My main interests are in homotopy theory, number theory, and related fields, especially the relationship between the two fields. In particular, I am interested in chromatic homotopy theory, power operations and topological Hochschild homology, as well as learning p-adic geometry.
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.
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.
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.
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.
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 Kab of a local field K. In 1965, Lubin and Tate constructed an explicit form of the local Artin map and Kab 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 ➝ En 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.
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.
This is the final paper for the Algebraic Topology course taken in UCB taught by 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 Kn 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.
This is a report for the 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.