
How does OF appear generally in linear boundary value and transmission problems and why is it useful to think about this question?     
What are adequate choices of function(al) spaces and symbol classes in order to analyse the well-posedness of problems and to use deeper results of factorisation theory?      
A sharp logical concept for equivalence and reduction of linear systems (in terms of OF) – why is it needed and why does it simplify and strengthen the reasoning?
Where do we need other kinds of operator relations beyond OF?
What are very practical examples for the use of the preceding ideas, e.g., in higher dimensional diffraction problems? 

Historical remarks and corresponding references are provided at the end of each section.



Presentation Material: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons