Unfortunately\ , multidimensional complex analysis seems to be wa y more complicated than complex analysis of a sing le variable. There exists a number of powerful the orems in it\, but they are organised into several disjoint theories\, and\, generally all of them ar e far from the needs of WH. In this mini-lecture course\, we hope to introduce topics in complex an alysis of several variables that we think are impo rtant for a generalisation of the WH technique. We will focus on the similarities and differences be tween functions of one complex variable and functi ons of two complex variables. Elements of differen tial forms and homotopy theory will be addressed.

We will start by reviewing some k nown 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 i ntroduced\, leading us naturally to discuss multid imensional integration contours and their possible deformations. One of our main focus will be on po lar and branch singularity sets and how to describ e how a multidimensional contour bypasses these si ngularities. We will explain how multidimensional integral representation can be used in order to pe rform an analytical continuation of the unknowns o f a 2D functional equation and why we believe it t o be important. Finally\, time permitting\, we wil l discuss the branching structure of complex integ rals depending on some parameters and introduce th e so-called Picard-Lefschetz formulae.&rdquo\;

LOCATION:Seminar Room 1\, Newton Institute CONTACT:INI IT END:VEVENT END:VCALENDAR