| Abstract |
<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,
indeed, is the matrix WH). A 2D WH method, could potentially be used e.g. for
finding a solution to the canonical problem of diffraction by a quarter-plane.
<br></span><br><span>Unfortunately,
multidimensional complex analysis seems to be way more complicated than complex
analysis of a single variable. There exists a number of powerful theorems in
it, but they are organised into several disjoint 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 analysis of
several variables that we think are important for a generalisation of the WH
technique. We will focus on the similarities and differences between functions
of one complex variable and functions of two complex variables. Elements 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 transforms and analytic
functions in 2D will be introduced, leading us naturally to discuss
multidimensional integration contours and their possible deformations. One of
our main focus will be on polar and branch singularity sets and how to describe
how a multidimensional contour bypasses these singularities. We will explain
how multidimensional integral representation 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 complex integrals depending on some parameters and
introduce the so-called Picard-Lefschetz formulae.”
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