活动地点：

活动类型：研究生课程

主讲人：徐浩

活动时间：

活动内容：

 

    We discuss Hodge Integrals on moduli spaces of Curves.

Ref:   C.Faber and R. Pandharipande,  Hodge integrals and Gromov-Witten Theory

几何与物理讨论班

讨论时间:   每周一 18：30-21：00,  每周三 10：00-11：30。其中 每周一 20：00-21：00为自由讨论时间。

地点： cms201

 

2006-3-29（星期三， 10：00-11：30）

  杨晓奎: Vanishing theorems: the heat kernel approach.

Ref: Thierry Bouche, Asymptotic results for hermitian line bundles

over complex manifolds：the heat kernel approach.

2006-3-27（星期一， 18：30-21：00）

    李逸：An introduction to string theory II

2006-3-22（星期三， 10：00-11：30）

    李逸： An introduction to string theory I

2006-3-20（星期一， 18：30-21：00）

     杨晓奎：Curvature of vector bundles associated to holomorphic fibrations

      Ref:  Bo Berndtsson, Curvature of vector bundles associated to holomorphic fibrations, math.CV/0511225

2006-3-15（星期三， 10：00-11：30）

    杨晓奎：Holomorphic Morse inequalities and applications

Abstract: We discuss the analytic proof of the holomorphic

Morse inequalities and some applications to algebraic geometry.

Ref：1. J.P. Demailly,  Holomorphic Morse inequalities, Proceedings of Symposia in Pure Mathematics, Volume 52(1991), PART2.

      2. J.M.Bismut, Demailly’s asymptotic Mosre inequalities: a heat equation proof, J. Funct. Anal. 72(1987), No.2, 263-278.

      3. Y.T. Siu,  Calculus inequalities derived from holomorphic Morse inequalities. Math. Ann. 286 (1990), no. 1-3, 549--558.

      4.  Y.T.Siu  Asymptotic Morse inequalities for analytic sheaf cohomology. Séminaire Bourbaki, Vol. 1985/86. Astérisque No. 145-146 (1987), 5, 283—297.

2006-3-13（星期一， 18：30-21：00）

徐浩：Computation with Localization

 

  Abstract: Localization technique is one of

the most powerful mathematical tools in mirror

symmetry. We will give several working

examples on the projective spaces.

2006-3-8（星期三， 10：00-11：30）

 尹方亮：Holomorphic equivariant cohomology II

 

Abstract: Our report will base on Prof. Liu's paper " Holomorphic equivariant cohomology". Some interesting relationships between certain cohomology  (Dolbeault equivariant cohomology) and residue formula ( Bott  residue formula) are discussed.

 

 

2006-3-6（星期一， 18：30-21：00）

 尹方亮：Holomorphic equivariant cohomology I

 

Abstract: Our report will base on Prof. Liu's paper " Holomorphic equivariant cohomology". Some interesting relationships between certain cohomology  (Dolbeault equivariant cohomology) and residue formula ( Bott  residue formula) are discussed.

 

2006-3-1（周三，10：00-11：30）         徐浩： Basic facts about Moduli of Curves       Abstract：We will give a short introducton to the Moduli of Curves $M_{g,n}$, including their definitions and combinatorial structures。

2006-2-27(周一 18：30-21：00)，

       杨晓奎： Various notions of positivity

       Abstract：We will introduce various notions of positivity: ample, Griffiths positive, Nakano positive, dual Nakano positive, strongly positive and so on. The relations of them will also be discussed.

        Ref: 5 and 7.

参加者及其主要内容简介：

1. 徐浩：镜像对称与代数几何

 

讨论镜像对称中的基本问题，如曲线计数，Gromov-Witten不变量等。介绍曲线模空间，Calabi-Yau流形，局部化技巧等。

 

部分参考文章

Localization and conjectures from string duality.  by Kefeng Liu

A proof of a conjecture of Marino-Vafa on Hodge integrals.

by  Chiu-Chu Melissa Liu, Kefeng Liu, and Jian Zhou

Enumeration of Rational Curves via Torus Actions.  by M. Kontsevich

Mathematical Aspects of Mirror Symmetry.  by David R. Morrison

 

主要参考书

Mirror Symmetry and Algebraic Geometry.  by David A. Cox and Katz

Mirror Symmetry (Clay Mathematics Monographs, V. 1).  by Vafa. et al

Mirror Symmetry (Smf/Ams Texts and Monographs, V. 1).  by Claire Voisin

 

2. 杨晓奎：代数几何与复几何中的超越方法

    主要讨论代数几何与复几何中的嵌入及消没性质， 包括复向量丛的正性，稳定性等。

主要参考文献：

1. Bo Berndtsson, Curvature of vector bundles associated to holomorphic fibrations, math.CV/0511225.
2. J. Eells  & L. Lemaire, Another report on harmonic maps. Bull. London Math. Soc. 20(1988), no. 5, 385--524.
3.  P. Griffiths, J. Harris, Principles of algebraic geometry, John Wiley, New York, 1978.
4. K. Liu,  Holomorphic equivariant cohomology,  Math.  Ann,  303(1995), 125-148.
5. B. Shiffman, A.J. Sommese, Vanishing theorems on complex manifolds, Birkhauser, 1985.
6.  Y.-T, Siu, The complex-analyticity of harmonic maps and the strong rigidity of compact Kahler manifolds. Ann. of Math. (2) 112 (1980), no. 1, 73--111.
7. Y.-T, Siu, Complex-analyticity of harmonic maps, vanishing and Lefschetz theorems. J. Differential Geom. 17 (1982), no. 1, 55--138.
8. Y.-T, Siu, A vanishing theorem for semipositive line bundles over non-Kahler manifolds. J. Differential Geom. 19 (1984), no. 2, 431--452.
9. Y.-T, Siu,  & S.-T, Yau,  Compact Kahler manifolds of positive bisectional curvature. Invent. Math. 59 (1980), no. 2, 189--204.

3.  李逸

    I will give some lectures about vertex operator algebra in physics and differential geometry, Hodge integrals.

 

4.尹方亮

  介绍代数几何中的一些基本概念和基本定理，例如概型等

主要参考文献：

1. Hartshorne,  Algebraic geometry  

2. Milnor,  Lectures on the h-cobordism theorem.

3.Kefeng, Liu Holomorphic equivariant coomology.

 

其他参加者的简介添加中————.