作为学术会议“Perspectives in Geometric Analysis at XJTU”的延续，7月8日—7月12日将举办几何分析暑期学校，邀请相关领域的专家学者报告学科基础理论以及学术前沿动态，欢迎有兴趣的老师和同学参加。
课程授课地点：理科楼202
授课时间安排：
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7月8日 (星期五) |
7月19日 (星期三) |
7月11日(星期一) |
7月12日(星期二) |
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上午 9:00-11:00 |
Xujia Wang |
Qing Han (9:00-10:20) |
Xujia Wang |
Xujia Wang |
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Gang Liu (10:30-11:50) |
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下午3:00—5:00 |
Albert Chau |
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Albert Chau |
Gang Liu |
课程简介:
Title: lectures on the parabolic Monge ampere equation and applications
Author：Albert Chau(UBC)
Abstract: In these two lectures I will review some basic estimates and results for the parabolic Monge Ampere equation on a complete Kahler manifold. As special cases, I will then discuss the Kahler Ricci flow and the flow of singular metrics including the conical Kahler Ricci flow.
Title: Boundary expansions for geometric PDEs
Author: Qing Han (Univ. Notre Dame)
Abstract: In some geometric problems, we need to discuss the asymptotic expansions of solutions near boundary and estimate the remainders. The list of such problems includes the singular Yamabe problem, the regularity of minimal surface near the asymptotic infinity in the hyperbolic space and the complete Kahler-Einstein metrics in strictly pseudo-convex domains. Usually, the underlying equations become degenerate along boundary. In this talk, we present a PDE approach for remainder estimates, which are referred to as the polyhomogeneity, and for the global regularity up to boundary.
Title: On some analytic and geometric applications of three circle theorems on Kahler manifolds I(II)
Author：Gang Liu(UCB)
Abstract: We derive a sharp three circle theorem on Kahler manifolds with nonnegative holomorphic sectional curvature. Then as applications, we derive sharp dimension estimates for polynomial growth holomorphic functions. We also use this as a tool to confirm a conjecture of Ni Lei.
Title: Monge’s mass transport problem
Speaker: Xujia Wang (ANU)
Abstract: The optimal transportation problem can be formulated as a Monge-Ampere type equation, and the existence and regularity of optimal mappings have been established under certain conditions. Monge’s original problem is one of the most interesting cases and is at the borderline of these conditions. With my collaborators Qi-Rui Li and Filippo Santambrogio, we recently studied the regularity of Monge’s problem and observed some delicate results. We proved that in a smooth approximation, the eigenvalues of the Jacobian matrix of the optimal mapping are uniformly bounded but the mapping itself may not be Lipschitz continuous. But in dimension two the mapping is continuous. In this talk I will discuss recent development in this direction.
