Research Interests

My research interests broadly lie in the connections among stable homotopy theory, higher algebra, algebraic geometry and tt-geometry, like chromatic homotopy theory, elliptic cohomology.

Specifically, I am interested in orientations from Thom spectra, rewritings of stable homotopy theory by higher algebra, reformulations of chromatic homotopy theory by spectral algebraic geometry (SAG), spectral moduli problems, and higher algebra in t-structured tensor triangulated $\infty$-categories.

Preprints

• Higher algebra in t-structured tensor triangulated $\infty$-categories (joint with Xiangrui Shen; in preparation)
  The flatness of a module over an $\mathbb{E}_\infty$-ring can be described as the t-exactness of the tensor product functor. This statement inspires us that many concepts from Lurie’s Higher Algebra can be generalized to the setting of t-structured tt-∞-categories, including (faithful) flat and étale morphisms and so on. Under suitable conditions on a t-structured tt-∞-category, we can establish Lazard’s theorem, étale rigidity, and the universal property of the derived category, among others.
• $\infty$-macrotopoi (in preparation)
  Lars Hesselholt and Piotr Pstrągowski introduced the notion of ∞-Macrotopoi in the appendix of Dirac Geometry II. It provides an excellent categorical framework–large enough to encompass a rich class of geometric objects while remaining not too large and well-behaved, making it an ideal setting for addressing fpqc sheafification, and perfectly incorporating SAG’s geometric stacks and even condensed mathematics. We will develop more and prove some basic properties of $\infty$-macrotopoi such as Giraud’s axioms.

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

Notes

• 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.