个人简介
教育经历
2004.09--2008.06:浙江大学,获学士学位。
2008.09--2014.03:浙江大学,获博士学位。
2011.12--2014.06:澳大利亚国立大学,获博士学位。
工作经历
2014.03--2019.06:澳大利亚国立大学,博士后。
2019.07起:浙江大学,百人计划研究员。
教学与课程
2021年春夏学期
实变函数,面向数学与应用数学(求是科学班);
参考教材:实变函数,周性伟、孙文昌编著,第三版,科学出版社。
黎曼几何,面向数学学院高年级本科生;
参考教材:黎曼几何初步,白正国等编,高等教育出版社。
2020年春夏学期
实变函数,面向数学与应用数学(求是科学班,金融学交叉创新平台);
参考教材:实变函数,周性伟、孙文昌编著,第三版,科学出版社。
黎曼几何,与盛为民老师联合教学,面向数学学院高年级本科生;
参考教材:黎曼几何初步,白正国等编,高等教育出版社。
2019年秋冬学期
偏微分方程(英文课程),面向海宁国际校区;
参考教材:Partial Differential Equations:An introduction 2nd edition,by Walter A.Strauss.World Publishing Corporation,2019。
研究兴趣:
非线性方程及其应用,Monge最优运输问题,黎曼流形中的分析问题,曲率流,凸几何分析。
指导学生
招收研究生
欢迎对几何与分析有兴趣同学做研究生。名额情况:每年博士生(1名),硕士生(1名)。
要求:数学基础扎实,科研兴趣较强,英文读写过关。
指导本科生毕业设计
2020年春夏学期:陈天宇(数学与应用数学2016级),刘明泽(数学与应用数学2016级),金泽(数学与应用数学2016级)
组织会议
浙江大学非线性偏微分方程与几何分析研讨会;时间:2019年12月,地点:浙大紫金港校区。会议安排PDF。
2021春夏实变函数教学安排
实变函数
课程考试:2021年7月3日(8:00-10:00)。课堂时间:周二第1、2节,周四第3、4节。课堂地点:紫金港西1-507。
主要参考教材:实变函数,周性伟、孙文昌编著,第三版,科学出版社。
实变函数的核心内容是测度和积分理论,是近代分析领域的基础知识。它是数学分析的继续、深化和推广,是数学系的重要基础课。
预修要求:数学分析,高等代数。
该课程的课堂内容主要包括:
朴素集合论与点集拓扑基本知识;
外侧度,可测集,勒贝格测度,可测集基本性质;
可测函数,可测函数基本性质、收敛性,可测函数构造;
勒贝格积分,积分性质,积分序列的极限,傅比尼定理;
函数的积分与微分,有界变差函数,绝对连续函数,勒贝格微分定理;
L^p空间中的收敛,L^p空间的完备性、可分性。
课堂安排(根据实际教学进度适当调整)
教学周
课堂内容
春学期第1周
集合及其运算;集合的极限;集合的基数;开集、闭集及其性质
春学期第2周
Cantor集;Baire纲定理;不可测集的构造
春学期第3周
外测度及其性质;习题课1
春学期第4周
可测集、测度及其性质;sigma代数;抽象测度简介
春学期第5周
可测函数及其基本性质;可测函数的逼近(用简单函数、阶梯函数);习题课2
春学期第6周
Egorov定理;Lusin定理;可测函数的收敛;Riesz定理
春学期第7周
依测度收敛、a.e.收敛、近一致收敛的比较;可测集、可测函数的更多性质
春学期第8周
Lebesgue积分的引入;习题课3
春学期结束夏学期开始
夏学期第1周
中期测试(随堂测试);积分的性质
夏学期第2周
有界收敛定理、单调收敛定理、Fatou引理、控制收敛定理
夏学期第3周
L^1空间及其若干性质、L^1收敛;Lebesgue积分与Riemann积分的比较
夏学期第4周
Fubini定理,Tonelli定理,及其若干应用;习题课4
夏学期第5周
卷积;函数逼近(mollification方法);Fourier变换及其应用
夏学期第6周
Lebesgue微分定理;有界变差与几乎处处可微
夏学期第7周
绝对连续函数与微积分基本定理;积分换元;Rademacher可微定理
夏学期第8周
L^p空间基本概念与性质,完备性与分离性;L^2空间与Hilbert空间简介
其他参考资料:[1]Real Analysis,E.Stein&R.Shakarchi,世界图书出版公司。
[2]实变函数论,第三版,周民强编著,北京大学出版社。
课后作业:
教材第一章习题6、8、10、14、18、21、23、26、36、39、40、44、46、48、52、55、58;
教材第二章习题8、9、10、11、12、15、16、17、18、19、20、21、27、28、31、32、36;
教材第三章习题1、2、4、6、10、11、12、14、17、19、23、24、25、26、28、30、31、33;
教材第四章习题2、3、6、7、8、10、16、18、19、20、23、25、26、34、36、41、45、46、47、52、53、60;
教材第五章习题1、4、6、10、17、18、20、21、23、26、27;
教材第六章习题1、2、6、7、8、10;
另外补充其它非教材上的习题若干。
2021春夏黎曼几何教学安排
黎曼几何
课堂时间:周二第6、7、8节。课堂地点:紫金港西2-314。
主要参考教材:黎曼几何初步,白正国等编,高等教育出版社。
“一般公认,黎曼几何是从德国数学家Riemann的有名就职演说《论作为几何学基础的假设》(1854年)发端的。后经Christoffel,Bianchi及Ricci等数学家进一步完善和拓广,成为Einstein创立广义相对论(1915年)的有力数学工具”(摘自主要参考教材前言)。黎曼几何的研究从局部发展到整体,产生了许多深刻的并在其他数学分支及现代物理学中有重要作用的结果,其在基础理论和实际应用上都有巨大价值。该课程是数学系高年级本科生的选修课。
预修要求:数学分析,高等代数,微分几何,微分流形。
该课程的课堂内容主要包括:
张量场,黎曼度量,黎曼联络,共变微分;
截面曲率,Ricci曲率,数量曲率;
调和形式与Hodge理论简介;
测地线与测地完备,指数映照,法坐标,弧长变分,Jacobi场;
曲率与拓扑:基本指标引理,Myers定理,Synge定理,Cartan-Hadamard定理;
Hessian比较定理,Laplacian比较定理,Bishop-Gromov体积比较定理;
黎曼子流形,超曲面,极小子流形。(由课程进度决定)
课堂安排(根据实际教学进度适当调整)
教学周
课堂内容
春学期第1周
微分流形简介,欧式空间中子流形的仿射联络
春学期第2周
微分流形上的仿射联络,绕率与曲率
春学期第3周
黎曼联络
春学期第4周
共变微分
春学期第5周
曲率张量
春学期第6周
截面曲率、Ricci曲率、纯量曲率
春学期第7周
共形变换
春学期第8周
外微分、Hodge星算子
春学期结束夏学期开始
夏学期第1周
Laplace-Beltrami算子,Hodge理论简介
夏学期第2周
测地线、指数映照、法坐标系
夏学期第3周
测地线完备性,Hopf-Rinow定理
夏学期第4周
弧长的第一、第二变分,Jacobi场
夏学期第5周
共轭点与距离极小测地线
夏学期第6周
基本指标引理、Myers定理、Synge定理
夏学期第7周
Cartan-Hadamard定理、割点
夏学期第8周
Hessian与Laplacian比较定理、Bishop-Gromov体积比较定理
其他参考资料:黎曼几何引论(上册),陈维恒、李兴校编,北京大学出版社。
Riemannian Geometry,2nd edition,by Peter Petersen,科学出版社。
Riemannian Geometry and Geometric Analysis,6th edition,by Jurgen Jost,世界图书出版公司。
2019秋冬偏微分方程教学安排
MATH3021:Partial Differential Equations
Final Exam:3 Jan 2019(14:00--16:00)
Class Times:Tuesday 1-3 pm(Lecture)/Friday 11 am-12 pm(Tutorial)
Venue:Teaching Building North A-204
Textbook:Partial Differential Equations:An introduction 2nd edition,by Walter A.Strauss.World Publishing Corporation,2019.
Prerequisites:Calculus,Linear Algebra and Ordinary Differential Equations.
Assessment:Homework:30%(5%x 6);Quizzes:30%(10%x 3);Final Exam:40%.
Quiz Dates:Quiz#1 October 8,2-3 pm;Quiz#2 November 12,2-3 pm;Quiz#3 December 17,2-3 pm.
Homework Policy
Homework consists of 6 written assignments that will be released beforehand.Assignments should be submitted to your tutor at the end of the tutorials on September 27,October 18,November 1,November 15,November 29,December 20.
Keep your solutions neat,succinct,elegant and make the main points clear.
Late assignment will not be accepted without a documented excuse.
Tutorials
Tutorials will be focussed on practising the methods for solving PDEs by solving problems similar to those discussed in the lectures.New ideas and skills are first introduced and demonstrated in lectures.You are expected to develop these skills by applying them to specific tasks in tutorials.
Prescription and goals
Partial differential equations(PDEs)play a central role in modern mathematics because they allow us to describe a wide variety of real-world systems.PDEs have applications in biology(spread of species),medicine(growth of tumours),sociology(emigration rates),economics(option pricing),chemistry(reaction rates),physics(radiation of electromagnetic waves),engineering(optimal transportation),and artificial intelligence(learning theories).Fundamental concepts of PDEs and methods for solving them are therefore important for understanding nature and technology.
This course provides an introduction to the basic properties of PDEs and to the techniques that have proved useful in analysing them.We will discuss the first order PDEs and also the second order PDEs with a focus on wave equation,diffusion/heat equation and Laplace equation.The solution methods studied in this course will include the method of characteristics,separation of variables,Fourier series and integral transforms.
This course will be useful for students in their further study in economics,mathematical finance,engineering and physics.We will stress the importance of both theory and applications of PDEs.
Having successfully completed this course,students should demonstrate competency in the following skills:
Explain the fundamental concepts of PDEs and their role in modern mathematics and applied contexts;
Be able to use the method of characteristics,separation of variables,Fourier series and integral transform techniques to solve some specific PDEs;
Demonstrate capacity for mathematical reasoning,analytical and logical thinking;
Apply problem-solving with PDEs to diverse situations in physics,engineering and other mathematical contexts.
Contents:An approximate list of topics is as follows.
First order PDEs;
Wave equation;
Diffusion equation;
Possion’s equation;
Separation of variables,Fourier series,Integral transforms.
Teaching Calender(subject to change!)
Week
Topics
1
Sections 1.1,1.2,1.5.Overview;First order PDEs
2
Sections 1.3,1.4,1.6.Second order PDEs:examples and types
3
Sections 2.1,2.2.Wave equation:d’Alembert’s formula,Causality and Energy
4*
Section 2.3.Diffusion equation:Maximum principle,Uniqueness,Stability
5
Section 2.4.Diffusion on the whole line
6
Section 3.1,3.2.Reflections:Diffusion on the half-line,Reflection of waves
7
Sections 3.3,3.4.Sources:Diffusion/Wave with a source
8
Sections 4.1--4.3 Boundary problems:Separation of variables
9*
Section 5.1.Fourier series(I)
10
Sections 5.2-5.4.Fourier series(II)
11
Sections 12.3,12.5.Integral transforms
12
Sections 6.1--6.3.Harmonic functions
13
Sections 7.1-7.3.Green’s identities and Possion’s equation
14*
Section 7.4.Green functions
Note:Sections indicated above are the ones in textbook.Week with an asterisk is along with a quiz.
Other References
Introduction to Partial Differential Equations(2nd edition),G.B.Folland.Princeton University Press,1995.
Partial Differential Equations,L.C.Evans.American Mathematical Society,2010.
Usage of Blackboard System
Lecture Notes,Course materials and announcements will be posted in the Blackboard System(BB system)http://c.zju.edu.cn.
Assignments and tutorial materials will be announced in the BB system.You may also find the solutions to assignments,tutorials and quizzes(after being assessed)in the system.
近期论文
查看导师新发文章
(温馨提示:请注意重名现象,建议点开原文通过作者单位确认)
The Minkowski problem in the sphere (with Q. Guang and X.-J. Wang). Preprint.PDF
The Lp dual Minkowski problem and related parabolic flows (with H. Chen). Preprint. PDF
Non-uniqueness of solutions to the Lp dual Minkowski problem (with J. Liu and J. Lu). Accepted by Int. Math. Res. Not. PDF
Variations of a class of Monge-Ampere type functionals and their applications (with H. Chen and S. Chen). Accepted by Analysis&PDE. PDF
The Lp-Brunn-Minkowski inequality for p<1 (with S. Chen, Y. Huang and J. Liu). Adv. Math. 368 (2020), 107166, 21 pp. Article Link
Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems (with W. Sheng and X.-J. Wang). J. Eur. Math. Soc. (JEMS) 22 (2020), no. 3, 893--923. Article Link
The Christoffel problem by the fundamental solution of the Laplace equation (with D. Wan and X.-J. Wang). Sci. China Math. (2020). https://doi.org/10.1007/s11425-019-1674-1.
Asymptotic convergence for a class of fully nonlinear curvature flows (with W. Sheng and X.-J. Wang). J. Geom. Anal. 30 (2020), no. 1, 834--860. Article Link
Continuity for the Monge mass transfer problem in two dimensions (with F. Santambrogio and X.-J. Wang). Arch. Rational Mech. Anal. 231 (2019), no. 2, 1045--1071. Article Link
The logarithmic Minkowski problem for non-symmetric measures (with S. Chen and G. Zhu). Trans. Amer. Math. Soc. 371 (2019), no. 4, 2623--2641. Article Link
Infinitely many solutions for centro-affine Minkowski problem. Int. Math. Res. Not. IMRN 2019, 5577--5596. Article Link
On the planar dual Minkowski problem (with S. Chen). Adv. Math. 333 (2018), 87--117. Article Link
A class of optimal transportation problems on the sphere, Dedicated to Professor G.C. Dong on the occasion of his 90th birthday (with X.-J. Wang). Sci Sin Math 48 (2018), no. 1, 181--200. Article Link
Two dimensional Monge-Ampere equations under incomplete Holder assumptions. Math. Res. Lett. 24 (2017), no. 2, 485--506. Article Link
On the Lp Monge-Ampere equation (with S. Chen and G. Zhu). J. Differential Equations 263 (2017), no. 8, 4997--5011. Article Link
Multiple solutions of the Lp-Minkowski problem (with Y. He and X.-J. Wang). Calc. Var. Partial Differential Equations 55 (2016), no. 5, Paper No. 117, 13 pp. Article Link
Regularity of the homogeneous Monge-Ampere equation (with X.-J. Wang). Discrete Contin. Dyn. Syst. 35 (2015), no. 12, 6069--6084. Article Link
Regularity in Monge's mass transfer problem (with F. Santambrogio and X.-J. Wang). J. Math. Pures Appl. 102 (2014), no. 6, 1015--1040. Article Link
Positivity of Ma-Trudinger-Wang curvature on Riemannian surfaces (with S.-Z. Du). Calc. Var. Partial Differential Equations 51 (2014), no. 3-4, 495--523. Article Link
Closed hypersurfaces with prescribed Weingarten curvature in Riemannian manifolds (with W. Sheng). Calc. Var. Partial Differential Equations 48 (2013), no. 1-2, 41--66. Article Link
Some Dirichlet problems arising from conformal geometry (with W. Sheng). Pacific J. Math. 251 (2011), no. 2, 337--359. Article Link
Surfaces expanding by the power of the Gauss curvature flow. Proc. Amer. Math. Soc. 138 (2010), no. 11, 4089--4102. Article Link