# Research

## Differential Equations

### Wednesday, November 26, 2014

Title: Spectral properties of the Schröedinger operator with a periodic potential
Speaker: Solomon Manukure
Time: 4:00pm‐5:00pm
Place: CHE 302

#### Abstract

I will talk about the the Bloch spectrum of the Schrödinger operator with real periodic potential. In particular, I will show, with some examples, the importance of the so-called monodromy matrix in computing the Bloch spectrum. The case of the finte-gap potential will also be considered.

### Wednesday, November 19, 2014

Title: Reductions of the KP equation and spectral properties of its finite-gap solutions
Speaker: Hongcai Ma, Department of Applied Mathematics
Donghua University
Shanghai, PR China
Time: 4:00pm‐5:00pm
Place: CHE 302

#### Abstract

We will talk about reductions of finite-gap solutions of the KP equation to the KdV and Boussinesq equations, and spectral properties of those finite-gap solutions.

### Wednesday, November 12, 2014

Title: Finite-gap solutions of the KP equation
Speaker: Yuqin Yao, Department of Applied Mathematics
China Agricultural University, Beijing, PR China
Time: 4:00pm‐5:00pm
Place: CHE 302

#### Abstract

First, we will discuss the differential equations for the Baker-Akhiezer functions. Second, we introduce finite-gap solutions of the KP equation. Third, we will generate real non-singular solutions of the KP 1 and KP 2 equations.

### Wednesday, November 5, 2014

Title: Hyperelliptic Curves and Using Theta Functions to Construct Functions and Differentials on Riemann Surfaces
Speaker: Morgan McAnally
Time: 4:00pm‐5:00pm
Place: CHE 302

#### Abstract

We will consider an algebraic curve $$X$$ of genus $$g$$ and describe two classical methods for constructing on $$X$$ meromorphic functions and meromorphic Abelian differentials, as well as period functions with essential singularities. Then we will consider the hyperelliptic curve $$X$$ of genus $$g$$ and describe meromorphic functions on it.

### Wednesday, October 22, 2014

Title: Abelian functions and Theta functions on Riemann surfaces
Speaker: Yujuan Zhang, College of Mathematics and Statistics
Lanzhou University, PR China
Time: 4:00pm‐5:00pm
Place: CHE 302

#### Abstract

Abelian functions are the bases to obtain the algebraic-geometrical solutions of completely integrable nonlinear equations and integrable finite-dimentional dynamical systems. In this seminar, we will talk about Abelian tori and Abelian functions, and introduce Theta functions to construct Abelian functions. Basic properties of Theta functions will be discussed, including the Riemann Theta formula and the Koizumi formula.

### Wednesday, October 15, 2014

Title: Integrable multi-component nonlinear wave equations from symplectic structures
Speaker: Stephen Anco, Department of Mathematics and Statistics
Time: 4:00pm‐5:00pm
Place: CHE 302

#### Abstract

I will show how the bi-Hamiltonian structure of integrable systems can be used to derive various related integrable wave equations, in particular, a hyperbolic equation and a peakon equation, as well as a Schrödinger equation and peakon-like kink equation if the original integrable system is unitarily invariant.

### Wednesday, October 8, 2014

Title: Elliptic curves and Abelian differentials and integrals on Riemann surfaces, II
Speaker: Yuan Zhou
Time: 4:00pm‐5:00pm
Place: CHE 302

### Wednesday, October 1, 2014

Title: Elliptic curves and Abelian differentials and integrals on Riemann surfaces
Speaker: Yuan Zhou
Time: 4:00pm‐5:00pm
Place: CHE 302

#### Abstract

First, we will introduce elliptic curves and elliptic functions. Second, we will discuss Abelian functions, differentials and integrals on Riemann surfaces, which include Jacobian varieties, divisors and the Riemann-Roch theorem.

### Wednesday, September 24, 2014

Title: Biological networks and its spatiotemporal dynamics
Speaker: Xiaoli Yang, Shaanxi Normal University
PR China
Time: 4:00pm‐5:00pm
Place: CHE 302

#### Abstract

Complex networks have been used to model many self-organizing systems such as food webs, genetic control networks, neural networks and social networks. The background and development of complex networks will be firstly introduced. Then, for several biological networks such as genetic regulatory network and neuronal network, we present the key roles of random noise, time delay and network topology in shaping its collective dynamics of resonance and synchronization.

### Wednesday, September 17, 2014

Title: Riemann surfaces, coverings and elliptic curves, II
Speaker: Xiang Gu
Time: 4:00pm‐5:00pm
Place: CHE 302

### Wednesday, September 10, 2014

Title: Riemann surfaces, coverings and elliptic curves
Speaker: Xiang Gu
Time: 4:00pm‐5:00pm
Place: CHE 302

#### Abstract

This is the first presentation of a series of DE seminars aiming to introduce fundamental techniques about the algebro-geometric approach to nonlinear integrable equations. In this very first talk, basic concepts of Riemann surfaces, coverings, and elliptic curves will be brought to the attention of the audience, together with certain examples, to discuss about nonlinear mathematical structures.

### Wednesday, September 3, 2014

Title: The mathematics behind complexitons to integrable equations
Speaker: Wen-Xiu Ma
Time: 4:00pm‐5:00pm
Place: CMC 116

#### Abstract

We talk about a solution-focused approach to classification of integrable differential equations. Solitons have the property of exponentially decay and positons describe quasi-periodic phenomena. The two kinds of corresponding mathematics subjects are the Hirota bilinear theory and algebraic curves, respectively. Complexitons combine two kinds of nonlinear phenomena and use complex eigenvalues of associated spectral problems. But what is the mathematical subject that we need to start from to analyze such solutions? We will talk about a few possible answers.