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Mathematics & Statistics

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**Title**

**Speaker**

**Time**

**Place**

Spectral properties of the Schröedinger operator with a periodic potential

Solomon Manukure

4:00pm-5:00pm

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.

**Title**

**Speaker**

**Time**

**Place**

Reductions of the KP equation and spectral properties of its finite-gap solutions

Hongcai Ma

Department of Applied Mathematics

Donghua University

Shanghai, PR China

4:00pm-5:00pm

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.

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**Speaker**

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Finite-gap solutions of the KP equation

Yuqin Yao

Department of Applied Mathematics

China Agricultural University

Beijing, PR China

4:00pm-5:00pm

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.

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**Speaker**

**Time**

**Place**

Hyperelliptic Curves and Using Theta Functions to Construct Functions and Differentials on Riemann Surfaces

Morgan McAnally

4:00pm-5:00pm

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.

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**Speaker**

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Abelian functions and Theta functions on Riemann surfaces

Yujuan Zhang

College of Mathematics and Statistics

Lanzhou University

PR China

4:00pm-5:00pm

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.

**Title**

**Speaker**

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Integrable multi-component nonlinear wave equations from symplectic structures

Stephen Anco

Department of Mathematics and Statistics

Brock University

Canada

4:00pm-5:00pm

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.

**Title**

**Speaker**

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**Place**

Elliptic curves and Abelian differentials and integrals on Riemann surfaces, II

Yuan Zhou

4:00pm-5:00pm

CHE 302

**Title**

**Speaker**

**Time**

**Place**

Elliptic curves and Abelian differentials and integrals on Riemann surfaces

Yuan Zhou

4:00pm-5:00pm

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.

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**Speaker**

**Time**

**Place**

Biological networks and its spatiotemporal dynamics

Xiaoli Yang

Shaanxi Normal University

PR China

4:00pm-5:00pm

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.

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**Speaker**

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Riemann surfaces, coverings and elliptic curves, II

Xiang Gu

4:00pm-5:00pm

CHE 302

**Title**

**Speaker**

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**Place**

Riemann surfaces, coverings and elliptic curves

Xiang Gu

4:00pm-5:00pm

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.

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**Speaker**

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The mathematics behind complexitons to integrable equations

Wen-Xiu Ma

4:00pm-5:00pm

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.