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SUMMARY:Smooth Infinitesimal Analysis - Filip Bár (University of Cambridg
 e)
DTSTART:20120208T160000Z
DTEND:20120208T170000Z
UID:TALK36262@talks.cam.ac.uk
CONTACT:Filip Bár
DESCRIPTION:This talk provides an introduction to Smooth Infinitesimal Ana
 lysis (SIA) using the Kock-Lawvere axiom. We will consider first the one-d
 imensional case and then\, after introducing the notion of Kock-Lawvere mo
 dule (called 'Euclidean R-module' in Lavendhomme)\, we will move on to _ar
 bitrary_ dimensions. \n\nK-L modules are stable under exponentiation and w
 e shall see that in the smooth world a Gâteaux-differential is in fact al
 ready a Fréchet differential. In particular\, our calculus works on  spac
 es of functionals considered in the calculus of variations in the same way
  as in finite dimensions.\n\nFinally\, we will introduce a compatible preo
 der on our ring R and the integration axiom. The usual rules to calculate 
 integrals will be obtained very easily from this and the K-L axiom. Higher
 dimensional integrals will be defined via iterated integration and Fubini'
 s theorem.\n\nThe corresponding sections in Lavendhomme’s book are 1.1.3
 \,1.1.4\, 1.2 and 1.3.
LOCATION:Centre for Mathematical Sciences\, MR4
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