Results

**1 - 6**of**6**### Research Statement

, 2013

"... I do research in number theory and arithmetic geometry, and particularly in the area related to Galois representations and deformation of abelian varieties. It is a celebrated theorem of Serre and Tate that the deformation theory of an abelian variety A over a field of characteristic p> 0 is equi ..."

Abstract
- Add to MetaCart

(Show Context)
I do research in number theory and arithmetic geometry, and particularly in the area related to Galois representations and deformation of abelian varieties. It is a celebrated theorem of Serre and Tate that the deformation theory of an abelian variety A over a field of characteristic p> 0 is equivalent that of its Barsotti-Tate group. When the abelian variety A is ordinary, there is a structure of formal group on the infinitesimal deformation space of A/k, and any deformation of A is determined by the Serre-Tate coordinates on the deformation space ([5]). Moreover, the Serre-Tate deformation theory can be applied to determine the local structure of Shimura varieties of Hodge type at ordinary locus ([11],[14]). My research has been forcused on applying the Serre-Tate theory to two types of questions in arithmetic geometry: the first is the study of the local behavior of Galois representations attached to nearly ordinary Hilbert modular forms; the second is the study of special cases of the Mumford-Tate conjecture. I will give a summary of my work ([19],[20]) toward these two directions below. 1 Local behavior of Hilbert modular Galois representations Let F be a totally real field inside an algebraic closure Q ̄ of Q and let FA be the adele ring of F. Fix a rational prime p and a prime p of F over p. Let m be an integral ideal of F and Sk(m,C) be the space of Hilbert modular cusp forms of parallel weight k and level m. For each prime q of F and fractional ideal n of F prime to m, let T (q) and 〈n 〉 be the Hecke operator acting on Sk(m,C). Suppose that we have a Hilbert modular form f ∈ Sk(m,C) such that f is an eigenvector for all the above Hecke operators, i.e. there exist c(q, f), d(n, f) such that T (q)(f) = c(q, f)f and 〈n〉(f) = d(n, f)f. Let Kf be the field generated over Q by these eigenvalues c(q, f), d(n, f), which is known to be a number field. Let λ be a prime of Kf over the rational prime p and let Kf,λ be the completion of Kf at λ. Then it is well known that there is a continuous representation ρf,λ: Gal(Q̄/F) → GL2(Kf,λ) attached to f. Moreover, if f is nearly p-ordinary in the sense that c(p, f) ∈ Kf ⊂ Kf,λ is a p-adic unit, then the restriction of ρf,λ to the decomposition group Dp of Gal(Q̄/F) at p is reducible and of the following shape: ρf,λ|Dp ∼

### Distributed

"... optimization decomposition for joint economic dispatch and frequency regulation ..."

Abstract
- Add to MetaCart

optimization decomposition for joint economic dispatch and frequency regulation

### 1Connecting Automatic Generation Control and Economic Dispatch from an Optimization View

"... Abstract—Automatic generation control (AGC) regulates me-chanical power generation in response to load changes through local measurements. Its main objective is to maintain system frequency and keep energy balanced within each control area so as to maintain the scheduled net interchanges between con ..."

Abstract
- Add to MetaCart

Abstract—Automatic generation control (AGC) regulates me-chanical power generation in response to load changes through local measurements. Its main objective is to maintain system frequency and keep energy balanced within each control area so as to maintain the scheduled net interchanges between control areas. The scheduled interchanges as well as some other factors of AGC are determined at a slower time scale by considering a centralized economic dispatch (ED) problem among different generators. However, how to make AGC more economically efficient is less studied. In this paper, we study the connections between AGC and ED by reverse engineering AGC from an optimization view, and then we propose a distributed approach to slightly modify the conventional AGC to improve its economic efficiency by incorporating ED into the AGC automatically and dynamically. I.