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Towards a multivariable Wiener-Hopf method: Lecture 1

Presented by: 
Raphael Assier
Andrey Shanin
Tuesday 6th August 2019 - 12:00 to 13:15
INI Seminar Room 1
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.  

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.

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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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons