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SUMMARY:From Sommerfeld diffraction problems to operator factorisation: Le
 cture 3 - Frank Speck (Universidade de Lisboa)
DTSTART:20190809T144500Z
DTEND:20190809T160000Z
UID:TALK128320@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:This<br>lecture series is devoted to the interplay between dif
 fraction and operator<br>theory\, particularly between the so-called canon
 ical diffraction problems<br>(exemplified by half-plane problems) on one h
 and and operator factorisation theory on the other hand. It<br>is shown ho
 w operator factorisation concepts appear naturally from applications<br>an
 d how they can help to find solutions rigorously in case of well-posed<br>
 problems as well as for ill-posed problems after an adequate normalisation
 .<br><br><br><br>The<br>operator theoretical approach has the advantage of
  a compact presentation of<br>results simultaneously for wide classes of d
 iffraction problems and space<br>settings and gives a different and deeper
  understanding of the solution<br>procedures. <br><br><br><br>The<br>main 
 objective is to demonstrate how diffraction problems guide us to operator 
 factorisation concepts and how useful those<br>are to develop and to simpl
 ify the reasoning in the applications.<br><br> <br>In<br>eight widely inde
 pendent sections we shall address the following questions:<br><br> How can
  we consider the classical Wiener-Hopf procedure as an operator<br>factori
 sation (OF) and what is the profit of that interpretation?<br>What are the
  characteristics of Wiener-Hopf operators occurring in<br>Sommerfeld half-
 plane problems and their features in<br>terms of functional analysis? <br>
 What are the most relevant methods of constructive matrix<br>factorisation
  in Sommerfeld problems? How does OF appear generally in linear boundary v
 alue and transmission<br>problems and why is it useful to think about this
  question?<br>What are adequate choices of function(al) spaces and symbol 
 classes in<br>order to analyse the well-posedness of problems and to use d
 eeper results of factorisation theory?<br>A sharp logical concept for equi
 valence and reduction of linear<br>systems (in terms of OF) &ndash\; why i
 s it needed and why does it simplify and<br>strengthen the reasoning? Wher
 e do we need other kinds of operator relations beyond OF? What are very pr
 actical examples for the use of the preceding ideas\,<br>e.g.\, in higher 
 dimensional diffraction problems? Historical<br>remarks and corresponding 
 references are provided at the end of each section.
LOCATION:Seminar Room 1\, Newton Institute
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