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: TBD
Date: TBD
The goal of this talk is to prove Proposition 5.6 and then Theorem 5.1. This shows how the abstract theory provides a concrete sheaf of (overconvergent holomorphic) functions, and also gives a first hint of GAGA.
Speaker: TBD
Date: TBD
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: TBD
Date: TBD
The goal of this talk is to continue the formulation of GAGA in the language of categorified locales and to set up the comparison between the algebraic and analytic sides.
Speaker: TBD
Date: TBD
The goal of this talk is to complete the proof of GAGA: for a proper scheme $X$ over $\mathbb{C}$, there is a natural equivalence:
$$\mathcal{C}^{an}(X)\simeq \mathcal{C}^{alg}(X)$$
Speaker: TBD
Date: TBD
Fall 2025, Basics of Liqiud analytic stucture