| 【2021.07.11-07.13 北京】2021几何与分析研讨会 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2021-06-28 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
 2021几何与分析研讨会会议手册
时间：2021年7月11-13日 地点：中科院数学与系统科学研究院思源楼813
报告人：（按拼音顺序） 葛剑（北京大学） 李逸 （东南大学） 李展（南方科技大学） 孙京洲（汕头大学） 王成波（浙江大学） 向昭银（成都电子科技大学） 许斌（中国科技大学） 尧小华（华中师范大学） 张蓥莹（清华大学）
会议组织（按拼音顺序） 陈亦飞（中科院数学与系统科学研究院） 葛剑（北京大学） 王作勤（中国科技大学）
主办单位 中科院数学与系统科学研究院
会议议程
报告题目和摘要（按拼音顺序）
n 葛剑（北京大学） 报告题目：Riemannian manifolds without conjugate points 摘要: In this talk, we discuss classical geometric properties for Riemannian manifolds without conjugate points. Surprisingly there are still open questions for this kind of surfaces, which is easy to understand yet hard to attack.
n 李逸 （东南大学） 报告题目：复几何中一类完全非线性偏微分方程 摘要：在本次报告中，我们将讨论一类复Monge-Ampere型偏微分方程及其在复几何中的应用。
n 李展（南方科技大学） 报告题目：Invariance of plurigenera for the generalized pair with abundant nef part 摘要：Generalized pairs naturally arise from canonical bundle formula and many other situations in algebraic geometry. The invariance of plurigenera for generalized pairs is false in general.However, when the nef part is abundant, then the invariance property holds as a straightforward application of the analytic approach of Siu and Berndtsson-Paun. This is an ongoing joint project with Zhiwei Wang.
n 孙京洲（汕头大学） 报告题目：asymptotics of Bergma kernels and applications 摘要：We will talk about the asymptotics of Bergman kernels from smooth cases to non-smooth cases. Then will talk about some of their applications.
n 王成波（浙江大学） 报告题目：径向拟线性波动方程的低正则适定性 摘要：本报告中，我将报告球对称条件下拟线性波动方程的具有最优正则性条件的局部适定性结果。该正则性条件无法通过Strichartz估计的球对称改善而得到。做为基本的理论工具，我们提出并证明了具有低正则系数要求的Morawetz型局部能量估计，以及与局部能量估计相适应的各类线性与非线性加权估计。我们的结果也能用于证明小初值低正则高维整体适定性，及三维几乎整体适定性。
n 向昭银（成都电子科技大学） 报告题目：趋化-流体耦合模型的研究进展 摘要：趋化-流体耦合模型的数学理论是最近十多年才引起数学家、生物学家、物理学家广泛关注的多学科课题。本报告拟对趋化-流体耦合模型的适定性理论、大时间行为、稳定性等最新研究进展作一简要介绍。
n 许斌（中国科技大学） 报告题目：Cone spherical metrics 摘要：Cone spherical, at and hyperbolic metrics are conformal metrics with constant curvature +1; 0 and -1, respectively, and with finitely many conical singularities on compact Riemann surfaces. The Gauss-Bonnet formula gives a necessary condition for the existence of such three kinds of metrics with prescribed conical singularities on compact Riemann surfaces. The condition is also sufficient for both at and hyperbolic metrics. However, it is not the case for cone spherical metrics, whose existence has been an open problem over twenty years. I will introduce the respectful audience some progress on this problem by using both Complex Analysis and Algebraic Geometry. l We obtained on compact Riemann surfaces a correspondence between meromorphic one-forms with simple poles and real periods and cone spherical metrics whose developing maps have monodromy in U(1), called reducible metrics. As an application, we found the cone angle constraint of reducible metrics on compact Riemann surfaces. l We obtained on compact Riemann surfaces a correspondence between meromorphic Jenkins-Strebel differentials with real periods and cone spherical metrics with monodromy in the semidirect product of U(1) and l We obtained on compact Riemann surfaces a correspondence between irreducible metrics with cone angles in This talk is based my joint works with Qing Chen, Yiran Cheng, Bo Li, Lingguang Li,Jijian Song and Yingyi Wu.
n 尧小华（华中师范大学） 报告题目：Kato smoothing and Strichartz estimates for fractional operators with Hardy potentials 摘要: Let $0<\sigma<n/2$ and $H=(-\Delta)^\sigma +a|x|^{-2\sigma}$ be Schrodinger type operators on $\R^n$ with a sharp coupling constant $a\le -C_{\sigma,n}$ ( $C_{\sigma,n}$ is the best constant of Hardy's inequality of order $\sigma$). In the present talk, we will address that sharp global estimates for the resolvent and the solution to the time-dependent Schrodinger equation associated with $H$. In the case of the subcritical coupling constant $a>-C_{\sigma,n}$, we first prove the uniform resolvent estimates of Kato--Yajima type for all $0<\sigma<n/2$, which turn out to be equivalent to Kato smoothing estimates for the Cauchy problem. We then establish Strichartz estimates for $\sigma>1/2$ and uniform Sobolev estimates of Kenig--Ruiz--Sogge type for $\sigma\ge n/(n+1)$. In the critical coupling constant case $a=-C_{\sigma,n}$ , we show that the same results as in the subcritical case still hold for functions orthogonal to radial functions. This is a joint-work with Haruya Mizutani.
n 张蓥莹（清华大学） 报告题目：Deformation of Fano manifolds 摘要：In this talk, we will discuss the existence of the K？hler Einstein metrics on a Fano manifold under the deformation of complex structures, this leads to the understanding of Weil-Petersson etric on the moduli space of Fano K？hler Einstein manifolds. We will also talk about a plurisubharmonic function the Teichmull？r space of K？hler-Einstein manifolds of general types. This is based on a recent work joint with H.-D. Cao, X. Sun and S.-T. Yau. |
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, called quasi-reducible metrics. Moreover, by using the Mumford-Thurston correspondence, we could construct new quasi-reducible metrics by drawing certain connected metric ribbon graphs.
and line sub-bundles of rank two stable vector bundles. As an application, we found new existence and uniqueness results about irreducible metrics on compact Riemann surfaces with genus greater than one.