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 mainly interested in higher algebra in general prestable categories, dualizable categories, and rewritings of chromatic homotopy theory in a modern way.

Preprints & Publications

arXiv
Introduces the atomic smashing frame to generalize the Balmer Spectrum to any presentably symmetric monoidal $\infty$-category $\mathcal{V}$, 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.
arXiv
Characterizes dualizable additive $\infty$-categories as separated Grothendieck prestable $\infty$-categories satisfying $\mathrm{AB4}^*$ and $\mathrm{AB6}$, identifies them with connective almost modules over almost connective $\mathbb{E}_{1}$-rings. Characterizes connective nuclear modules $\mathrm{Nuc}(R)_{\ge 0}$ in the sense of Clausen--Scholze as the additive rigidification of $\mathrm{Mod}_{R,\geq 0}^{\mathrm{cpl}}$. Also introduces prestable motives.
arXiv
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 $\infty$-category of spaces is a mode.
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

Writings

PDF
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.
PDF
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.
PDF
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.