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11 July 2004

16:12

 

Lecture Three

 

Linear differential equations drive= n by paths of finite length - series solutions and motivation for considering iterated integrals.

Factorial decay of the iterated int= egrals p<2. Do these methods prove that the solution has finite  p-variati= on? Linear differential equations as entire analytic functions.

Applications: rolling a sphere, and solving time inhomogeneous second order odes. Simple data compression.

The abstract object: the full signa= ture of a path segment - as an element of the tensor series. The Chen-multiplica= tive property.

Linear functionals on the Tensor al= gebra as co-ordinate iterated integrals on path space. The shuffle product. Group like elements in the tensor algebra.

The range of the signature map is a group. The truncated signature as a map into the free n-step nilpotent Lie group.

The universal properties of the Ten= sor Algebra.

 

 

 

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11 July 2004

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12 July 2004

07:53

 

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12 July 2004

08:36

 

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Terry Lyons, Mathematical Institute, 24-29 St. Giles, Oxford

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