报告题目：On the Fermat-type partial differential-difference equations on C^n

报告人：Zhuan John Ye教授，University of North Carolina Wilmington

报告摘要：

We start from the Pythagorean Theorem (gou gu Theorem in China) to the Fermat Last Theorem, then, to our current research results on partial differential and partial difference equations (PDDEs) in the complex space of dimension $n$.

Assume that $n$ is a positive integer, $p_j (j = 1,2,··· ,6)$ are polynomials, $p$ is an irreducible polynomial, and $f$ is an entire function on C^n . Let $L(f) =\sum_{ j=1}^n q_{tj} f_{z_{t_ j}} and $f(z) = f(z_1 +c_1,...,z_n +c_n)$, where $q_{tj} (j = 1,2,··· ,s ≤ n)$ are non-zero polynomials on C^n and $c\in C^n\{0}$. We show the structures of all entire solutions to the non-linear PDDE

$$(p_1L(f) + p_2 \overline{f} + p_5 f) ^2 + (p_3L(f) + p_4 \overline{f} + p_6 f)^2 = p, $$

which is called a Fermat-type partial differential-difference equation. Further, we find many sufficient conditions and/or necessary conditions for the existence, as well as the concrete representations, of entire solutions to the Fermat-type PDDE. We also demonstrate several examples on C^2 with non-constant coefficients to verify that all representations in our theorems exist and are accurate and that the entire solutions to the Fermat-type PDDEs could have finite or infinite growth order. Many previous publications are corollaries of our theorems.

时间：2024年5月29日下午3点

报告地点：腾讯会议：713-258-142

报告人简介：

Zhuan John Ye（叶专），博士、教授，复分析领域的国际著名数学家。1992年博士毕业于美国普渡大学。1992-2016年在北伊利诺伊大学任教。2016年至今在美国北卡罗来纳大学威尔明顿分校任教，曾担任数学与统计系主任等职务。研究方向涉及复分析，随机解析函数论，复解析动力系统，复微分和差分方程等多个领域。在《Invent． Math．》、《Duke J. Math.》、《Trans of AMS》、《Math Z.》等顶尖一流数学杂志上发表40多篇创造性学术论文。出版学术专著《Nevanlinna’s theory of value distribution》。受邀在美国、中国、加拿大、德国、西班牙、芬兰、英国等国家做了40多次学术报告。