This workshop will bring together leading international researchers to discuss recent advances in the arithmetic, Hodge‑theoretic, p‑adic, and computational aspects of p‑curvatures. The goal is to create an intimate and collaborative environment, fostering informal discussions and new research directions.
Time: October 27–31, 2025
Location: R610, Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS)
Organizer: Thomas Bitoun (SIMIS)
Speakers:
- Fangu Chen — UC Berkeley
- Javier Fresán — Sorbonne University & IMJ‑PRG, Paris, France
- Josh Lam — University of Bonn (Bonn Junior Fellow) & Max Planck Institute for Mathematics, Bonn, Germany
- Jae Hee Lee — Stanford University, Stanford, USA
- Arthur Ogus — University of California, Berkeley, USA
- Mao Sheng — Yau Mathematical Sciences Center (Tsinghua University) & BIMSA, Beijing, China
- Daxin Xu — Morningside Center of Mathematics, AMSS, Chinese Academy of Sciences, Beijing, China
- Kang Zuo — Wuhan University, Wuhan, China
Program: One talk in the morning and one in the afternoon each day, with extended time for questions, background, and informal discussion.
Monday
10:00-12:00 Mao Sheng (Zoom Meeting ID: 423 317 8953, Passcode: SIMIS)
14:00-16:00 Josh Lam (Zoom Meeting ID: 830 9896 9308, Passcode: SIMIS)
Tuesday
10:00-12:00 Jae Hee Lee (Zoom Meeting ID: 830 9896 9308, Passcode: SIMIS)
14:00-16:00 Javier Fresan (Zoom Meeting ID: 830 9896 9308, Passcode: SIMIS)
Wednesday
10:00-12:00 Jae Hee Lee (Zoom Meeting ID: 830 9896 9308, Passcode: SIMIS)
14:00-16:00 Kang Zuo (Zoom Meeting ID: 830 9896 9308, Passcode: SIMIS)
Thursday
10:00-12:00 Javier Fresan (Zoom Meeting ID: 830 9896 9308, Passcode: SIMIS)
13:00-14:30 Fangu Chen (Zoom Meeting ID: 830 9896 9308, Passcode: SIMIS)
15:00-16:30 Daxin Xu (Zoom Meeting ID: 830 9896 9308, Passcode: SIMIS)
Friday
10:00-12:00 Arthur Ogus (Zoom Meeting ID: 830 9896 9308, Passcode: SIMIS)
14:00-16:00 Josh Lam (Zoom Meeting ID: 830 9896 9308, Passcode: SIMIS)
Title: Nonlinear Hodge correspondence in positive characteristic
Speaker: Mao Sheng
Abstract: I shall report on my recent work on nonlinear Hodge correspondence in positive characteristic. It is a correspondence between certain category of nonlinear connections with nilpotent p-curvatures and certain category of nonlinear Higgs fields with nilpotency condition. I shall also introduce the notion of a nonlinear Fontaine module, and explain that the relative nilpotent de Rham moduli, equipped with the nonabelian Gauss-Manin connection, the nonabelian Hodge filtration and the nonabelian Frobenius structure induced by the Ogus-Vologodsky correspondence, is naturally a nonlinear Fontaine module.
Title: Non-abelian cohomology and p-curvature
Speaker: Josh Lam
Abstract: One central aspect of algebraic geometry is that the cohomology of algebraic varieties admit many extra structures as compared with the cohomology of, say, topological spaces. This has to do with the many different cohomology theories that we have at our disposal, such as Betti, de Rham, \’etale, and even crystalline. The goal of my talks is to discuss these in the setting of non-abelian cohomology and the many different realizations we have for it, in parallel with the setting of usual cohomology. The story over the complex numbers was pioneered by Simpson, whereas I will focus mostly on the arithmetic side of the story, namely mod p and p-adic aspects. We will see that this subject has close ties to classical questions about differential equations such as the Schlesinger system. This is based on a joint work and works in progress with Daniel LItt, following also earlier ideas of Kisin, Papaioannou, and Menzies.
Title: p-curvatures in symplectic topology
Speaker: Jae Hee Lee
Abstract: To a symplectic manifold with sufficient positivity properties (e.g. a smooth Fano or Calabi–Yau variety), one can associate a system of compatible first order differential equations with integer coefficients using counts of genus 0 holomorphic curves. This system is called the quantum D-module, and the p-curvature of its mod p reduction is related to Steenrod operations on mod p cohomology. I will survey these ideas, including a discussion of a q-deformation involving Adams operations in K-theory if time permits.
Title: Arithmetic Gevrey series
Speaker: Javier Fresan
Abstract: Arithmetic Gevrey series are power series with algebraic coefficients that satisfy a linear differential equation and certain growth conditions of arithmetic nature. Depending on the specific shape of this condition, they come into two main classes: G-functions and E-functions. I will survey on the state-of-the-art on the motivic origin of these functions, their differential equations, and their values at algebraic numbers.
Title: p-adic Simpson correspondence and Grothendieck anabelian Geometry
Speaker: Kang Zuo
Abstract: Classical Teichmüller space plays a central role in geometry, connecting Riemann surfaces, hyperbolic geometry, and complex analysis. Hitchin gave a modern perspective on this space using the non-abelian Hodge correspondence, which provides a striking interaction between complex geometry and topology via differential equations. In the p-adic setting, we propose a parallel program that can be developed through the p-adic Simpson correspondence. In this talk, I will describe an approach to constructing a p-adic analogue of Teichmüller space by applying this correspondence to certain special Higgs bundles. This p-adic Teichmüller space is expected to reveal deep connections with anabelian geometry, a field that studies how the action of Galois group on the geometric fundamental groups encode the geometric information of varieties. I will also speculate more classes of varieties for anabelian geometry.
The talk will be based on joint works with Mao Sheng and Jinbang Yang, as well as ongoing collaborations with Xiaotao Sun and Jinbang Yang.
Title: A generalization of Elkies’ theorem on infinitely many supersingular primes
Speaker: Fangu Chen
Abstract: In 1987, Elkies proved that every elliptic curve defined over $\mathbb{Q}$ has infinitely many supersingular primes. In this talk, I will present an extension of this result to certain abelian fourfolds in Mumford’s families and more generally, to certain families of Kuga-Satake abelian varieties. I will review Elkies’ proof and explain how his strategy of intersecting with CM cycles can be adapted to our setting. I will also discuss some of the techniques in our proof to study the local properties of the CM cycles.
Title: Symmetry of Hitchin and p-Hitchin maps and their p-adic analogues
Speaker: Daxin Xu
Abstract: The Hitchin map, defined on the moduli of Higgs bundles by the characteristic polynomials of Higgs fields, plays a central role in the study of Higgs bundle moduli. In parallel, Bezrukavnikov–Braverman introduced the p-Hitchin map on the moduli of de Rham bundles via the p-curvature. Chen–Zhu established a non-abelian Hodge theory based on the symmetry of these maps. In this talk, we will review these theories and discuss their p-adic analogues. The talk is based on joint work with Ben Heuer.
Title: Divided powers, p-curvature, and diffraction
Speaker: Arthur Ogus
Abstract: I will give an overview of an ongoing project with Vadim Vologodsky. We explain a geometric interpretation of p-curvature which reveals a “hidden” homogeneous structure of divided power envelopes in characteristic p. This can be used to give an explicit and purely crystalline construction of Drinfeld’s Sen operator on mod p de Rham cohomology.
Attendance: Space is limited. We welcome participation from students and researchers, especially those based in China. To attend, please email the organizer with your name, affiliation, and which days you plan to join. Please note that funding is not available for non‑speaking attendees.
Contact: For any questions regarding the workshop, please contact Thomas Bitoun — thomas@simis.cn.
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