题目： Summability and Resurgence in One Complex Variable

报告人： David SAUZIN 教授，法国

Abstract : Since the exposition requires only a bit of familiarity with analytic functions of one complex variable, reminders will be made at the beginning of the course, about analytic functions and convergent power series. In these reminders, the proofs of the main results are omitted, however a lot explanations are provided so as to acquaint the attendants with useful practical tools.

The theories of summability and resurgence deal with the mathematical use of certain divergent power series. Resurgence has recently become the subject of the attention of many mathematicians and physicists working in geometry and quantum physics. In the context of analytic differenceor-differential equations, it sheds new light on the Stokes phenomenon (for linear and nonlinear problems). It has also been successfully applied in the context of the exact WKB method. Stokes phenomenon and WKB method are important and classical topics, very useful for the students’ curriculum; Resurgence is now becoming more and more popular among researchers.

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. an analytic 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.

Classical examples will be considered: the Euler series, the Stirling series, a less known example by Poincaré, and the Riemann-Hurwitz zeta function, which will offer the occasion of an excursion in Analytic Number Theory (introduction to the theory of the Riemann zeta function, resurgent proof of the functional equation, evocation of the Prime Number Theorem and the Riemann Hypothesis).

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. Application to WKB expansions in quantum mechanics will also be outlined

课程计划：

Reminder on the convergence of power series
Reminder on analytic functions of one complex variable
Divergent series as asymptotic expansions
Laplace transform and Borel-Laplace summation
Introduction to resurgence theory
Examples: Stirling series, Riemann-Hurwitz zeta function
Application to analytic ordinary differential equations, linear and nonlinear Stokes phenomenon
Application to WKB expansions in quantum mechanics.

参考文献: C. Mitschi and D. Sauzin. “Divergent Series, Summability and Resurgence I, Monodromy and Resurgence”. Lecture Notes in Mathematics 2153, Springer, 2016, 298 pp. The second part of this book (pp. 121-293, by D. Sauzin) is particularly relevant for the course; however, reminders will be made at the beginning of the course, about analytic functions and

convergent power series, and along the course in accordance with the needs of the attendants.

课程说明：Course is aimed at undergraduate & master & PhD students. The course is given in

English, written on the blackboard in full details, the level of the exposition being adjusted to the

background of the attendants.

时间： 2019年3月26日-6月15日每周二上午9：00-12：00

地点：565net必赢客户端本部教二楼 713 教室

联系人：孙善忠

欢迎全体师生积极参加！