q-difference equation and q-Borel Laplace
This course is an introduction to the analytic theory of linear q-difference equations, centered on q-Borel–Laplace summation. The main theme is how divergent formal solutions of q-difference equations can be analyzed through q-Gevrey asymptotics and then summed into actual analytic or meromorphic solutions. Along the way, the course will introduce the basic local structure of linear q-difference systems, the meaning of slopes, and the role of connection problems in concrete examples. A secondary theme will be the relation between q-summation and q-Stokes phenomena: in particular, we will give a gentle introduction to the idea of q-alien derivations, viewed as residues of q-Stokes operators in the irregular theory. This part will remain introductory and motivational rather than fully technical.
Lecturer
Date
2nd April ~ 18th June, 2026
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Thursday | 17:05 - 18:40 | A3-3-301 | ZOOM 14 | 712 322 9571 | BIMSA |
| Thursday | 19:20 - 20:55 | A3-3-301 | ZOOM 14 | 712 322 9571 | BIMSA |
Prerequisite
Some familiarity with power series methods, asymptotic expansions, and classical Borel–Laplace summation is helpful, but not strictly required; I will briefly review the necessary classical background before introducing the q-analogues.
Syllabus
The course will introduce the basic analytic theory of linear q-difference equations, including formal solutions, local behavior, and q-Gevrey asymptotics. Its main focus will be q-Borel and q-Laplace transforms, q-summability, and the construction of analytic or meromorphic solutions from divergent formal series. We will also discuss a small number of concrete examples, briefly touch on q-Stokes phenomena and q-alien ideas. Depending on time, we may conclude with a brief outlook on q-WKB, Galois theory and Habiro ring.
Reference
Jean-Pierre Ramis, Jacques Sauloy, Changgui Zhang, Local analytic classification of q-difference equations.
Lucia Di Vizio, Changgui Zhang, On q-summation and confluence.
Thomas Dreyfus, Confluence of meromorphic solutions of q-difference equations.
Jean-Pierre Ramis, Jacques Sauloy, The q-analogue of the wild fundamental group (I).
Jean-Pierre Ramis, Jacques Sauloy, The q-analogue of the wild fundamental group and the inverse problem of the Galois theory of q-difference equations.
Takeshi Morita, A connection formula between the Ramanujan function and the q-Airy function.
Virginie Bugeaud, Local Galois group of irregular q-difference equations.
Lucia Di Vizio, Changgui Zhang, On q-summation and confluence.
Thomas Dreyfus, Confluence of meromorphic solutions of q-difference equations.
Jean-Pierre Ramis, Jacques Sauloy, The q-analogue of the wild fundamental group (I).
Jean-Pierre Ramis, Jacques Sauloy, The q-analogue of the wild fundamental group and the inverse problem of the Galois theory of q-difference equations.
Takeshi Morita, A connection formula between the Ramanujan function and the q-Airy function.
Virginie Bugeaud, Local Galois group of irregular q-difference equations.
Video Public
Yes
Notes Public
Yes
Language
Chinese
, English
Lecturer Intro
During the course of his doctoral studies, Yong LI primarily focused on resurgence theory and its applications. His recent work involves exploring the application of resurgence theory to various subjects, including modular forms, topological invariants of 3-manifolds, quantum mechanics, etc.