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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Towards a multivariable Wiener-Hopf method: Lectur
 e 2 - Raphael Assier (University of Manchester)\; 
 Andrey Shanin (Moscow State University)
DTSTART;TZID=Europe/London:20190807T090000
DTEND;TZID=Europe/London:20190807T101500
UID:TALK127987AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/127987
DESCRIPTION:<span>A multivariable\, in particular two complex 
 variables (2D)\, Wiener-Hopf (WH) method is one of
  the desired generalisations of the classical and 
 celebrated WH technique that are easily conceived 
 but very hard to implement (the second one\, indee
 d\, is the matrix WH). A 2D WH method\, could pote
 ntially be used e.g. for finding a solution to the
  canonical problem of diffraction by a quarter-pla
 ne.   &nbsp\; <br></span><br><span> Unfortunately\
 , multidimensional complex analysis seems to be wa
 y more complicated than complex analysis of a sing
 le variable. There exists a number of powerful the
 orems in it\, but they are organised into several 
 disjoint theories\, and\, generally all of them ar
 e far from the needs of WH.  In this mini-lecture 
 course\, we hope to introduce topics in complex an
 alysis of several variables that we think are impo
 rtant for a generalisation of the WH technique. We
  will focus on the similarities and differences be
 tween functions of one complex variable and functi
 ons of two complex variables. Elements of differen
 tial forms and homotopy theory will be addressed. 
   <br></span><br><span>We will start by reviewing 
 some known attempts at building a 2D WH and explai
 n why they were not successful. The framework of F
 ourier transforms and analytic functions in 2D wil
 l be introduced\, leading us naturally to discuss 
 multidimensional integration contours and their po
 ssible deformations. One of our main focus will be
  on polar and branch singularity sets and how to d
 escribe how a multidimensional contour bypasses th
 ese singularities. We will explain how multidimens
 ional integral representation can be used in order
  to perform an analytical continuation of the unkn
 owns of a 2D functional equation and why we believ
 e it to be important. Finally\, time permitting\, 
 we will discuss the branching structure of complex
  integrals depending on some parameters and introd
 uce the so-called Picard-Lefschetz formulae.&rdquo
 \;  <b>&nbsp\;</b>  </span><br><br><br><br>
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:INI IT
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