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.
讲师
日期
2026年03月24日 至 06月18日
位置
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| 周二,周四 | 09:50 - 11:25 | Shuangqing-A731 | Zoom 16 | 468 248 1222 | BIMSA |
参考资料
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].
听众
Undergraduate
, Advanced Undergraduate
, Graduate
视频公开
公开
笔记公开
公开
语言
英文
讲师介绍
赵博文博士于2020年12月获得耶鲁大学博士学位，随后加入北京雁栖湖应用数学研究院（BIMSA）在丘成桐教授指导下从事博士后研究，并于2025年起任助理教授。她的研究方向聚焦于广义相对论与数学物理领域，致力于探索时空几何、引力理论及相关数学结构的深层联系。