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SUMMARY:Towards a multivariable Wiener-Hopf method: Lecture 3 - Raphael As
 sier (University of Manchester)\; Andrey Shanin (Moscow State University)
DTSTART:20190808T110000Z
DTEND:20190808T121500Z
UID:TALK128233@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:<span>A multivariable\, in particular two complex variables (2
 D)\, Wiener-Hopf (WH) method is one of the desired generalisations of the 
 classical and celebrated WH technique that are easily conceived but very h
 ard to implement (the second one\, indeed\, is the matrix WH). A 2D WH met
 hod\, could potentially be used e.g. for finding a solution to the canonic
 al problem of diffraction by a quarter-plane.   &nbsp\;  <br></span><br><s
 pan>Unfortunately\, multidimensional complex analysis seems to be way more
  complicated than complex analysis of a single variable. There exists a nu
 mber of powerful theorems in it\, but they are organised into several disj
 oint theories\, and\, generally all of them are far from the needs of WH. 
  In this mini-lecture course\, we hope to introduce topics in complex anal
 ysis of several variables that we think are important for a generalisation
  of the WH technique. We will focus on the similarities and differences be
 tween functions of one complex variable and functions of two complex varia
 bles. Elements of differential forms and homotopy theory will be addressed
 .   <br></span><br>We will start by reviewing some known attempts at build
 ing a 2D WH and explain why they were not successful. The framework of Fou
 rier transforms and analytic functions in 2D will 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 sin
 gularity sets and how to describe how a multidimensional contour bypasses 
 these singularities. We will explain how multidimensional integral represe
 ntation can be used in order to perform an analytical continuation of the 
 unknowns of a 2D functional equation and why we believe it to be important
 . Finally\, time permitting\, we will discuss the branching structure of c
 omplex integrals depending on some parameters and introduce the so-called 
 Picard-Lefschetz formulae.&rdquo\;  <br><br><br><br>
LOCATION:Seminar Room 1\, Newton Institute
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