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

• Dualizable additive $\infty$-categories (joint with Ishan Levy and Vova Sosnilo; in progress)
  We investigate dualizable additive $\infty$-categories and proved that dualizable additive $\infty$-categories are precisely those separated Grothendieck prestable $\infty$-categories satisfying AB4* and AB6. Also we introduce prestable motives and prestable K-theory.
• Unification of Zariski, Balmer and smashing spectrum (joint with Changhan Zou; in progress)
  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 (Draft) (Comments Welcome!)
  Many concepts from higher algebra—such as finitely presented, flat, and étale morphisms of $\mathbb E_\infty$-rings—can be naturally generalized to the setting of $t$-structured tensor triangulated $\infty$-categories ($ttt$-$\infty$-categories). Under a natural structural condition we call ``projective rigidity’’, we establish analogues of Lazard’s theorem, étale rigidity, and the universal property of the derived category. We show that projective rigidity holds in many familiar examples, including the $\infty$-categories of spectra, filtered spectra, graded spectra, genuine $G$-spectra for finite groups $G$, and Artin–Tate motivic spectra over a perfect field—all equipped with their standard $t$-structures.

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.