课程简介:

• 课程１

  课程名：Lectures on Non-compact Shrinking Gradient Ricci Solitons

  主讲人：Peng Lu (Oregon University)

  简介： The lectures will include the following theorems related with gradient Ricci solitons and the equations: Cao-Zhou's estimate of potential function, the estimate of scalar curvature and its consequence; entropy, volume growth, curvature decay and splitting at infinity. Some topics will to be chosen.

• 课程2

  课程名: Introduction to volume comparisons, gradient bounds and degenerations of Riemannian metrics

  主讲人: Qi Zhang (U. C. Riverside)

  简介： We will start with the classical volume comparisons theorems and gradient bound for harmonic-caloric functions on manifolds. Then we will describe some basic results of Colding and Cheeger-Colding on the degenerations of Riemannian metrics with point-wise lower Ricci bounds under Gromov-Hausdorff convergence. These include continuity of volume, splitting theorem and properties of tangent cones. Connections to current research topics on relaxing the Ricci condition will be given. These lectures are coordinated with Profess Zhenlei Zhang's lectures on related but more recent results.

• 课程3

  课程名：Spectrum theory and its applications in Kahler geometry

  主讲人: Zhiqin Lu (U.C. Irvine)

  简介： In this serious of talks, I will present some recent development of the spectrum theory of complete non-compact manifolds and non-complete manifolds. We will use these results in Kähler geometry. In particular, we shall estimate the upper bound of the off diagonal Bergman kernel and study the L^2 estimates on Kähler manifolds.

• 课程4

  课程名: Co-dimension 4 conjecture and its applications

  主讲人：Zhenlei Zhang (Capital Normal University)

  简介： I will give a series of talks about Cheeger-Naber's paper "Regularity of Einstein manifolds and the co-dimension 4 conjecture". The main result is the proof of the co-dimension 4 conjecture for manifolds with bounded Ricci curvature. It states that the non-collapsing limit of such manifolds has at least co-dimension 4 singularities. Applications to 4-manifolds will also be discussed. Some pre-knowledge on Gromov-Hausdorff convergence and harmonic radius are required.

• 课程5

  课程名: From index theory to Bergman kernel: a heat kernel approach

  主讲人: Xiaonan Ma (Institut de Mathématiques de Jussieu, France)

  简介： We explain first the Hirzebruch-Riemann-Roch theorem and introduce the characteristic classes, Chern-Weil theory, then we explain its heat kernel proof: the local index theorem for HRR theorem. We try finally to understand the Bergman kernel from heat kernel inspired by the heat kernel proof of HRR theorem.

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