Research Interests

My research interests broadly lie in the connections among stable homotopy theory, higher algebra, algebraic geometry and algebraic K-theory.

Currently, I am interested in higher algebra in generalized prestable framework, and rewritings of stable (chromatic) homotopy theory by higher algebra.

Preprints

• Dualizable additive $\infty$-categories (joint with Ishan Levy and Vova Sosnilo; in progress)
  Characterizes dualizable additive $\infty$-categories as separated Grothendieck prestable $\infty$-categories satisfying AB4* and AB6, and identifies them with connective almost modules over almost connective $\mathbb{E}_{1}$-rings. Also introduces prestable motives and prestable K-theory.
• Generalized Telescope Conjecture (in progress)
  Introduces the atomic smashing frame to generalize the Balmer frame to presentably symmetric monoidal $\infty$-categories $\mathcal{V}$, and formulates the associated telescope conjecture. Resolves this conjecture in the settings of $\infty$-topoi and connective modules over connective $\mathbb{E}_\infty$-rings, establishes a recollement theorem in the stable setting, and introduces the Serre smashing frame.
• Smashing, Balmer, Zariski spectra: an ideal approach, arxiv (joint with Changhan Zou)
  We introduce a general Zariski frame functor to unify Zariski, Balmer and smashing spectrum. We introduce the notion of $\Sigma$-triviality for a pointed $\infty$-category, which allows the quotient by an ideal in it. We show that the $\Sigma$-trivialization of the \infcat of spaces is a mode.
• Higher algebra in $t$-structured tensor triangulated $\infty$-categories, arxiv
  Extends higher algebra concepts (finitely presented, flat, and étale morphisms) to $t$-structured tensor triangulated $\infty$-categories. Under “projective rigidity”—a condition shown to hold for spectra, filtered/graded spectra, genuine $G$-spectra, and Artin–Tate motivic spectra—we establish analogues of Lazard’s theorem, étale rigidity, and the universal property of the derived category.

Talks and slides

• $\infty$-topoi and parametrized homotopy theory
  Graduate Topology Seminar at SUSTech, 2024/6/17
• Picard $\infty$-groupoids, Picard groups of $E_\infty$-rings and generalized Thom spectra
  Graduate Topology Seminar at SUSTech, 2024/4/23
  In this talk, we will introduce the Picard $\infty$-groupoids and calculate Picard groups of several $\mathbb{E}_\infty$-rings, like K-theory spectra and topological modular forms. Besides, we also introduce how to generalize the Thom spectrum functor into any presentably symmetric monoidal $\infty$-category, by using the universal property of Picard $\infty$-groupoids.
• Barr-Beck Theorem, Morita theory and Brauer groups in $\infty$-categories
  Graduate Topology Seminar at SUSTech, 2024/3/19
  In this talk, we will introduce the $\infty$-categorical version of the Barr-Beck theorem, Morita theory and Brauer groups. And we will see that the Brauer group $\operatorname{Br}(\mathbb{S})$ of the sphere spectrum is zero using the spectral sequence involving the etale cohomology by Antieau–Gepner’s work in 2012.
• An overview of $\infty$-categories and higher algebra
  Graduate Topology Seminar at SUSTech, 2023/12/26
• The σ-orientation and its AHR $\mathbb{E}_{\infty}$-refinement $MString\to tmf$
  IWoAT Summer School 2023: Operads, spectra, and multiplicative structures, BIMSA, Beijing, China, 2023/08/17
• Thom spectra, infinite loop spaces, generalized cocycles, and the $\sigma$-orientation
  Graduate Topology Seminar at SUSTech, 2023/05/23
• Sites, Sheaves, Formal Groups and Stacks
  Graduate Topology Seminar at SUSTech, 2022/12/06
• Elliptic curves and Abelian varieties
  Undergraduate topology seminar, Sichuan University, 2022/05/30
• The Stable homotopy theory and EKMM framework
  Graduate Topology Seminar at SUSTech, 2021/11/18

Writings

• Elliptic cohomology theories and the $\sigma$-orientation
  Ando-Hopkins-Strickland found a special orientation from $MU\langle 6\rangle$ to elliptic cohomology theories, called $\sigma$-orientation. In this note we will give both topological and algebro-geometric settings of $\sigma$-orientation. Furthermore, we will introduce the precise definitions of formal groups, line bundles on a formal group, and particularly the $n$-connective cover of an $E_{\infty}$-space, which seems not well-described in ordinary references.
• Postnikov-type convergence in $\infty$-categorical framework
  Lurie introduced an $\infty$-categorical framework for the postnikov tower in a presentable $\infty$-category in his HTT. Here we will generalize it to the case of any ascending sequence of reflective full subcategories in an $\infty$-category.
• The Right Adjunction of Thom spectrum Functor
  A specific description of the right adjoint functor to Thom spectrum functor, which is given by the total space of a fiber bundle with fibers infinite loop spaces.
• Notes on elliptic curves and abelian varieties
  This note will provide an introduction to formal groups, elliptic curves and abelian varieties. We first how to get a natural formal group from a smooth group variety. Second we prove that any elliptic curve admits a natural structure of group variety by a technique about relative effective Cartier divisor. After that, we introduce étale-local decomposition and the quotient scheme. In the last chapter we will see that elliptic curves are exactly abelian varieties of $\operatorname{dim}=1$ and that any abelian variety is automatically commutative, smooth and projective. Furthermore we can see that the group structure on an abelian variety is unique under a prescribed unit.