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SUMMARY:From Sommerfeld diffraction problems to operator factorisation: Le
 cture 1 - Frank Speck (Universidade de Lisboa)
DTSTART:20190807T144500Z
DTEND:20190807T160000Z
UID:TALK128182@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:<span>This lecture series is devoted to the interplay between 
 diffraction and operator theory\, particularly between the so-called canon
 ical diffraction problems (exemplified by half-plane problems) on one hand
  and operator factorisation theory on the other hand. It is shown how oper
 ator factorisation concepts appear naturally from applications and how the
 y can help to find solutions rigorously in case of well-posed problems as 
 well as for ill-posed problems after an adequate normalisation.  &nbsp\;  
 <br></span><br><span>The operator theoretical approach has the advantage o
 f a compact presentation of results simultaneously for wide classes of dif
 fraction problems and space settings and gives a different and deeper unde
 rstanding of the solution procedures.   &nbsp\;  <br></span><br><span>The 
 main objective is to demonstrate how diffraction problems guide us to oper
 ator factorisation concepts and how useful those are to develop and to sim
 plify the reasoning in the applications.  &nbsp\; <br></span><br><span> In
  eight widely independent sections we shall address the following question
 s:  &nbsp\; <br></span><span>How can we consider the classical Wiener-Hopf
  procedure as an operator factorisation (OF) and what is the profit of tha
 t interpretation?&nbsp\;<span>&nbsp\;&nbsp\;&nbsp\;&nbsp\;&nbsp\;&nbsp\;&n
 bsp\; <br></span></span><span>What are the characteristics of Wiener-Hopf 
 operators occurring in Sommerfeld half-plane problems and their features i
 n terms of functional analysis? <br></span><span> What are the most releva
 nt methods of constructive matrix factorisation in Sommerfeld problems? <b
 r></span><span>How does OF appear generally in linear boundary value and t
 ransmission problems and why is it useful to think about this question?<sp
 an>&nbsp\;&nbsp\;&nbsp\;&nbsp\;&nbsp\; <br></span></span><span>What are ad
 equate choices of function(al) spaces and symbol classes in order to analy
 se the well-posedness of problems and to use deeper results of factorisati
 on theory?<span>&nbsp\;&nbsp\;&nbsp\;&nbsp\;&nbsp\;&nbsp\; <br></span></sp
 an><span>A sharp logical concept for equivalence and reduction of linear s
 ystems (in terms of OF) &ndash\; why is it needed and why does it simplify
  and strengthen the reasoning? <br></span><span>Where do we need other kin
 ds of operator relations beyond OF? <br></span><span>What are very practic
 al examples for the use of the preceding ideas\, e.g.\, in higher dimensio
 nal diffraction problems?&nbsp\; <br></span><br>Historical remarks and cor
 responding references are provided at the end of each section.  <br>
LOCATION:Seminar Room 1\, Newton Institute
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