活动日期：2024年11月16日
活动地点：东华大学松江校区2号学院楼237报告厅
三胞胎素数与陈景润的思想
时间：09:00
主讲人：李嘉旻
主讲人简介：
李嘉旻，山东大学国家高层次数学人才培养中心在读博士，导师是刘建亚教授。研究领域是解析数论，研究兴趣集中在各种形式的素数分布问题，现主持国家自然科学基金一项，在《Science China Mathematics》、《Journal of Number Theory》等期刊发表SCI三篇。
内容摘要：
Hardy和Littlewood猜测对于任意k个线性多项式，如果不存在局部障碍，便可以同时表素数无穷多次。著名的孪生素数猜想是其取k＝2的特例。在本次报告中，我们将关注k＝3的情形，研究所谓三胞胎素数猜想，我们证明了存在无穷多素数p，使得p+2不超过3个素因子，p+6不超过6个素因子，记作“1+3+6”；更进一步，如果假设Elliott-Halberstam猜想成立，我们可以证明“1+3+3”。
1-3-5猜想的证明
时间：09:40
主讲人：周广良
主讲人简介：
周广良，同济大学博士后，合作导师是蔡迎春教授；博士毕业于南京大学，导师是孙智伟教授。研究方向为数论，目前发表或接受学术论文9篇。主持“中国博士后科学基金第73批面上项目”和“国家自然科学基金青年科学基金资助项目”。此外还获得了上海市“超级博士后”激励资助计划。
内容摘要：
2016年4月, 南京大学孙智伟教授提出猜想: 任意的正整数可以表示成x^2+y^2+z^2+t^2的形式, 并且x+3y+5z为平方数, 此处, x,y,z,w均为自然数. 2020年, A. Machiavelo, N. Tsopanidis证明了上述猜想. 这场报告我们将详细介绍该猜想的论证过程.此外, 我们还会介绍一些相关工作与相关猜想.
Asymptotics of hypergeometric functions
时间：10:30
主讲人：杭鹏程 (本校研究生)
主讲人简介：
杭鹏程，东华大学在读博士，导师是胡良剑教授。研究兴趣是渐近分析及其与特殊函数、数论等的相互作用，主要工作集中在超几何型函数的渐近行为与zeta函数的解析理论两方面，目前在JMAA和SIGMA期刊发表论文两篇。
内容摘要：
In this talk, we make a survey of the results on asymptotics of hypergeometric functions. Further, we shall list some relevant applications and research problems.
Large gap asymptotics of the hard edge tacnode process
时间：11:10
主讲人：刘竣文
主讲人简介：
刘竣文, 复旦大学在读博士, 导师: 张仑教授. 本科毕业于南开大学, 研究兴趣为随机矩阵及其在p-adic上的应用.
内容摘要：
A special type of geometric situation in ensembles of non-intersecting paths occurs when the non-intersecting trajectories are required to be nonnegative so that the limit shape becomes tangential to the hard-edge 0. The local fluctuation is governed by the universal hard edge tacnode process, which also arises from some tiling problems. It is the aim of this work to explore the integrable structure and asymptotics for the gap probability of the hard edge thinned/unthinned tacnode process over (0,s). We establish an integral representation of the gap probability in terms of the Hamiltonian associated with a system of differential equations.In this talk, some applications of our results are discussed as well.
Some results from a quadratic summation of Gasper and Rahman
时间：13:00
主讲人：徐畅
主讲人简介：
现就读于华东师范大学，是二年级博士生，导师为刘治国教授，研究方向是q-级数，先后在Proc. Amer. Math. Soc., Results. Math, Rocky Mountain J. Math. 等杂志发表7篇论文。
内容摘要：
Applying a quadratic summation of Gasper and Rahman, we verify two q-supercongruences conjectured by Guo and refined a q-supercongruence of Guo. Moreover, we get some new supercongruences modulo p^2 or p^3 where p is a prime.
The dynamic behavior of elliptic function solutions for the focusing mKdV equation
时间：13:40
主讲人：孙暄 (本校教师)
主讲人简介：
孙暄，2024年6月毕业于华南理工大学数学学院，现就职于东华大学数学与统计学院数学系。主要从事可积非线性孤子方程椭圆函数解的相关研究工作，现已经在 Adv. Math，Phy. D, 等期刊发表多篇学术论文。
内容摘要：
We examine the spectral and orbital stability of elliptic function solutions for the focusing modified Korteweg–de Vries (mKdV) equation, constructing corresponding breather solutions to reveal stable and unstable dynamic behaviors. These elliptic function solutions and the fundamental solutions of the Lax pair are precisely represented using theta functions. By applying the modified squared wavefunction (MSW) method, we construct all linearly independent solutions of the linearized mKdV equation and provide a necessary and sufficient condition for spectral stability of the elliptic function solutions with respect to subharmonic perturbations. When spectral stability is confirmed, we establish the orbital stability of these solutions within a suitable Hilbert space. Additionally, using the Darboux-Bäcklund transformation, we construct breather solutions to further illustrate the dynamics, both stable and unstable.
On the long-time asymptotics of the modified Camassa-Holm equation with step-like initial data
时间：14:30
主讲人：李高瞻
主讲人简介：
李高瞻，复旦大学2020级基础数学博士生，导师：范恩贵教授。研究方向是可积系统与孤立子理论，主要工作集中在可积偏微分方程的长时间渐近行为以及Riemann-Hilbert问题表示方面。相关成果发表在国内著名期刊《Science China Mathematics》。
内容摘要：
We study the long-time asymptotics for the solution of the modified Camassa-Holm (mCH) equation with step-like initial data.
\begin{align}
&m_{t}+\left(m\left(u^{2}-u_{x}^{2}\right)\right)_{x}=0, \quad m=u-u_{xx}, \nonumber\\
& {u(x,0)=u_0(x)\to \left\{ \begin{array}{ll}
1/c_+, &\ x\to+\infty,\\[5pt]
1/c_-, &\ x\to-\infty,
\end{array}\right.\nonumber}
\end{align}
where $c_+$ and $c_-$ are two positive constants. It is shown that the solution of the step-like initial problem can be characterized via the solution of a matrix Riemann-Hilbert (RH) problem in the new scale $(y,t)$. A double coordinates $(\xi, c)$ with $c=c_+/c_-$ is adopted to divide the half-plane $\{ (\xi,c): \xi \in \mathbb{R}, \ c> 0, \ \xi=y/t\}$ into four asymptotic regions. Further applying the Deift-Zhou steepest descent method, we derive the long-time asymptotic expansions of the solution $u(y,t)$ in different space-time regions with appropriate g-functions. The corresponding leading asymptotic approximations are given with the slow/fast decay step-like background wave in genus-0 regions and elliptic waves in genus-2 regions. The second term of the asymptotics is characterized by the Airy function or parabolic cylinder model. Their residual error order is $\mathcal{O}(t^{-1})$ or $\mathcal{O}(t^{-2})$, respectively.
Fractional Calculus: From Classical Theory to Forefront
时间：15:10
主讲人：罗旻杰 (本校教师)
主讲人简介：
罗旻杰，副教授，硕士生导师。2017年毕业于华东师范大学数学系，基础数学专业，获理学博士学位。2017年9月至2018年9月在香港城市大学开展博士后研究。2018年9月起任职于东华大学数学与统计学院。主要研究方向为特殊函数与分数阶微积分，已在《J. Math. Anal. Appl.》、《Proc. Amer. Math. Soc.》、《Frac. Calc. Appl. Anal.》、《Integral Transforms Spec. Funct.》等国际知名 SCI 期刊上发表论文 20 余篇，并先后获得上海市青年科技英才扬帆计划项目1项，上海市核心数学与实践重点实验室开放课题1项以及国家自然科学基金青年项目1项。
内容摘要：In this talk, we first give a brief review of the early development of the Fractional Calculus. Then, based on the well-known results of Saigo and Kiryakova, we show the core idea of the Generalized Fractional Calculus and illustrate how it has evolved over the last few decades to become an important part of the modern theory of Fractional Calculus. We also report our latest work on a certain class of fractional integral operators whose kernels involve a very special class of generalized hypergeometric functions. Finally, we point out some problems for further research.
矩阵变量 Gauss 超几何函数的 Erdelyi 型积分及分数阶积分算子的 Laplace 变换
时间：16:00
主讲人：郭靓佳 (本校学生)
主讲人简介：郭靓佳，东华大学在读硕士，导师是罗旻杰副教授，研究方向为分数阶特殊函数。
内容摘要：This report presents the study of Gaussian hypergeometric functions of matrix argument and the Laplace transforms of fractional integral operators for matrix variables. We derive an Erdelyi-type integral for the Gaussian hypergeometric function, extending its application to matrix calculus. Additionally, we explore the Laplace transforms of fractional integral operators. These findings have implications for control theory and signal processing.
Applications of Babenko's method
时间：16:20
主讲人：周雪林 (本校学生)
主讲人简介：
周雪林，东华大学在读硕士，导师是罗旻杰副教授，研究方向为分数阶特殊函数。
内容摘要：
This paper extends a class of fractional integral operators to their q-analog forms and analyzes the boundedness of these operators within the L^p_q(0, a) space. We investigate the semigroup properties of these operators and employ Babenko's method to solve integral equations. Rigorous proofs are provided for the existence and uniqueness of solutions to the corresponding nonlinear integral equations.