# Qingyuan Jiang

(Pronunciation: "Ch'ing --Yuen, Jyahng". IPA: [tʃɪŋ--jʊˈan, dʒjaŋ] )

## Chinese characters: 姜清元

I am currently an Assistant Professor (Tenure-Track) at the Hong Kong University of Science and Technology (HKUST) since December 2023.

Before joining HKUST, I was a visiting scholar at the Hausdorff Research Institute for Mathematics in Bonn from September to December 2023, participating in the program Algebraic geometry: derived categories, Hodge theory, and Chow groups.

From 2019 to 2023, I was a postdoctoral researcher at the University of Edinburgh working with Arend Bayer, under the EPSRC grant Enhancing Representation Theory, Noncommutative Algebra And Geometry Through Moduli, Stability And Deformations and the ERC Consolidator grant WallCrossAG, no. 819864.

I was a postdoctoral member of the Institute for Advanced Study from 2018 to 2019.

I completed my PhD at the Chinese University of Hong Kong under the supervision of Conan Leung in 2018.

I received my Bachelor's degree and Master's degree from Peking University, Beijing, China, under the supervision of Huijun Fan.

Here is my CV (dated: July 2023).

## Contact

Office: Room 3491, Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong

Email: jiangqingyuan (at) gmail (dot) com

jiangqy (at) ust (dot) hk

Phone: Extn. 7437 from 57986757

## Research

I am interested in algebraic geometry and geometric representation theory, especially in topics related to derived categories and derived algebraic geometry.

### Research Papers:

[13]. Continuum envelops on Fargues--Fontaine curves and elliptic curves (joint with Heng Du and Yucheng Liu). 94 pages, 17 figures. (See also arXiv: 2404.04551)

This paper explores the applications of Bridgeland stability conditions in studying Fargues-Fontaine curves. We introduce the continuum envelopes on these curves and analyze their homological properties. Additionally, we investigate SL(2,Z) variants of Colmez--Fontaine's division algebra. Fargues--Fontaine curves exhibit strong similarities with elliptic curves and noncommutative tori from these perspectives.

[12]. Derived Categories of Derived Grassmannians. 35 pages. Submitted. (See also arXiv: 2307.02456)

(See also the poster for Imperial Conference on July 7, 2023, and here for a simple worked-out example.)

This paper verifies the author's Quot formula conjecture, generalizing and strengthening Toda's result in [Tod23]. Applications: blowups of determinantal ideals, reducible schemes, varieties of linear series on curves, etc. Our approach uses DAG, allowing us to work over arbitrary base over Q, and providing concrete descriptions of Fourier-Mukai kernels in terms of derived Schur functors.

[11]. Brill—Noether theory for Hilbert schemes of points on surfaces (joint with Arend Bayer and Huachen Chen). Int. Math. Res. Not. Volume 2024, Issue 10, May 2024, Pages 8403–8416 DOI. (See also arXiv 2304.12016);

(See also the handwritten slides and supplementary notes for more examples.)

This paper presents two proofs that the Brill--Noether loci in the Hilbert scheme of points on a smooth connected surface are non-empty whenever their expected dimension is positive, and that they are irreducible and have expected dimensions.

[10]. Derived Grassmannians and derived Schur functors. 99 pages. Submitted. (See also arXiv: 2212.10488.)

(References and typos will be updated timely here; arXiv updates will be slightly delayed.)

This paper develops two theories, the geometric theory of derived Grassmannians (and flag schemes) and the algebraic theory of derived Schur (and Weyl) functors, and establishes their connection, a derived generalization of the Borel--Weil--Bott theorem.

[9]. Derived projectivizations of complexes. 100 pages. Submitted. (See also arXiv: 2202.11636; handwritten slides.)

(References and typos will be updated timely here; arXiv updates will be slightly delayed.)

This paper develops the counterparts of Grothendieck’s theory of projectivizations in the realm of derived algebraic geometry (DAG). The main results include generalized Serre’s theorem, a derived version of Beilinson’s relations, and semiorthogonal decompositions. Applications: classical situations such as reducible schemes, stabilization maps of prestable curves, certain singular threefolds, and moduli situations such as Hecke correspondences.

[8]. Derived categories of Quot schemes of locally free quotients. 102 pages. Submitted. (See also arXiv: 2107.09193.)

This paper systematically studies the Grothendieck Quot schemes of locally free quotients, proposes a conjecture on their structures (the Quot formula), and verifies it in various cases. Our approach unifies various known formulae, such as formulae for blowups, Grassmannian bundles, standard flips/flops, and projectivizations, into a simple framework, and removes smoothness assumptions. It also leads to new results such as formulae for virtual flips, $\mathrm{Quot}_2$-formula, blowup formulae for determinantal ideals of codimension $\le 4$, etc.

[7]. On the Chow theory of Quot schemes of locally free quotients. 39 pages. Submitted. (See also arXiv:2010.10734.)

This paper establishes the Chow-theoretical Quot formula. Applications include the Chow-theoretical blowup formula for blowups of determinantal ideals, decompositions of Chow groups for moduli of linear series on curves, and for generalized nested Hilbert schemes.

[6]. On the Chow theory of projectivizations. J. Inst. Math. Jussieu., 22(3), 1465--1508 (2023), DOI. (See also arXiv: 1910.06730).

This paper establishes the Chow-theoretical projectivization formula. Applications: symmetric powers of curves, nested Hilbert schemes, and varieties resolving Voisin’s maps.

[5]. Derived category of projectivization and flops (joint with Conan Leung). Adv. Math. 396: 108--169 (2022), DOI. (See also arXiv:1811.12525).

This paper establishes (i) the projectivization formula for derived categories and (ii) ``flop--flop=twist" results for flops from Springer-type resolutions for determinantal hypersurfaces. Applications: a blowup formula for blowup along Cohen--Macaulay codimension-$2$ subschemes, symmetric powers of curves, $\Theta$-flops, and nested Hilbert schemes on surfaces.

[4]. Derived category of projectivization and generalized linear duality. 10 pages. See also arXiv: 1812.05685.

This paper establishes the duality between the projectivization formula and the blowup formula, in the sense of HPD.

[3]. Categorical duality between joins and intersections (joint with Conan Leung). 39 pages. See also arXiv:1811.05135.

This paper establishes the categorical duality between joins and intersections, in the sense of homological projective duality (HPD).

[2]. Blowing up linear categories, Refinements, and Homological Projective Duality with base locus (joint with Conan Leung). 39 pages. See also arXiv: 1811.05132.

This paper defines blowups of linear categories and their refinements in the sense of Lefschetz category theory and establishes its behavior under homological projective duality (HPD).

[1]. Categorical Pl\"ucker Formula and Homological Projective Duality (joint with Conan Leung and Ying Xie). J. Eur. Math. Soc. (JEMS) 23 (2021), no. 6, 1859--1898.

This paper generalizes the fundamental theorem of homological projective duality (HPD) of Kuznetsov from linear sections to general nonlinear intersections.

### Works in progress:

(1) . Abel maps for integral curves: a derived perspective. In preparation.

This paper is an application of the framework of derived projectivizations. For a family of integral curves, we will define the derived relative Hilbert schemes of points on the curves and study their Abel maps to compactified Jacobians. The framework of derived projectivizations allows us to completely describe these Abel maps' (coherent-)homological properties. Moreover, we will establish semiorthogonal decompositions of their derived categories in terms of the derived categories of the relative compactified Jacobians.

(2). "Galois descent and ascent for Fargues--Fontaine curves." (title subject to changes; joint with Heng Du and Yucheng Liu). In preparation.

This paper series continues our investigation into the applications of Bridgeland stability conditions in the study of Fargues-Fontaine curves. We primarily examine their equivariant categories, explore the relations between their stability conditions and tilted hearts, investigate generalizations of Banach-Comez spaces, and establish Galois descent and ascent theories for these curves.

(3). Triangle varieties from LLSVS eigthfolds. (joint with Arend Bayer and Huachen Chen). In preparation.

This paper constructs two different triangle varieties in the sense of Voisin for LLSvS eightfolds associated with very general cubic fourfolds

(4). "Heckec correspondences for surfaces revisited." In preparation.

This paper revisits the Hecke correspondences for surfaces, studied by Andrei Neguts, from the perspective of derived projectivizations. We will establish the relation between the algebra from Hecke correspondences and the elliptic Hall algebra for any surface.

## Teaching:

### MATH 3131: Honors in Linear and Abstract Algebra II. HKUST, 2024 Spring.

Course Description:

This course is designed for honor students who have completed MATH 2131: Honors in Linear and Abstract Algebra I.

Delve into the captivating realm of abstract algebra, exploring fundamental algebraic structures such as Groups, Rings, Modules, and Fields.

Course Details:

Lectures: Wednesday and Friday, 13:30 – 14:50, Room 4502.

Tutorials: Monday, 19:00 – 19:50, Room 2463.

Course Resources:

Lecture Notes (continuously expanded and updated throughout the course).

Lecture Videos: Available on Canvas.

Midterm Exam: Problems and Solutions.

Final Exam: May 27th, 2024, 04:30 PM - 07:30 PM, at LG5 Multi-function Room.

### Algebraic Geometry, University of Edinburgh, 2023 Spring.

In the spring of 2023, I co-lectured the undergraduate course algebraic geometry with Pavel Safronov. Based on lecture notes of Pavel, Miles Reid's book "Undergraduate Algebraic Geometry", and William Fulton's book "Algebraic curves: an introduction to algebraic geometry" (available from their websites). I am teaching the following lectures:

Lecture 1. Quadric surfaces. (Written notes.)

Lecture 2. Segre embedding. An algorithm for finding lines on surfaces. (Written notes.)

A supplementary note explaining why there are ten lines passing through a line on a cubic surface.

Lecture 3. Cubic surfaces and 27 lines. (Written notes.)

A supplementary note on double sixers.

Lecture 4. Rationality of cubic surfaces. (Written notes.)

Lecture 5. Blowups and resolutions of singularities. (Written notes.)

Lecture 6. Blowups, eliminations of indeterminacy of rational maps, and Cremona involutions. (Written notes.)

### Groups, Rings & Modules, SMSTC class, Edinburgh, 2022 Fall.

In the fall of 2022, I teach the second half of the SMSTC course Groups, Rings & Modules.

Lecture 1. Introduction to Modules. (In-class written notes.)

Lecture 2. Noetherian properties and Hilbert basis theorem. (In-class written notes.)

Lecture 3. Integral domains (Euclidean domains, PIDs, UFDs). (In-class written notes.)

Lecture 4. Algebraic numbers and rings of integers. (In-class written notes.)

Lecture 5. Introduction to Algebraic Geometry. (In-class written notes.)

Assignments and Suggested Solutions are available in the SMSTC system.

### Homological Algebra, University of Edinburgh, 2021 Spring.

In the spring of 2021, I co-taught the GlaMS course on Homological Algebra, organized by Ben Davison. I am in charge of the second block: examples and applications, and here are the links to the materials of this block:

Lecture 2.1. Tor and Ext functors. notes, video, and real-time written notes.

Lecture 2.2. Hochschild (co)homology. notes, video, and real-time written notes.

Exercise for the second block. Solutions to the exercise.

## Some Notes:

Presentation Notes: "What is Algebraic Geometry? ", given at the course "SCIE 1500 Guided Study on Research" in HKUST on March 19, 2024.

Lecture notes on Introduction to Derived Algebraic Geometry, given at HKUST in April 2023.

Notes on Linear Algebras and Schur Functors, written in 2022.

Notes (handwritten) on infinity categories written in Aug 2021, which could serve as a visual introduction to Lurie’s book Higher Topos Theory.

Notes (handwritten) on geometric categorification for the seminar on Braid group actions on derived categories in Dec 2020.

Notes (handwritten) on homological mirror symmetry for projective planes for Nick Sheridan's seminar on HMS in Nov 2019.