Organizer: Xuetangban | Term: Spring 2026
The goal of the seminar is to prove GAGA in complex analytic geometry following Clausen–Scholze. The outline of the topics is as follows:
liquid categories → liquid tensor product → locales and idempotent algebras → holomorphic functions → analytification → GAGA.
Main References:
Clausen–Scholze, Condensed Mathematics and Complex Geometry
Scholze, Lectures on Analytic Geometry
Ko Aoki, The sheaves–spectrum adjunction
Camargo, Notes on Solid Geometry
Speaker: Heng Du
Date: Mar. 10th
Heng will give an overview of the plan of the seminar.
Note: In Talks 2, 3, and 4, the goal is to prove Theorem 3.11 in Condensed Mathematics and Complex Geometry, or Theorem 6.11 in Lectures on Analytic Geometry.
Speaker: Hanqi Wang
Date: Mar. 24th
We will review the notation appearing in the statements of Theorems 6.10 and 6.11 in Lectures on Analytic Geometry. We will also recall the basics of analytic rings and explain the relation between Theorem 6.10 and Theorem 6.11 using the appendix to Lecture VI.
Speaker: Yifan Yao
Date: Mar. 31st
The goal of this talk is to prove Theorem 6.9 in Lectures on Analytic Geometry, following Lecture VII.
Speaker: Zhixing Huang
Date: Apr. 7th
The goal of this talk is to explain the chain of reductions:
Theorem 8.14 ⇒ Theorem 8.12 ⇒ Theorem 8.11 ⇒ Theorem 8.7 ⇒ Theorem 8.4 ⇒ Theorem 6.11.
Speaker: Mingyu Bai
Date: Apr. 14th
We will complete the proof of Theorem 6.11, and also of Theorem 3.11 in Condensed Mathematics and Complex Geometry.
Speaker: Fanyi Li
Date: Apr. 21st
The goal of this talk is to isolate the geometric language of locales used by Clausen–Scholze. We will go through the first part of Lecture VII in Condensed Mathematics and Complex Geometry. Good supplementary references are Section 5.1.2 of Camargo's notes and Aoki's paper. This can be split into two talks if necessary.
Speaker: Heng Du
Date: May 12th
The goal of this talk is to prove Proposition 5.6 and then Theorem 5.1, based on results in Section 4. This shows how the abstract theory provides a concrete sheaf of (overconvergent holomorphic) functions, and also gives a first hint of GAGA.
Speaker: Mingyu Bai
Date: May 19th
In this talk, we introduce the following notions:
Categorified locale $(X,\mathcal{C})$
The algebraic categorified locale $(X,\mathcal{C}^{alg}(X))$
The analytic categorified locale $(X(\mathbf{C}),\mathcal{C}^{an}(X))$
The valuation-theoretic space $\mathrm{Spa}(A,\mathbf{C})'$
The goal of this talk is to formulate the global comparison problem in the language of (categorified) locales in the sense of Clausen–Scholze. We will explain the natural map:
$$S(A)\to \mathrm{Spa}(A,\mathbf{C})'$$
introduced in Lecture VII, and why the valuation-theoretic space $\mathrm{Spa}(A,\mathbf{C})'$ serves as the key intermediary between the algebraic and analytic sides in the proof of GAGA. Aoki's The sheaves–spectrum adjunction is especially useful as a conceptual companion, since it clarifies the meaning of categorified locales and the relation between a locale and its associated sheaf category.
Speaker: Yingdi Qin (SIMIS), Jiahao Niu (Stanford)
Date: June 11
牛嘉豪（Stanford）
Title: From Liquid Mathematics to Analytic Stacks
Abstract: This talk gives a brief overview of how liquid analytic structures, together with the general formalism of algebraic geometry, lead to the theory of analytic stacks. Analytic stacks provide a fertile ground on which different geometric objects—schemes, analytic spaces, smooth manifolds, and beyond—can grow, meet, and interact within a single framework. As guiding examples, I will discuss Betti and de Rham stacks, and explain how their comparison gives rise to a complex analytic Riemann–Hilbert correspondence.
秦瑛迪（SIMIS）
Title: Condensed Mathematics and Universal Analytic Ring
Abstract: Condensed mathematics is a framework designed to unify the treatment of algebra and analysis, providing a more convenient way to work with algebraic objects with topology, such as topological abelian groups and topological vector spaces. It combines algebra and analysis by introducing the abelian category of condensed abelian groups, which contains (compactly generated) topological abelian groups as a full subcategory. Clausen and Scholze developed new proofs of GAGA, Riemann-Roch and Grauert Coherent theorems in complex geometry under this framework.
An analytic ring is a condensed ring A equipped with the additional data of a notion of “complete A-modules” and a corresponding “complete tensor product,” which plays a fundamental role in analytic geometry. Two paradigmatic examples are the solid integers and the gaseous reals. The solid integers have applications in non-Archimedean geometry, while the gaseous reals apply to Archimedean geometry.
In this talk, I will introduce a universal analytic ring that unifies the solid integers and the gaseous reals. I will then define a universal closed unit disc over the affine line. This universal disc specializes to the classical closed unit disc both in complex geometry and in p-adic geometry, thereby providing a unified perspective on these two geometric worlds.
Fall 2025, Basics of Liqiud analytic stucture