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 undergraduate 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.
  
  
• Power operations in Morava E-theory
  
  This is my note for the talk on the course Topics in Algebraic Topology at University of Copenhagen, taught by Lars Hesselholt . 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.
  
  
• Explicit Local Class Field Theory à la Lubin and Tate with an Application to Algebraic Topology
  
  Local class field theory was originally proved via global class field theory, and there was no explicit description of the local Artin map and the maximal abelian extension K^ab of a local field K. In 1965, Lubin and Tate constructed an explicit form of the local Artin map and K^ab from formal group laws. In 1979, Coleman proved an interpolation theorem on division values in local fields by constructing a norm operator depending on Lubin-Tate formal group laws. On the other hand, in topology, Ando established an algebraic criterion on when a complex orientation MU ➝ E_n for Morava E-theory is an H_∞-map. The criterion relates desired orientations to a specific property of formal group laws.
  
  This is my undergraduate thesis and this thesis has two parts. Firstly, we prove explicit local class field theory following of Lubin and Tate. Secondly, we give a new proof of Ando's theorem in topology via Coleman's norm operator from explicit local class field theory.
  
  
• Chromatic Homotopy Theory
  
  This is a survey of chromatic homotopy theory. In this survey, we first show the global picture of chromatic homotopy theory. Then we will introduce the theory following Ravenel's orange book . We will first introduce the periodicity theorem and telescope conjecture, which is a geometric model of the algebraic chromatic filtration, a filtration of homotopy groups that is easier to compute. Then we talk about the thick category theorem, which gives a filtration of the category of finite p-local complexes.
  
  
• Quillen's Plus Construction and Algebraic K-theory
  
  This is the final paper for the Algebraic Topology course taken in UCB taught by Prof. Constantin Teleman . The survey gives an introduction to algebraic K-theory. We first introduce the classical definition of K₀, K₁ groups and some properties of them. Then we apply Quillen’s plus construction to construct general K_n and show that they are in accord with the previous definitions when n = 0, 1. Finally, we give a brief introduction to the K-theory Ω spectrum.
  
  
• Ellipltic Curves over ℂ
  
  This is a report for the Berkeley Mathematics Directed Reading Program in UCB. We talk about elliptic curves over complex numbers in this survey. First we introduce elliptic curves from the calculation of the arc lengths of ellipses, which is the historical origin of elliptic curves. Then we present some general basics of elliptic curves, like the group law and isogenies. Finally, we prove the equivalence of categories of elliptic curves over complex numbers and lattices in the complex plane.