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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Towards a multivariable Wiener-Hopf method: Lectur
e 4 - Raphael Assier (University of Manchester)\;
Andrey Shanin (Moscow State University)
DTSTART;TZID=Europe/London:20190809T090000
DTEND;TZID=Europe/London:20190809T101500
UID:TALK128296AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/128296
DESCRIPTION: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.
Unfortunately\, multidi
mensional complex analysis seems to be way more co
mplicated than complex analysis of a single variab
le. There exists a number of powerful theorems in
it\, but they are organised into several disjoint
theories\, and\, generally all of them are far fro
m the needs of WH. In this mini-lecture course\,
we hope to introduce topics in complex analysis of
several variables that we think are important for
a generalisation of the WH technique. We will foc
us on the similarities and differences between fun
ctions of one complex variable and functions of tw
o complex variables. Elements of differential form
s and homotopy theory will be addressed.
We will start by reviewing some known atte
mpts at building a 2D WH and explain why they were
not successful. The framework of Fourier transfor
ms and analytic functions in 2D will be introduced
\, leading us naturally to discuss multidimensiona
l integration contours and their possible deformat
ions. One of our main focus will be on polar and b
ranch singularity sets and how to describe how a m
ultidimensional contour bypasses these singulariti
es. We will explain how multidimensional integral
representation can be used in order to perform an
analytical continuation of the unknowns of a 2D fu
nctional equation and why we believe it to be impo
rtant. Finally\, time permitting\, we will discuss
the branching structure of complex integr
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:INI IT
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