Venue:
Wednesdays: Shuangqing C548
Fridays: Shuangqing B627
Description:
One-month long lectures on novel techniques in birational geometry. Equipped with this new idea and method, participants may be able to discover and prove new results of their own.
This will be a working seminar series on Universal Characteristic-free Resolution of Singularities.
The lectures will be based on the materials of the article with the same title posted on arXiv.
https://arxiv.org/pdf/2507.21400. In what follows, by the article, we mean this paper.
This seminar series has two central goals.
1. To examine and verify the key details of the article, thereby ensuring that its arguments rest on a rigorous foundation.
2. To introduce new techniques and perspectives in birational geometry, with the hope that participants will carry them forward — discovering and proving new theorems of their own.
The seminar runs twice per week, each last at a minimum of 2 hours. The lecture series will be composed of 8 sessions whose topics are as follows.
Lecture 1. The Grassmannian Gr (3, n) as a Universe for All Possible Singularities
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1a. Innovative ideas and novel approaches that fundamentally depart from existing methods. These are drawn primarily from Sections 2 and 9 (the appendix) of the article.
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1b. Elementary yet useful material on the Plücker relations of Gr (3, n), based on Section 3 of the article.
(Slogan: “Anything new you can state and prove about Grassmannians is important.” I will reveal who said this in my first lecture.)
Lectures 2 and 3. Constructing a New and Improved Universe for All Singularities
These lectures correspond to Section 4 of the article and mark the first major step of the work. We will devote two sessions: the first to the actual construction of the new universe, and the second to its key properties, which will serve as a foundation throughout the article. Proof of several central propositions will be presented in detail.
(Why these matters for the audience: the field is full of results and theorems still waiting to be discovered and proved. This is not a promise, but a reflection of the state of the art.)
Lecture 4. The First Universal Blowups of the New Universe: Universal Theta-Blowups
This lecture corresponds to Section 5 of the article. It will be a relatively quick and streamlined session. The central idea and method are both transparent, and the audience should be able to grasp them readily by the end.
Lectures 5 and 6. The More Decisive Universal Blowups: Universal p- and l-Blowups
These lectures correspond to Section 6 of the article, one of its two most technically demanding parts. By this stage, the audience will, I hope, already be immersed in the new framework and eager to dive deeper, ready for more, and nothing but more!
Lecture 7. Birational Transforms of All Singularities under the Universal Blowups
This lecture corresponds to Section 7 of the article, the next technically intensive stage. However, following Section 6 (covered in the previous two sessions), much of the material should feel more straightforward. This section also begins to clarify why our resolution process can be called
universal.
Lecture 8. The Finale: Smoothness via the Jacobian Criterion
Despite the diversity of approaches to resolving singularities, the underlying principle is the same: replace the ill-behaved Jacobian matrix Jac(I) of a problematic ideal with the well-behaved Jacobian matrix Jac(J) of a better birational ideal.
In this concluding lecture, we will do exactly that—compute a single, beautiful Jacobian matrix that works for all singularities.
Prerequisite: One-year course on Algebraic Geometry
Reference: Universal Characteristic-free Resolution of Singularities, arXiv
Target Audience: Young algebraic geometers, postdocs, and Graduate students
Teaching Language: English, Chinese
Registration: https://www.wjx.top/vm/mbzy8Ik.aspx#
