活動日期：2024年11月16日
活動地點：東華大學松江校區2号學院樓237報告廳
三胞胎素數與陳景潤的思想
時間：09:00
主講人：李嘉旻
主講人簡介：
李嘉旻，山東大學國家高層次數學人才培養中心在讀博士，導師是劉建亞教授。研究領域是解析數論，研究興趣集中在各種形式的素數分布問題，現主持國家自然科學基金一項，在《Science China Mathematics》、《Journal of Number Theory》等期刊發表SCI三篇。
内容摘要：
Hardy和Littlewood猜測對于任意k個線性多項式，如果不存在局部障礙，便可以同時表素數無窮多次。著名的孿生素數猜想是其取k＝2的特例。在本次報告中，我們将關注k＝3的情形，研究所謂三胞胎素數猜想，我們證明了存在無窮多素數p，使得p+2不超過3個素因子，p+6不超過6個素因子，記作“1+3+6”；更進一步，如果假設Elliott-Halberstam猜想成立，我們可以證明“1+3+3”。
1-3-5猜想的證明
時間：09:40
主講人：周廣良
主講人簡介：
周廣良，同濟大學博士後，合作導師是蔡迎春教授；博士畢業于南京大學，導師是孫智偉教授。研究方向為數論，目前發表或接受學術論文9篇。主持“中國博士後科學基金第73批面上項目”和“國家自然科學基金青年科學基金資助項目”。此外還獲得了上海市“超級博士後”激勵資助計劃。
内容摘要：
2016年4月, 南京大學孫智偉教授提出猜想: 任意的正整數可以表示成x^2+y^2+z^2+t^2的形式, 并且x+3y+5z為平方數, 此處, x,y,z,w均為自然數. 2020年, A. Machiavelo, N. Tsopanidis證明了上述猜想. 這場報告我們将詳細介紹該猜想的論證過程.此外, 我們還會介紹一些相關工作與相關猜想.
Asymptotics of hypergeometric functions
時間：10:30
主講人：杭鵬程 (本校研究生)
主講人簡介：
杭鵬程，東華大學在讀博士，導師是胡良劍教授。研究興趣是漸近分析及其與特殊函數、數論等的相互作用，主要工作集中在超幾何型函數的漸近行為與zeta函數的解析理論兩方面，目前在JMAA和SIGMA期刊發表論文兩篇。
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
時間：11:10
主講人：劉竣文
主講人簡介：
劉竣文, 複旦大學在讀博士, 導師: 張侖教授. 本科畢業于南開大學, 研究興趣為随機矩陣及其在p-adic上的應用.
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
時間：13:00
主講人：徐暢
主講人簡介：
現就讀于華東師範大學，是二年級博士生，導師為劉治國教授，研究方向是q-級數，先後在Proc. Amer. Math. Soc., Results. Math, Rocky Mountain J. Math. 等雜志發表7篇論文。
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
時間：13:40
主講人：孫暄 (本校教師)
主講人簡介：
孫暄，2024年6月畢業于華南理工大學數學學院，現就職于bevictor伟德官网數學系。主要從事可積非線性孤子方程橢圓函數解的相關研究工作，現已經在 Adv. Math，Phy. D, 等期刊發表多篇學術論文。
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
時間：14:30
主講人：李高瞻
主講人簡介：
李高瞻，複旦大學2020級基礎數學博士生，導師：範恩貴教授。研究方向是可積系統與孤立子理論，主要工作集中在可積偏微分方程的長時間漸近行為以及Riemann-Hilbert問題表示方面。相關成果發表在國内著名期刊《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
時間：15:10
主講人：羅旻傑 (本校教師)
主講人簡介：
羅旻傑，副教授，碩士生導師。2017年畢業于華東師範大學數學系，基礎數學專業，獲理學博士學位。2017年9月至2018年9月在香港城市大學開展博士後研究。2018年9月起任職于bevictor伟德官网。主要研究方向為特殊函數與分數階微積分，已在《J. Math. Anal. Appl.》、《Proc. Amer. Math. Soc.》、《Frac. Calc. Appl. Anal.》、《Integral Transforms Spec. Funct.》等國際知名 SCI 期刊上發表論文 20 餘篇，并先後獲得上海市青年科技英才揚帆計劃項目1項，上海市核心數學與實踐重點實驗室開放課題1項以及國家自然科學基金青年項目1項。
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.
矩陣變量 Gauss 超幾何函數的 Erdelyi 型積分及分數階積分算子的 Laplace 變換
時間：16:00
主講人：郭靓佳 (本校學生)
主講人簡介：郭靓佳，東華大學在讀碩士，導師是羅旻傑副教授，研究方向為分數階特殊函數。
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
時間：16:20
主講人：周雪林 (本校學生)
主講人簡介：
周雪林，東華大學在讀碩士，導師是羅旻傑副教授，研究方向為分數階特殊函數。
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.