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【2021.06.01-06.25 北京】数学暑期学校
2021-06-02 | 编辑:
 主办单位:中国科学院数学与系统科学研究院

协办单位:中国科学技术大学几何与物理研究中心

基金支持:华罗庚数学科学中心

 

短期课程列表

 

  • 课程名称:Introduction to Hermitian-Yang-Mills and Kahler-Einstein equations
  • 主讲人:陈杲(中国科学技术大学)
  • 时间: 9:00-11:00am, 6月1、2、3日
  • 地点:数学院南楼219多媒体报告厅
  • 摘要:          

1.Introduction to manifold and vector bundles       

2.Introduction to Kahler geometry

3.Kahler - Einstein, Hermitian-Yang-Mills and other related equations

  

  • 课程名称:Introduction to 2d harmonic maps
  • 主讲人:宋翀(厦门大学)
  • 时间:2:30-4:30pm, 6月3、4日; 9:00-11:00am, 6月7日
  • 地点:数学院南楼219多媒体报告厅
  • 摘要:This 6-hour mini-course aims to give a survey on the theory of blow-up analysis of 2d harmonic maps. We will follow the vein of a lecture given by the late Professor Weiyue Ding more than twenty years ago, while focusing on some new developments in this area. The course is divided into three parts:

1. We will start from basic definitions, examples, some properties and formulas of harmonic maps. Then I will present Eells-Sampson’s theorem on harmonic map heat flows.

2. We will give a sketch of Sacks-Uhlenbeck’s method in finding 2d harmonic maps. Then we will study the convergence and blow-up behavior of a sequence of 2d harmonic maps, including the energy identity and bubble-tree construction.

3. In this part we will focus on the blow-up analysis of 2d harmonic map heat flow. In particular, I will introduce a recent result on bubble-tree convergence at finite singularities of 2d harmonic map flow into K?hler manifolds.

   

  • 课程名称:Bergman kernel and applications
  • 主讲人:孙京洲(汕头大学)
  • 时间:9:00-11:00am, 6月8、10、11日
  • 地点:6月8/11日数学院南楼219多媒体报告厅;6月10日数学院南楼202室
  • 摘要:In this 6-hour mini-course we talk about the Bergman kernels on complex manifolds and their applications .

1.We talk about the Bergman kernel on open domains in the Euclidean space. The construction and basic properties
2. We talk about the asymptotic expansion of the Bergman kernel on complex manifolds, the peak section technique and a sketch of the proof.
3. In this part we first talk about some applications of the Bergman kernels, then we talk about the Bergman kernel in the singular settings.

 

  

  • 课程名称:Convergence theory in Ricci flow
  • 主讲人:李宇(中国科学技术大学)
  • 时间:9:00-11:00am, 6月15、17日
  • 地点:数学院南楼219多媒体报告厅
  • 摘要:The Ricci flow has been used to resolve long-standing open questions in Riemannian geometry and three-dimensional topology. In this mini-course, we focus on the convergence theory for the Ricci flow in general dimensions and its application to the Sphere Theorem.   

1. We discuss the maximum principle arguments and the preserved curvature conditions such as the positive isotropic curvature, PIC1, PIC2, under the Ricci flow.

2. We present the convergence theorems of the Ricci flow under various curvature conditions and their applications.

 

 

  • 课程名称:Homogenous Complex Monge-Ampere Equation
  • 主讲人:胡京辰(上海科技大学)
  • 时间:9:00-11:00am, 6月14、16、18日
  • 地点:数学院南楼219多媒体报告厅
  • 摘要:In this 6-hour mini-course we discuss some basic techniques for the homogenous complex Monge-Ampere equation.

1. We present the C^{1,1} estimate for the homogenous complex Monge-Ampere equation, and show that this estimate is optimal.
2.We introduce the holomorphic foliation method for the homogenous complex Monge-Ampere equation, developed by S. Donaldson. And using this method, we obtain some regularity results.
3. In this part we present the classical result of Laszlo Lempert on the the holomorphic foliation of convex bodies in complex space.

  

 

  • 课程名称:Introduction to stable minimal hypersurfaces
  • 主讲人:李皓昭(中国科学技术大学)
  • 时间:9:00-11:00am, 6月21、22日
  • 地点:数学院南楼219多媒体报告厅
  • 摘要:In this mini-course, we will discuss some classic results on stable minimal hypersurfaces.

1. We present the curvature estimates of minimal hypersurfaces and the non-existence of stable n-dim minimal cones with $2\leq n\leq 6$.
2. We present the result of Schoen-Simon on the regularity of stable minimal hypersurfaces, and sketch some ideas of Wickramasekera's work on general regularity theory for stable codim 1 integral varifolds.

  

 

  • 课程名称:Introduction to Willmore surfaces in $R^n$
  • 主讲人:李宇翔(清华大学)
  • 时间:9:00-11:00am, 6月23、24、25日
  • 地点:数学院南楼219多媒体报告厅
  • 摘要:In this 6-hour mini-course, we will discuss some classic results on Willmore surfaces in $R^n$:

1.We will present some basic properties of Willmore surfaces.

2.We will introduce the compensated compactness of conformal immersions from the unit 2-disk into $R^n$ with small $\|A\|_{L^2}$, and the Helein’s convergence theorem.

3.We will use the Helein’s convergence theorem to prove the existence of minimizing Willmore torus in $R^n$, which was first obtained by L. Simon.

 

  

报告列表

  • 报告题目:Global regularity and long time dynamics of Schrodinger map flows into Kahler manifolds
  • 主讲人:黎泽(宁波大学)
  • 时间:9:00-10:00am, 6月4日
  • 地点:数学院南楼219多媒体报告厅
  • 摘要:In this talk, we introduce our recent works on global regularity and long time dynamics of Schrodinger map flows into compact Kahler manifolds with small data in critical spaces. In the first part, we briefly recall the works on Schrodinger map flows and then explain the essential difficulty of general targets compared with the constant curvature case. In the second part, we explain the main idea of our iteration-bootstrap argument which successfully solves the general target case.

 

 

  • 报告题目:Recent progress on gravitational instantons
  • 主讲人:陈杲(中国科学技术大学)
  • 时间:10:30-11:30am, 6月4日
  • 地点:数学院南楼219多媒体报告厅
  • 摘要:In this talk, I review previous works on gravitational instantons and explain my recent joint work with Viaclovsky and Zhang on ALG and ALG^* gravitational instantons.

 

 

  • 报告题目: On singular csck metrics
  • 主讲人:郑恺(同济大学)
  • 时间:9:00-10:00am, 6月9日
  • 地点:数学院南楼219多媒体报告厅
  • 摘要:In this talk, we first will present recent progress on the Yau-Tian-Donaldson conjeture in the logarithmic setting. Then we will carry out a comparison of different K-stability notions and show various results when the underlying class is merely big.

 

 

  • 报告题目: On the subharmonicity of the Mabuchi energy
  • 主讲人:李龙(上海科技大学)
  • 时间:10:30-11:30am, 6月9日
  • 地点:数学院南楼219多媒体报告厅
  • 摘要:In this talk, we will discuss the (strict) convexity of the Mabuchi functional along a weak geodesic, via the method of \epsilon-geodesic approximation. It will be proved that the fiberwise volume element of the \epsilon-geodesic converges to the volume element of the weak geodesic in the L^2 sense, provided the affine Mabuchi energy. Moreover, the weak geodesic is fiberwise uniformly non-degenearte, if the Mabuchi energy is \epsilon-affine.

 

 

  • 报告题目:The regularity of a class of semilinear elliptic equaitons
  • 主讲人:江瑞奇(湖南大学)
  • 时间:2:30-3:30pm, 6月11日
  • 地点:数学院南楼219多媒体报告厅
  • 摘要:In this talk, we will show the decay estimate for a class of semilinear elliptic equations which admit a good divergence structure in critical dimension and deduce the Holder regularity of weakly biharmonic almost complex structures.

 

 

  • 报告题目:Strictly nef vector bundles and characterization of projective spaces
  • 主讲人:杨晓奎(清华大学)
  • 时间:2:30-3:30pm, 6月16日
  • 地点:数学院南楼219多媒体报告厅
  • 摘要:In this talk, we will present some recent progress on the geometry of strictly nef vector bundles, with focus on the characterization of projective spaces.
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