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八月份的活動:(尚無任何基礎)
Time & Place:
 8/8–8/19 (1st and 2nd weeks): Monday to Friday.

pm: 1:30 – 3:30.

Lecture Room B, 4th Floor 3rd General Building,

National Tsing-Hua University, Hsinchu
 
8/22–9/2 (3rd and 4th weeks): Monday to Friday.

am: 10:00 – 12:00.

Room 308, National Taiwan University, Taipei.

Organizer:
 Jungkai Alfred Chen (National Taiwan University)
Speakers:
 Andrea D'Agnolo (Universita di Padova, Italy)

Pierre Schapira (Universite Paris VI, France)

Program:
 << 1st week: Homological algebra >>

Andrea D'Agnolo

Review on categories, limits, abelian categories,
triangulated categories, localization, derived categories.

<< 2nd week: Sheaves >>

Pierre Schapira

Grothendieck topologies, presheaves and sheaves, direct and inverse
images,internal operations on abelian sheaves, derived categories of
abelian sheaves, a glance to unboundedderived categories of sheaves,
proper direct images and duality.

<< 3rd week and 4th week: D-modules theory >>

Pierre Schapira

Construction of the filtered sheaf of rings D, characteristic variety of
coherent D-modules, operations, a glance to the microlocal theory of
sheaves, constructible sheaves, holomorphic solutions of D-modules,
elliptic pairs, holonomic system and Riemann-Hilbert correspondence.
A glance to irregular holonomic D-modules.
=======================================================================
2005/7/9
talk about vector bundle,connection,curvature of complex
manifolds. in NCTS.

About the topic:
主持人:
 林惠雯 (中央大學數學系) linhw@math.ncu.edu.tw

時間:
 2005年 6/29(三)、7/5 (二)、7/12 (二)、7/19 (二)

       7/26 (二)、8/1(一) 共六次

上午10:00 ~ 11:30 & 下午1:00 ~ 4:00

內容:
 Complex manifolds

Sheaf Cohomology

Kahler geometry and Hodge theorem

Divisors and Line bundles

Kodaira vanishing theorem

Kodaira embedding theorem
 
方式:
 由學生輪流報告

6/29    7/5    7/12    7/19    7/26    8/1

Place:
 Lecture Room B of National Center for Theoretical Sciences,

4th Floor, The 3rd General Building, National Tsing Hua University

參考書籍:
 P.Griffiths&J.Harris, Principles of algebraic geometry
=============================================================

2005/5,6
final examination for differential geometry:
talk about Plateau problem.
什麼事都不做,全心全意讀了一個月。
這個問題用到了PDE、複變及高等微積分的一些推論。
這個月,讓我看到自己的極限。

===============================================================
數學家 科學家

數學家也是科學家的一份子,不應被排除在外,
學數學的過程中,應時時吸收外界的養份,多去看看別人如何用數學。
倘若一個讀數學的人,成日只沉溺鑽研於繁鎖的數學結構,卻對生活
所發生的事情陌不關心,也難怪別人會看不起學數學的人。

學數學應該是搭配各種科學學習的,這也是為什麼數學系必須要在一個大
學裡,而不是在一個公寓裡。一個數學系的學生,不應該只待在數學系,
更應該走出自己的系館,看看別人如何使用數學。
                                                             6/4
================================================================
[do carmo] 4-3.1

Show that if x is an orthogonal parametrization,that is F=0,then

K = (-1/2√EG) {(E_v/√EG)_v + (G_v/√EG)_u}

pf:      eg-f^2    |x_u x_v x_uu||x_u x_v x_vv|-|x_u x_v x_uv||x_u x_v x_uv|
    K = -------- = -----------------------------------------------------------
         EG-F^2                                EG*EG

              │┌      ┐              │ │┌      ┐              │
         1    ││x_u^t │              │ ││x_u^t │              │
    = -------{││x_v^t │(x_u x_v x_vv)│-││x_v^t │(x_u x_v x_uv)│ }
       EG*EG  ││x_uu^t│              │ ││x_uv^t│              │
              │└      ┘              │ │└      ┘              │



          │                             │ │
     1    │E          F        x_u*x_vv │ │E          F       x_u*x_uv │
= -------{│F          G        x_v*x_vv │-│F          G       x_v*x_uv │}
    EG*EG │x_u*x_uu x_v*x_uu   x_uu*x_vv│ │x_u*x_uv x_v*x_uv  x_uu*x_vv│
          │                             │ │                            │


          │                                   │ │                        │
     1    │E          0       (-1/2)G_u       │ │E        0       1/2E_v │
= -------{│0          G       (1/2)G_v        │-│0        G     (1/2)G_u │}
   EG*EG  │1/2E_u   -1/2E_v  -1/2E_vv-1/2G_uu │ │1/2E_v  1/2G_u      0   │
          │                                   │ │                        │


      1
= -------- [(-1/2)EG(E_vv+G_uu)+(1/4)E_uG_uG + (1/4)E_vG_vE + (1/4)E_vE_vG
    EG*EG
                                                        + (1/4)EG_uG_u ]


        1                     E_v(E_vG+EG_v)         G_u(E_uG+EG_u)
 =  - ----- [E_vv+G_uu - (1/2)-------------- - (1/2) ---------------]
       2EG                          EG                      EG


     1           1           1            E_v(E_vG+EG_v)        G_u(E_uG+EG_u)
=- -----  [E_vv*---- + E_uu*----- - (1/2)*--------------- -(1/2)*-------------]
   2√EG       √EG         √EG             (EG)^3/2           (EG)^(3/2)



      1       E_v        E_u
= - ----  [(------)_v +(-----)_u ]
    2√EG    √EG        √EG

                                     □

[do carmo] 4-3.1(活動標架版)(by Franz)

Suppose

       X : D -> M

is an orthogonal parametrization of M around a point of M, and D is an open

set on R^2.

We know that

     E =<X_u,X_u> , F = <X_u,X_v> = 0 , G = <X_v,X_v>

Set E_1 = X_u/ √E , E_2= X_v/√G, then

 {E_1,E_2} is an orthonomal frame on X(D).

Suppose {θ_1,θ_2} is the dual coframe of {E_1,E_2}, then

        θ_1 = √E du, θ_2 = √G dv,

denote {w_12} the connection form of the frame mentioned above,

then
      dθ_1 = w_12 ^ θ_2 = -(√E)_v du ^ dv = (-(√E)_v/√G du )^θ_2

      dθ_2 =-w_12 ^ θ_1 = (√G)_u du ^ dv = -(√G)_u/√E dv ^ θ_1

therefore,

              w_12 =   -(√E)_v/√G du +(√G)_u/√E dv

The Gauss equation states

                  dw_12 =-K θ_1 ^ θ_2

We have

                 dw_12 = {((√E)_v/√G)_v + ((√G)_u/√E)_u) du ^ dv

                       =√EG{((√E)_v/√G)_v + ((√G)_u/√E)_u) θ_1 ^ θ_2

Thus

                  K = -√EG{((√E)_v/√G)_v + ((√G)_u/√E)_u)


====================================================

有關真理最明晰,最美麗的陳述,最終必以數學形式展現。

                        ─ 梭羅 (H.Thereau 1817-1862)

                                             2003/9/9
======================================================
                少思、少念、少欲、少事、少語、少笑、

                少愁、少樂、少喜、少怒、少好、少惡

                行此十二少,乃養生之都契也。

                多思則神怠,多念則精散,多欲則智損,

                ,多事則形疲,多語則氣促,多笑則肝傷,

                多愁則心懾,多樂則意溢,多喜則忘錯昏亂,

                多怒則百脈不定,多好則專迷不治,多惡則焦煎無寧,

                此十二多不除,喪生之本也。
                                              7/24

===========================================================
                 習慣成自然是個神奇的魔術師,

                 它對美麗的東西極為殘忍,

                 卻對醜陋的東西極為仁慈。

==============================================================
                 偏了方向,只會浪費時間,

                 原以為方向對了,原來卻是在繞遠路,

                 人生的各種抉擇,何嘗不是在決定要走的方向...

===============================================================

                 我們無法見到三維以上的空間,

                     但卻要時常想像它在數學世界的存在。

=================================================================