BEGIN:VCALENDAR
VERSION:2.0
PRODID:-//talks.cam.ac.uk//v3//EN
BEGIN:VTIMEZONE
TZID:Europe/London
BEGIN:DAYLIGHT
TZOFFSETFROM:+0000
TZOFFSETTO:+0100
TZNAME:BST
DTSTART:19700329T010000
RRULE:FREQ=YEARLY;BYMONTH=3;BYDAY=-1SU
END:DAYLIGHT
BEGIN:STANDARD
TZOFFSETFROM:+0100
TZOFFSETTO:+0000
TZNAME:GMT
DTSTART:19701025T020000
RRULE:FREQ=YEARLY;BYMONTH=10;BYDAY=-1SU
END:STANDARD
END:VTIMEZONE
BEGIN:VEVENT
CATEGORIES:Synthetic Differential Geometry Seminar
SUMMARY:Micro Squares\, Connections and the Lie Bracket o
f Vector Fields - Filip Bár (University of Cambrid
ge)
DTSTART;TZID=Europe/London:20120312T170000
DTEND;TZID=Europe/London:20120312T183000
UID:TALK36973AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/36973
DESCRIPTION:Using the exponential laws in the cartesian closed
category of microlinear spaces we obtain that the
iterated tangent bundle is the bundle of micro sq
uares. There are various ways to structure it as a
bundle over the tangent bundle. One way is to see
it as a bundle over ordered pairs of tangents by
mapping a micro square to its pair of principal ax
es. This bundle turns out to be an affine bundle o
ver the tangent bundle and plays an important (uni
fying) role for connections and Lie brackets of ve
ctor fields.\n\nGeometrically\, a (linear) connect
ion on a tangent bundle is a structure that 'conne
cts' infinitesimally neighbouring tangent spaces (
respecting their R-linear structures). Any section
of the affine bundle of micro squares is consider
ed a connection\, since passing from a pair of tan
gent vectors to the respective micro square amount
s to thickening the pair of tangent vectors to an
infinitesimal grid\, allowing infinitesimal 'paral
lel' transport of one tangent along the other. If
the section is homogeneous in its arguments\, then
this yields a linear connection.\n\nUsing exponen
tial adjunction there are three ways to consider v
ector fields. Defining vector fields as sections o
f the tangent bundle of M one can consider them eq
uivalently either as infinitesimal flows\, or tang
ents at the identity map of the microlinear space
of diffeomorphisms\, i.e.\, as elements of the Lie
algebra of the Lie group Diff(M). The Lie bracket
of vector fields can be defined by using this Lie
algebra representation and translating the intend
ed geometric meaning directly into algebra. Moreov
er\, using the affine bundle structure of micro sq
uares the Lie bracket can be written in a way rese
mbling the canonical Lie bracket from ring theory.
This is because any pair of vector fields determi
nes a micro square in a natural way.
LOCATION:Centre for Mathematical Sciences\, MR9
CONTACT:Filip Bár
END:VEVENT
END:VCALENDAR