I will speak about the quest ion of the mathematically

optimal reconstr uction of a function from a finite number of terms of its power

series at a point\, and on a ditional data such as: as domain of analyticity\,< br>

bounds or others.

\ ;

Aside from its intrinsic mathem atical interest\, this

question is importa nt in a variety of applications in mathematics and physics

such as the practical computation of the Painleve transcendents\, which I will

< br> use as an example\, and the reconstruction of functions from resurgent

perturbative seri es in models of quantum field theory and string th eory. Given

a class of functions which hav e a common Riemann surface and a common type of bo unds

on it\, we show that the optimal proc edure stems from the uniformization

theore m. A priori Riemann surface information and bounds exist for the Borel

transform of asymptot ic expansions in wide classes of mathematical prob lems

such as meromorphic systems of linear or nonlinear ODEs\, classes of PDEs and

m any others\, \; known\, by mathematical

theorems\, \; to be resurgent. \; I will also discuss some (apparently) new

unifor mization methods and maps. Explicit uniformization in Borel plane is

possible for all linear or nonlinear second order meromorphic ODEs.

\;

This optimal procedure is dramatically superior to the

existing (generally ad-hoc) ones\, both theoretica lly and in their effective

numerical appli cation\, which I will illustrate. The comparison w ith Pade approximants

is especially intere sting.

\;

W hen more specific information exists\, such as the nature

of the singularities of the functi ons of interest\, we found methods based on

convolution operators to eliminate these singula rities. The type of

singularities is known for resurgent functions coming from many problems in

analysis. With this addition\, the acc uracy is improved substantially with

respe ct to the optimal accuracy which would be possible in full generality.

\;

< br>

Work in collaboration with G. Dunne\, U. Conn.

LOCATION:Seminar Room 2\, Newton Institute CONTACT:info@newton.ac.uk END:VEVENT END:VCALENDAR