Writings

• A proof for Ando's theorem on norm-coherent coordinates via the Coleman norm operator     [arXiv]
  
  

  Ando established an algebraic criterion for when a complex orientation for a Morava E-theory is an H_∞-map. The criterion relates such an orientation to a specific property of the formal group associated to the E-theory, namely, a norm coherence condition on its coordinate. On the other hand, Coleman constructed a norm operator for interpolating division values in local fields, which depends on a Lubin--Tate formal group law. These formal group laws are important tools in explicit local class field theory.
  
  In this article, we give a conceptual proof for Ando's theorem using the Coleman norm operator via the bridge of formal group laws between topology and arithmetic.
  
  This article grows out from my Bachelor thesis below. Comparing to the thesis, it focuses more on the topological side and is in terms of formal groups. Moreover, my undergraduate thesis requires the Morava E-theory to classify deformations of a Honda formal group law, but this article only requires it to classify deformations of a formal group law of finite height.
  
  
• A partial description of the chromatic support of E_n-algebras
  
  

  For a commutative ring spectrum, using power operations, Hahn proved that its chromatic support is an interval containing zero. A natural question is how the chromatic support of E_n-algebras looks like. In this article, we give a partial description of the problem.
  
  
• Redshift conjecture for commutative ring spectra
  
  

  Redshift conjecture concerns about how algebraic K-theory interacts with chromatic homotopy theory. It says that the algebraic K-theory raises the chromatic complexity by 1. The conjecture for commutative ring spectra has been formulated and solved by a series of works by Burklund, Clausen, Hahn, Land, Matthew, Meier, Naumann, Noel, Schlank, Tamme and Yuan. In this note, we introduce and summarize the proof of the redshift conjecture for commutative ring spectra.
  
  
• Ambidexterity in chromatic homotopy theory
  
  

  This is a survey written with Xiansheng Li of paper with the same title by Carmeli, Schlank and Yanovski. In this survey, we state and prove the main result of the paper that the cateogory of T(n)-local spectra is ∞-semiadditive. The original paper sets up a general machinery to solve the ambidexterity problems. However, we want to depict the core of the proof and prove the main theorem more directly. Hence, the content of this survey is almost a subset of the original paper, extracting essential details without any diagram chasing and keeping specific to be intuitive.
  
  
• Power operations in Morava E-theory
  
  

  This is my note for the talk on the course on the course Topics in Algebraic Topology, KU . This talk contains two parts. The first one is a modular interpretation of power operations in Morava E-theory via deformations of Frobenius, for which the main reference is The congruence criterion for power operations in Morava E-theory . The second one is an application of power operations in Morava E-theory in the computation of homotopy groups, for which the main reference is On the rationalization of the K(n)-local sphere . We will also give a short introduction to this paper.
  
  
• p-adic Galois Representations and (phi, Gamma)-modules
  
  

  In this note we present a basic theory of p-adic Galois representations and (φ, Γ)-modules. In particular, we prove a series of equivalences between both (1-)categories over various rings following Fontaine and Cherbonnier--Colmez.