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2013年几何暑期学校系列讲座- Prof. Laurentiu Maxim 课程名称：Characteristic classes of singular toric varieties 授 课 人：Prof. Laurentiu Maxim (University of Wisconsin Madison, USA) 课时安排: 7月2日，周二，14:00~16:00 p.m. Room 1418 欢迎广大师生参加！

This series of short course mainly on some topics:the first a generalized Poisson equation with zero boundary value;the second topic is about Sobolev inequality, a new idea on Hardy-Littlewood-Sobolev-Lieb inequality;the third topic is about Yamebe problem.

In this series of lectures we will present a self-contained introduction to the theory of asymptotics for solutions of elliptic and parabolic equations developed by Leon Simon. We will discuss two important applications of the theory, namely: 1) The Cheeger-Tian technique for proving uniqueness of tangent cones at infinity for Ricci-flat metrics with quadratic curvature decay and Euclidean volume growth, 2) The study of Harmonic maps and the uniqueness problem for tangent maps with isolated singularities.

Kahler geometry is one of the basic tools of algebraic geometry and the study of complex manifolds. Also, Kahler geometry provides many examples for symplectic geometry and Riemannian geometry. I will give an introduction of Kahler geometry, starting with basics (de Rham differential and Levi-Civita connection). The topics to cover: 1. Vector bundles and analysis on manifolds. 2. Connections in vector bundles. 3. Levi-Civita connection and its holonomy. 4. Kaehler manifolds and their holonomy. Examples of Kaehler manifolds. 5. Supersymmetry on Kaehler manifolds and its applications. 6. Berger's classification of irreducible holonomies and its applications. Some knowledge of topology (manifolds,smooth maps) and linear algebra (tensor product,Grassmann algebra) will be helpful.

Our main goal is to prove some of the structure theorems for Riemannian manifolds with uniform lower bounds on their Ricci curvature due to Cheeger and Colding in the mid-1990's. This involves an interesting mix of Riemannian geometry and geometric analysis. In particular, we will use our main goal as an excuse to talk about many different ideas and techniques in the field. The only prerequisites are basic courses on manifolds and Riemannian geometry. Some previous experience with linear elliptic equations is helpful but not necessary.

Ideal in the polynomial rings, Hilbert Basis Theorem and Nullstellensatz. Affine algebraic varieties and their morphisms. Dimension. Projective varieties and homogeneous ideals. Vector bundles. Grassmannians and varieties determined by rank conditions. Hilbert Polynomials and Bezout’s Theorem. Algebraic Curves, linear systems and Riemann-Roch Theorem.

2013年几何学暑期学校系列讲座-Zhiyuan Li 报告人： Zhiyuan Li (Szego assistant professor at Stanford University ) 课时安排: 7月4日，7月8日，14:30-16:30 授课教室：1418 报告内容： 7月4日 Title: Noether-Lefschetz theory and applications (I): Special cubic fourfolds and modular forms Abstract: In this talk, I will talk about the degree of special cubic fourfoulds divisors. In particular, we show that the generating series of these degrees is a level three modular form. The modularity comes from Noether-Lefschetz theory and Borcherd's work, which I will introduce in this lecture. This work is joint with Letao Zhang. 7月8日 Title: Noether-Lefschetz theory and applications (II):Noether- Lefschetz conjecture for K3 surfacesAbstract: After introducing the Noether- Lefschetz divisors on moduli K3 surfaces, we prove the Noether-Lefschetz conjecture for low degree K3 surfaces. For the second half of this talk, I will also briefly talk about its close relation to automorphic represe

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 Aranda Pino Profesor of Department of Algebra, Geometry and Topology Science Faculty, University of Málaga, Spain. Research Interests: Leavitt path