学术报告
Introduction to 1-summability and resurgence
题目：Introduction to 1-summability and resurgence
报告人：Prof. David Sauzin（France）
摘要: The theories of summability and resurgence deal with the mathematical use of certain divergent power series.
The first part of the course is an introduction to 1-summability. The definitions rely on the formal Borel transform and the Laplace transform along an arbitrary direction of the complex plane. Given an arc of directions, if a power series is 1-summable in that arc, then one can attach to it a Borel-Laplace sum, i.e. a holomorphic function defined in a large enough sector and asymptotic to that power series in Gevrey sense.
The second part is an introduction to Ecalle's resurgence theory. A power series is said to be resurgent when its Borel transform is convergent and has good analytic continuation properties: there may be singularities but they must be isolated. The analysis of these singularities, through the so-called alien calculus, allows one to compare the various Borel-Laplace sums attached to the same resurgent 1-summable series. In the context of analytic difference-or-differential equations, this sheds light on the Stokes phenomenon.
A few elementary or classical examples will be considered (the Euler series, the Stirling series, a less known example by Poincaré). Special attention must be devoted to non-linear operations: 1-summable series as well as resurgent series form algebras which are stable by composition. An example of a class of non-linear differential equations giving rise to resurgent solutions will be analyzed.
The exposition requires only some familiarity with holomorphic functions of one complex variable.
The lectures are based on the second part of the Lecture Notes in Mathematics 2153 to appear soon http://www.springer.com/it/book/9783319287355 (an earlier version of the text is available as https://hal.archives-ouvertes.fr/hal-00860032)
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