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SUMMARY:Towards a multivariable Wiener-Hopf method: Lecture 4 - Raphael As
 sier (University of Manchester)\; Andrey Shanin (Moscow State University)
DTSTART:20190809T080000Z
DTEND:20190809T091500Z
UID:TALK128296@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. <br></span><br><span>  Unfor
 tunately\, multidimensional complex analysis seems to be way more complica
 ted than complex analysis of a single variable. There exists a number of p
 owerful theorems in it\, but they are organised into several disjoint theo
 ries\, and\, generally all of them are far from the needs of WH.  In this 
 mini-lecture course\, we hope to introduce topics in complex analysis of s
 everal variables that we think are important for a generalisation of the W
 H technique. We will focus on the similarities and differences between fun
 ctions of one complex variable and functions of two complex variables. Ele
 ments of differential forms and homotopy theory will be addressed.   <br><
 /span><br>We will start by reviewing some known attempts at building a 2D 
 WH and explain why they were not successful. The framework of Fourier tran
 sforms and analytic functions in 2D will be introduced\, leading us natura
 lly to discuss multidimensional integration contours and their possible de
 formations. One of our main focus will be on polar and branch singularity 
 sets and how to describe how a multidimensional contour bypasses these sin
 gularities. We will explain how multidimensional integral representation c
 an 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 complex in
 tegr<br><br><br><br>
LOCATION:Seminar Room 1\, Newton Institute
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