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SUMMARY:Cauchy-type integrals in multivariable complex analysis - Loredana
  Lanzani (Syracuse University)
DTSTART:20191029T160000Z
DTEND:20191029T170000Z
UID:TALK133309@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:This is joint work with Elias M. Stein (Princeton University).
 <br><br>The classical Cauchy theorem and Cauchy integral formula for analy
 tic functions of one complex variable give rise to a plethora of applicati
 ons to Physics and Engineering\, and as such are essential components of t
 he Complex Analysis Toolbox. <br><br>Two crucial features of the integrati
 on kernel of the Cauchy integral (Cauchy kernel\, for short) are  its ``an
 alyticity&rsquo\;&rsquo\; (the Cauchy kernel is an analytic function of th
 e output variable) and  its ``universality&rsquo\;&rsquo\; (the Cauchy int
 egral is meaningful for almost any contour shape). One drawback of  the Ca
 uchy kernel is that it lacks a good transformation law under conformal map
 s (with a few exceptions).<br><br>This brings up two questions: Are there 
 other integration kernels that retain the main features of the Cauchy kern
 el but also have  good transformation laws under conformal maps? And: is t
 here an analog of the Cauchy kernel for analytic functions of two (or more
 ) complex variables that retains the aforementioned crucial features? <br>
 <br>In this talk I will give a survey of what is known of these matters\, 
 with an eye towards enriching the Complex Analysis Toolbox as we know it\,
  and towards building a ``Multivariable Complex Analysis Toolbox&rsquo\;&r
 squo\;.
LOCATION:Seminar Room 1\, Newton Institute
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