几何分析在几何学的研究中起了重要的作用，它的兴起是在上世纪七十年代中后期和八十年代初期。在此期间，一系列重要几何问题得到了解决，这包括Yamabe问题，正质量猜想以及Calabi 猜想。椭圆型方程在这些问题的解决中起了关键性的作用。事实上，这三类问题分别对应了半线性、拟线性和完全非线性椭圆型方程。非线性椭圆型方程理论正是建立在这三类方程的基础上。

      随着（严格）椭圆型方程理论的完善，退化椭圆型方程已成为偏微分方程中重要的和活跃的课题之一，特别是与几何和物理有关的退化椭圆型方程更是引起了广泛的关注。由于退化性的多样化，到目前为止退化椭圆型方程并没有完整的理论，与之相关的一些重要问题也没有得到完全解决，这包括环簇上的预定标量曲率问题和Monge-Ampère方程的特征值问题。这些问题的共同特点如下：方程本身定义在紧带边流形上，方程在内部为严格椭圆，退化只发生在边界。这些问题的核心是方程的解在边界附近的渐进行为和几何性质的研究。

      为推动我国学者在相关问题上的研究，基金委天元基金资助于4月5日至4月22日在北京大学北京国际数学研究中心举办为期3周的专题讲习班，计划为来自全国大学、研究所的研究生和青年学者讲授4门共48学时与退化椭圆型方程相关的课程。讲习班期间主办方为京外学员提供住宿和饭卡，为京内学员提供饭卡。资助名额有限。请感兴趣的研究生和青年学者尽快报名。如有问题，请和刘赫老师联系。

组织者：葛剑（jge@math.pku.edu.cn）

            韩青（qhan@math.pku.edu.cn）

            史宇光（ygshi@math.pku.edu.cn）            

联系人：刘赫 (liuhe@math.pku.edu.cn )

资助单位： 中国自然科学基金委员会天元基金

                 北京大学北京国际数学研究中心

                 北京大学数学科学学院

                                                                                              研讨班教学大纲

I.   课程对象：基础数学和应用数学专业博士研究生和青年学者。

II.  课程目的：通过对几何分析和退化椭圆型方程的学习，使研讨班学员了解、掌握微分流形上退化椭圆型方程研究涉及的问题、方法以及国际上该领域研究的一些最新进展。推动我国学者在该领域的研究。

III. 课程内容：环簇上的几何与分析, Liouville 方程的变分特性， 非线性椭圆方程正解在孤立奇点的渐进分析，和具有锥型奇性的流型。

IV.  课程要求：具备黎曼几何、偏微分方程基础知识。

V. 课程简介：

1.  课程题目：环簇上的几何与分析

     授课教师：四川大学陈柏辉

     内容简介：It is known that a toric variety can be read from a Delzant convex polytope. In this lecture, we will explain that such constructions can be generalized to convex polytopes with certain non-degeneracy condition. Then we will study several related geometry problems. For example, the study of extremal metrics on toric manifolds can be generalized to such geometric objects.

     参考资料：

     [1] M. Audin, Toric Actions on Symplectic Manifolds, Progress in Math., Birkhauser, 2004.

     [2] V. Guillemin, Moment Maps and Combinatorial Invariants of Hamiltonian T^n-Spaces, Progress in Math., Birkhauser, 1994.

2．课程题目：Variational Aspects of Liouville Equations

     授课教师：Andrea Malchiodi, Scuola Normale Superiore

     内容简介：We consider a class of Liouville equations and systems which arise in differential geometry when prescribing the Gaussian curvature of a surface and in models of mathematical physics describing stationary Euler flows and self-dual Chern-Simons equations. We discuss methods, variational in nature, to derive general existence results from suitable improvements of the Moser-Trudinger inequality combined with topological methods.

    参考资料：

    [1] Z. Djadli, A. Malchiodi,  Existence of conformal metrics with constant Q-curvature, Ann. of Math. 168 (2008), 813-858.

    [2] A. Carlotto, A. Malchiodi, Weighted barycentric sets and singular Liouville equations on compact surfaces. J. Funct. Anal. 262 (2012), 409–450.

    [3] D. Bartolucc, A. Malchiodi, An improved geometric inequality via vanishing moments, with applications to singular Liouville equations, Comm. Math. Phys., 322 (2013), 415–452.

    [4] A. Malchiodi, D. Ruiz, A variational analysis of the Toda system on compact surfaces, Comm. Pure Appl. Math.,  66 (2013),  332–371.

    [5] L. Battaglia, A. Jevnikar, A. Malchiodi, D. Ruiz,  A general existence result for the Toda system on compact surfaces. Adv. Math. 285 (2015), 937–979.

3．课程题目：非线性椭圆方程正解在孤立奇点的渐进分析

     授课教师：北京师范大学熊金钢

     内容简介：Understanding asymptotic behaviors of solutions of PDEs near an isolated singularity is of the basic importance. In this lecture, we will report a classical result for positive solutions of the nonlinear PDEs, which include the Yamabe equation, due to Caffarelli-Gidas-Spruck. These authors proved that the isolated singular solutions have to be asymptotically radially symmetric and close to some ODE solutions in the whole space. We will also report refined results due to Korevaar-Mazzeo-Pacard-Schoen, which give the next order expansion. Refined results have been used crucially to glue singular complete conformal metrics with constant scalar curvature.

     参考资料：

     [1] L. Caffarelli, B. Gidas, J. Spruck, Asymptotic symmetry and local behavior of semi-linear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271–297.

    [2] Z. Han, Y. Li, E. Teixeira, Asymptotic behavior of solutions to the k-Yamabe equation near isolated singularities, Invent. Math., 182 (2010), 635–684.

    [3] N. Korevaar, R. Mazzeo, F. Pacard, R. Schoen, Refined asymptotics for constant scalar curvature metrics with isolated singularities, Invent. Math., 135 (1999), 233-272.

4．课程题目：Manifolds with conic singularities

     授课教师：首都师范大学张震雷

     内容简介：We will start with the basic notion of conic singularities. Then, following Donaldson, we introduce the linear analysis on manifolds with conic singularities and give a proof of his openness theorem for conic Kahler-Einstein manifolds on Fano manifolds with varying cone angles. Finally we discuss the Holder regularity of weak conic Kahler-Einstein metrics in the paper of Jeffres-Mazzeo-Rubinstein, following the idea of Tian. If time permits, we will also have a short discussion of Li-Sun's work on the existence of conic Kahler-Einstein metrics for small cone angles.

     参考资料：

    [1] S. Donaldson, Kahler metrics with cone singularities along a divisor, arXiv:1102.1196

    [2] T. Jeffres, R. Mazzeo and Y. Rubinstein, Kahler-Einstein metrics with edge singularities (with an appendix by C. Li and Y. Rubinstein), Annals of Mathematics, 183 (2016), 95-176.

    [3] C. Li and S. Sun, Conical Kahler-Einstein metrics revisited, Comm Math Physics, 331 (2014), 927-973.

VI. 课程安排