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 known attempts at building a 2D WH and explai n why they were not successful. The framework of F ourier transforms and analytic functions in 2D wil l be introduced\, leading us naturally to discuss multidimensional integration contours and their po ssible deformations. One of our main focus will be on polar and branch singularity sets and how to d escribe how a multidimensional contour bypasses th ese singularities. We will explain how multidimens ional integral representation can be used in order to perform an analytical continuation of the unkn owns of a 2D functional equation and why we believ e it to be important. Finally\, time permitting\, we will discuss the branching structure of complex integrals depending on some parameters and introd uce the so-called Picard-Lefschetz formulae.&rdquo \;

LOCATION:Seminar Room 1\, Newton Institute CONTACT:info@newton.ac.uk END:VEVENT END:VCALENDAR