Exploration into divergent series, resurgence and non-pertrubative analysis
Perturbative expansions in quantum field theory are generically divergent. Rather than being a flaw, this divergence reflects the presence of non-perturbative physics that cannot be captured by ordinary power series. This course will develop the mathematical framework for understanding such expansions, including asymptotic series, Borel summation, transseries, Stokes phenomena, and Écalle’s theory of resurgence. The goal is to understand how divergent series encode non-perturbative information and how one can systematically extract it.
A secondary motivation comes from recent developments in holography. In AdS/CFT, the gravitational path integral appears to compute quantities with an ensemble-like character, while the dual CFT is a single deterministic quantum system. This tension is highlighted by the factorization puzzle of Euclidean wormholes. While our focus will remain on the mathematics of divergent series, these developments provide a useful physical context where non-perturbative effects and large-N behavior play a central role.
The main references will be Costin’s Asymptotics and Borel Summability[2], and the physics-oriented primer of Aniceto–Başar–Schiappa[1], Mariño’s Instantons and Large N [4]. Additional background on holography and wormholes will be drawn from JT gravity as a matrix integral[6], No Ensemble Average[7], and Filtering CFTs at large N[3]. Mitschi–Sauzin’s Divergent Series, Summability and Resurgence I[4] may be also discussed.
A secondary motivation comes from recent developments in holography. In AdS/CFT, the gravitational path integral appears to compute quantities with an ensemble-like character, while the dual CFT is a single deterministic quantum system. This tension is highlighted by the factorization puzzle of Euclidean wormholes. While our focus will remain on the mathematics of divergent series, these developments provide a useful physical context where non-perturbative effects and large-N behavior play a central role.
The main references will be Costin’s Asymptotics and Borel Summability[2], and the physics-oriented primer of Aniceto–Başar–Schiappa[1], Mariño’s Instantons and Large N [4]. Additional background on holography and wormholes will be drawn from JT gravity as a matrix integral[6], No Ensemble Average[7], and Filtering CFTs at large N[3]. Mitschi–Sauzin’s Divergent Series, Summability and Resurgence I[4] may be also discussed.
Lecturer
Date
24th March ~ 18th June, 2026
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Tuesday,Thursday | 09:50 - 11:25 | Shuangqing-A731 | Zoom 16 | 468 248 1222 | BIMSA |
Reference
1. I. Aniceto, G. Başar, R. Schiappa, “A primer on resurgent transseries and their asymptotics,” Physics Reports 809 (2019) 1–135 [arXiv:1802.10441].
2. O. Costin, Asymptotics and Borel Summability, CRC Press, 2008/2009.
3. H. Liu, “Filtering CFTs at large N: Euclidean Wormholes, Closed Universes, and Black Hole Interiors,” 2025 [arXiv:2512.13807].
4. M. Mariño, Instantons and Large‑N: An Introduction to Non‑Perturbative Methods in QFT, Cambridge University Press, 2015.
5. C. Mitschi, D. Sauzin, Divergent Series, Summability and Resurgence I: Monodromy and Resurgence, Springer, 2016.
6. P. Saad, S. H. Shenker, D. Stanford, “JT gravity as a matrix integral,” 2019 [arXiv:1903.11115].
7. J.-M. Schlenker, E. Witten, “No Ensemble Average Below the Black Hole Threshold,” 2022 [arXiv:2202.01372].
2. O. Costin, Asymptotics and Borel Summability, CRC Press, 2008/2009.
3. H. Liu, “Filtering CFTs at large N: Euclidean Wormholes, Closed Universes, and Black Hole Interiors,” 2025 [arXiv:2512.13807].
4. M. Mariño, Instantons and Large‑N: An Introduction to Non‑Perturbative Methods in QFT, Cambridge University Press, 2015.
5. C. Mitschi, D. Sauzin, Divergent Series, Summability and Resurgence I: Monodromy and Resurgence, Springer, 2016.
6. P. Saad, S. H. Shenker, D. Stanford, “JT gravity as a matrix integral,” 2019 [arXiv:1903.11115].
7. J.-M. Schlenker, E. Witten, “No Ensemble Average Below the Black Hole Threshold,” 2022 [arXiv:2202.01372].
Audience
Undergraduate
, Advanced Undergraduate
, Graduate
Video Public
Yes
Notes Public
Yes
Language
English
Lecturer Intro
Bowen Zhao got her PhD from Yale University in December 2020. After a postdoc at BIMSA, she joined as Assistant Professor in 2025. She is interested in General relativity and Mathematical Physics and particularly problems about Black Hole.