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'uêeÚπ5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Hilbert 1
;K0
Riemann-Roch½n²;y² . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Á²Ê 6Introduction To Integrable Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . & 24What is algebraic topology about? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Marcel Bokstedt 49
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n/¥Zc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ÜÆÉ 57
êƤ
Joseph Liouville) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .¤C 68
'uêeÚπ5∗
David Hilbert
[Nachrichten der Gesellschaft der Wissenschaften zu Gottingen S.113-116 (1893).
Mathem. Annalen Bd. 43, S.216-219 (1893).]
bêe÷vng§
a+ a1e+ a2e2 + · · ·+ ane
n = 0,
ÙXêa, a1, . . . , anþknê"
§>¦±XeÈ©∫ ∞
0=
∫ ∞
0zρ[(z − 1)(z − 2) · · · (z − n)]ρ+1e−zdz,
Ù¥ρL«ê§KLª
a
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2
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∫ ∞
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TLª©)XeüLªÚµ
P1 = a∫∞0 + a1e
∫ ∞
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2
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2+ · · ·+ ane
n
∫ ∞
n,
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∫ 1
0+a2e
2
∫ 2
0+ · · ·+ ane
n
∫ n
0.
úª ∫ ∞
0zρe−zdz = ρ!
L²§È©∫∞0 ´ρ!Økn꧿§|^z = z′+ 1, z = z′+ 2, . . . , z =
z′ + n´§e∫∞1 , e2
∫∞2 , . . . , en
∫∞n ´(ρ+ 1)!Øknê"l P1´ρ!
Økn꧿¤áρ+ 1Óª
P1
ρ!≡ ±a(n!)ρ+1 (ρ+ 1). (1)
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©Ñ?IKe"Èö5"
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,¡§e©O^KÚkL«¼ê
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Ú
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|∫ 1
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0| < 2kKρ, . . . , |
∫ n
0| < nkKρ
¿P
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Kkت
|P2| < κKρ. (2)
y3(½êρ§ÙÄkUêa · n!اÙg÷vκKρ
ρ! < 1"u´duÓ
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«T§Ù§Kdu1 + eiπ0§Lª
(1 + eα1)(1 + eα2) · · · (1 + eαn) = 1 + eβ1 + eβ2 + · · ·+ eβN
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XMêβ1, . . . , βMØÓu0§ Ùu0§KùMêβ1, . . . , βM´Xe/
ªMg§µ
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òTLª¦±È© ∫ ∞
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2
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βM
,
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′dz′ = (ρ+ 1)!G(βi),
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ÑbρM+MØ"duβ1, . . . , βM´Xê§f(z) = 0§l ¦±ÄXêb¤
êꧤ±
G(β1) +G(β2) + · · ·+G(βM )
7knê"ddѧLªP1uρ!Økn꧿¤á
ρ+ 1ÓªP1
ρ!≡ abρM+Mbρ+1
M , (ρ+ 1). (3)
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|∫ βi
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¿P
κ = |β1eβ1 |+ |β2eβ2 |+ · · ·+ |βMeβM |k,
Kddت
|P2| < κKρ. (4)
3
y(½êρ§ÙÄkabbMاÙg÷vκKρ
ρ! < 1"u´duÓª(3)§P1ρ!´ØUρ + 1Ø! ØÓu0ê§qÏlت(4) P2
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£Ú,u2 Ȥ
1=Lindemann-Weierstrass½nµezi?¿pØÓêê§Ni½XdØ"§KXe/ª
ªØU¤áµ0 = N0ez0 + N1e
z1 + N2ez2 + · · · + Nre
zr"F. Lindemann, Uber die Zahl π. In:
Mathematische Annalen 20 (1882), S. 213 - 225"Èö5"2Ä58-Äê53.)§yÒÖua¬iénóÆ£Eidgenossische Technische Hochschule,
Zurich¤"
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de l’Academie des Sciences, 77, (1873), Paris, pp. 18-24, 226-233, 285-293¶5êÆû
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ÛÉå"XHermann Weyl3David Hilbert and his mathematical work
¥¤µ/A characteristic feature of Hilbert’s method is a peculiarly direct attack on
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§Ò¬õ5ù(J"~õ¼êX√z§log z",·±C
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m8§XC×−]−∞, 0[§¿Ùþü©|§ù*þw84¼ê
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] −∞, 0[þ§K·mX,f+f− 3XþÊÜëY¼êf"(X
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l ·mX,§3ÛÜþÑÓuCm8§Ïd±!ØÙþ
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/X(C2, Hausdorff)\E("Ù¥E(´X1wãþ§ÙØÓ
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OPΓ(U,E(k)X ),Γ(U,OX), Γ(U,MX),Γ(U,ΩX),Γ(U,M
(1)X )"X,YRiemann¡§ë
YNf : X → Y¡X§eÙ3ÛÜIeX§¡Ó§e∃g : Y → XX¦
f g = idY , g f = idX"
51. Γ(U,G ), (G = E(k)X ,OX ,MX ,ΩX ,M
(1)X ) þC−5m"
C ⊂ Γ(U,OX) ⊂ Γ(U,E(0)X ),Γ(U,OX) ⊂ Γ(U,MX) þC−ê"
7
Γ(U,MX)C*"
Γ(U,ΩX) ⊂ Γ(U,M(1)X ),Γ(U,ΩX) ⊂ Γ(U,E
(1)X )"
Γ(U,ΩX)Γ(U,OX)−"Γ(U,M(1)X )Γ(U,MX)−"
XRiemann¡§U ⊂ Xm8§∀p ∈ U§pNCÛÜIz = x + iy : V →W ⊂ C,s.t.p ∈ V ⊂ U"Kdx,dydz := dx + idy,dz := dx − idy´Γ(V,E
(1)X )
ü|Γ(V,E(0)X )− Ä" dz´Γ(V,ΩX)(½Γ(V,M
(1)X ))Γ(V,OX)(½Γ(V,MX)) -Ä" ∀f ∈
Γ(U,MX)§f3p:":§4:ê½ÂûÐ" ∀ω ∈ Γ(U,M(1)X ), ω|V = f(z)dz§é
,f ∈ Γ(U,MX)"ω3p:":!4:ê½Âf3p:":!4:ê"
~1. X = P1 = C ∪ ∞;Riemann¡"zC ⊂ P1þI§Kw = 1z∞NC
I"dzÑXþæX©/ª§Ï3Iwe§dz = d( 1w ) = − 1
w2 dw3w =
0(z = ∞)?æX"dz3∞k4:"3·cE√z 3C×þû½N¼ê§e3X+,X−¥Ñ\\0Ú∞§¿©Oòùü0Êܧü∞ÊܧK·;Riemann¡(Xeã)"Ù3ÿÀþ¥¡S2"
·òy²§?ÛÿÀþS2Riemann¡þÓuP1"
~2. τ ∈ H = z ∈ C | Im z > 0"Λ := Z⊕τZ ⊂ C:"X = C/Λ;Riemann
¡"℘éAuΛWeierstrass¼ê
℘(z) :=1
z2+
∑
w∈Λ w 6=0
[1
(z − w)2− 1
w2]
K℘ÑXþæX¼ê"CþX©/ªdz3Λ^eØC§ÏdÑXþ
X©/ª"ØÓu~1§éØÓτ§,Xτ3ÿÀþÑ´ü¡§
Riemann¡ ´ØÓ"
X;Riemann¡,KXÿÀmo´½4¡(½dE(
Ñ)"Ïdd4¡©a½nXÓg−¡"½ÂgXº"·kH1(X,Z) ∼=Z2g"duXN´mN§´Γ(X,OX) = C§=Xþvk²X¼ê"
XþæX¼ê(´k"d§·òy²dimC Γ(X,ΩX) = g§=ª
dimC Γ(X,OX)− dimC Γ(X,ΩX) = 1− g
8
ù´Riemann-Roch½nAÏ/"
öS1. y²dimC Γ(P1,ΩP1) = 0"=P1þÃ0X©/ª"
1.2 ;Riemann¡ê
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´êAÛ¥ü!Äé"oÑ/`§êÒ´^õª":½Â
AÛé§XF1êÆïÄIax2+bxy+cy2+dx+ex+f = 0§½3Lü
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ù§Xê±3?Û¥§XQ,Q(i),R,C,Fq"ùXê(½.)Ø´
ê4§·±Òê¯êدK§Xxn + yn = 1þ´Äk¦xy 6= 0kn
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ÄEK1wê"
½Â2. -Pn = (x0, · · · , xn) ∈ Cn | (x0, · · · , xn) 6= (0, · · · , 0)/ ∼§Ù¥d'X∼½Â(x0, · · · , xn) ∼ (y0, · · · , yn) ⇔ ∃λ ∈ C× s.t.(x0, · · · , xn) = (λy0, · · · , λyn)"
P(x0, · · · , xn)3Pn¥[x0 : · · · : xn]"½ÂDi = [x0 : · · · : xn] ∈ Pn | xi 6= 0§0 ≤ i ≤ n§KkV
ϕi : Di → Cn
[x0 : · · · : xn] 7→ (x0
xi, · · · , xi
xi, · · · , xn
xi)
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÷v
1)3kEXêàgõªfj(x0, · · · , xn) ∈ C[x0, · · · , xn], 1 ≤ j ≤ k"¦X =
[x0 : · · · : xn] ∈ Pn | fj(x0, · · · , xn) = 0, ∀j2) ∀0 ≤ i ≤ n. ∀[x0 : · · · : 1
1i : · · · : xn] ∈ Di ∩X±ek × nEXêÝn− 1"
Jac(fj(x0, · · · , 11i
, · · · , xn)) :=
(∂fj(x0, · · · , 11i
, · · · , xn)
∂xl
)
1≤j≤k 0≤l≤n l 6=i
~3. P1P1¥"
9
~4. £²¡I¤X = [x : y : z] ∈ P2|x2 + y2 + z2 = 0P2¥"
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3,Ù¥g2, g3 ∈ C ÷vg3
2 − 27g23 6= 0"XP2¥"
·½ÂmXeµ
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1)∀q ∈ Y ,ϕ−1(q)k8½ϕ−1(q) = X"
2)∀0 ≤ i ≤ m, 0 ≤ j ≤ n,Pϕµϕ−1(Dj)
⋂Di → Dj [x0 : ... : 1
1i : ... : xm] 7→ [f0(x0, ..., xi, ..., xm) : ... : 1
1j :
... : fn(x0, ..., xi, ..., xm)]
∀0 ≤ l ≤ n, l 6= j,∃õªgl(x0, ..., xi, ..., xm), hl(x0, ..., xi, ..., xm)
s.t.∀[x0 : ... : 1 : ... : xm] ∈ ϕ−1(Dj)⋂Di
gl(x0, ..., xi, ..., xm) 6= 0
fl(x0, ..., xi, ..., xm) = hl(x0,...,xi,...,xm)gl(x0,...,xi,...,xm)
ϕ¡Ó§e∃ψ : Y → X,s.t. ϕ ψ = idY , ψ ϕ = idX
öS2. y²~4¥X ∼= P1£)"
·rPnw´nE6/§ÙÛÜIdϕi : Di → CnÑ"éuPn¥
X,·#KXê( òÙwPnEf6/"¢6/Û¼ê½néN´í
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XN§Ïd·lÆ;Riemann¡Æ¼fµ
F : → ;Riemann¡
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Riemann¡Óa"
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10
Ïd§yF´essentially surjective,Iy²§z;Riemann¡ÑUXi
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;Riemann¡ÓuX"
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2 Riemann-Roch½nQãíØ
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^dy²íر9Riemann-Roch½n/§^Riemann-Roch½ny²;
Riemann¡±i\Km¥"
X;Riemann¡,XþØf´/X
D =∑
p∈Xnp[p] np ∈ ZA0
/ª5|Ü"ÙN¤gdAbel+§PDiv(X)"éuD =∑
p∈X np[p]§
½ÂdegD =∑
p∈X np"¡DkØf§PD ≥ 0,enp ≥ 0,∀p"PD > 0,eD ≥0D 6= 0"
£Γ(X,MX)"éuf ∈ Γ(X,MX)×,½Â
(f) =∑
p∈Xvp(f)[p], Ù¥vp(f) =
m f3p:km":
−m f3p:km4:
0 f3p:QÃ":Ã4:
K(f) ∈ Div(X)"¡/X(f)ØfXþÌØf"
öS3. ^3ê½ny²deg(f) = 0"
5¿Γ(X,MX)× → Div(X), f 7→ (f)+Ó"
aq/§éω ∈ Γ(X,M(1)X )− 0,½Â
(ω) =∑
p∈Xvp(ω)[p], Ù¥vp(ω) =
m ω3p:km":
−m ω3p:km4:
0 ω3p:QÃ":Ã4:
11
∀D1, D2 ∈ Div(X),¡D1, D25d§eD1 − D2 = (f)é,f ∈ Γ(X,MX)×,P
D1 ∼ D2"
∀ω1, ω2 ∈ Γ(X,M(1)X )−0,´∃f ∈ Γ(X,MX)× s.t.ω1 = f ·ω2,(ω1) = (f) + (ω2)§
=(ω1) ∼ (ω2)"¡/X(ω1)ØfXþ;KØf"·w§?¿ü;KØf5
d"
∀D ∈ Div(X),-
Γ(X,OX(D)) = f ∈ Γ(X,MX)×|(f) +D ≥ 0⋃0
Γ(X,ΩX(D)) = ω ∈ Γ(X,M(1)X )− 0|(ω) +D ≥ 0
⋃0
KÙþC−5m"-r(D) = dimC Γ(X,OX(D)), i(D) = dimC Γ(X,ΩX(−D))"
y3§·ÿØ´Är(D) <∞, i(D) <∞"D1 ∼ D2 = D1 + (f), f ∈ Γ(X,MX)×,KkC−5Óµ
Γ(X,OX(D2))→ Γ(X,OX(D1)), g 7→ f · g
Γ(X,ΩX(D2))→ Γ(X,ΩX(D1)), ω 7→ f · ω
r(D), i(D)þ6uD5da"
qK = (ω0)X;KØf§w0 ∈ Γ(X,M(1)X ), ∀D ∈ Div(X),kC- 5Ó:
Γ(X,OX(D))→ Γ(X,ΩX(D −K))
f 7→ f · ω0
r(D) = i(K −D) (1)
·kQãéD ≥ 0Riemann-Roch½nXeµ
½n1. X;Riemann¡,ºg"∀D ∈ Div(X), D ≥ 0§¤áµr(D), i(D) <
∞
r(D)− i(D) = 1− g + degD (∗)
·ò3©1nÜ©¥y²T½n"
ù½n`?3u(∗)à£ã´Xþ÷v½4:!":5æX¼ê!æX©/ªê§Ï XE(k'§ (∗)màXÿÀ¡ÿÀk'"ÏdoÑ`§Riemann-Roch½n£ã/ÿÀXÛAÛ0£,§ùp
12
/AÛ0EAÛ½êAÛ¤",áuùÆͶ½n´Gauss-Bonnet½n§
Ù£ã´ÿÀXÛÇ"
·±tl½n1íØ"
íØ1. X;Riemann¡,ºg,KdimC Γ(X,ΩX) = g"
yµD = 0,Kr(D) = dimC Γ(X,OX) = 1, degD = 0"d(∗)§i(D) = g,i(D) =
dimC Γ(X,ΩX)
52. 3©1nÜ©¥§·¬ÑíØ1Õáu½n1y²§ íØ1ò¬
^u½n1y²¥"
íØ2. X;Riemann¡§ºg,∀p ∈ X ∃~êXþæX¼êf ,Ù3X −pþX§3p?4:êuug + 1"
yµD = (g + 1)[p],Kr(D) ≥ 1− g + degD = 2§Γ(X,OX(D)) ' C
íØ3. X;Riemann¡,º0§KX ∼= P1"
yµfXíØ2§Kf3X − pþX§3p?Tk4:"òfwXNf : X → P1"
∀q ∈ P1,½Âτ(q) =∑
x∈f−1(q) vx(f − f(x))"
Ù¥§vx(f − f(x))¼êf(z)− f(x)3z = x?":ê"
´§τP1þÛÜ~¼ê§~ê§τ(∞) = 1"
fü§l Ó"
íØ4. X;Riemann¡§ºg§KXþ;KØf§KdegK = 2g − 2"
yµeg = 0§díØ3§X ' P1"·3~1¥®w§P1þkæX©/ªdz§
Ù3C = P1 − ∞þæX§3∞?k":"ÏK ∼ dz§deg5dak'§
degK = −2"
eg > 0§w ∈ Γ(X,ΩX)− 0"-D = (w)§KD ≥ 0"d'Xª(1)§
r(D) = i(D −D) = i(0) = dimC Γ(X,ΩX) = g§
i(D) = r(D −D) = r(0) = dimC Γ(X,OX) = 1"
\(∗)§g − 1 = 1− g + degD§degK = degD = 2g − 2"
dd§·±Qã¿y²/ªRiemann-Roch½nXeµ
½n2. (Riemann-Roch) X;Riemann¡§ºg"∀D ∈ Div(X)§¤á
r(D), i(D) <∞r(D)− i(D) = 1− g + degD" (∗)
13
yµ£â½n1¤
1Úµe½néD¤á§KéK −D¤á§Ù¥KXþ?;KØf"yµ-D′ = K −D"d'Xª(1)§·k
r(D′) = i(K −D′) = i(D) <∞§i(D′) = r(K −D′) = r(D) <∞"
r(D′)− i(D′) = i(D)− r(D) = −1 + g − degD"
degD′ = degK − degDdíØ4====== 2g − 2− degD§degD = 2g − 2− degD′"
l r(D′)− i(D′) = −1 + g − (2g − 2− degD′) = 1− g + degD′§=(∗)éD′¤á"1Úµ∀D ∈ Div(X)§er(D) > 0½i(D) > 0§K½néD¤á"
yµer(D) > 0§f ∈ Γ(X,OX(D))§KD′ := (f) +D ≥ 0"d½n1§
r(D′), i(D′) <∞r(D′)− i(D′) = 1− g + degD′"
r(D) = r(D′)§i(D) = i(D′)§degD = degD′§½néD¤á"
ei(D) > 0§i(D) = r(K −D) > 0§KX;KØf"
dþ½néK −D¤á"l d1Ú§½néD¤á"1nÚµ∀D ∈ Div(X)§½néD¤á"
yµPD = D1 −D2§Ù¥D1 ≥ 0§D2 ≥ 0Ñy3D1¥:ØÑy3D2¥"
d½n1§
r(D1) ≥ 1− g + degD1 = 1− g + degD + degD2§ (2)
´Γ(X,OX(D)) ⊂ Γ(X,OX(D1))§Γ(X,OX(D))´Γ(X,OX(D1))þdegD2
5§)m§
r(D) ≥ r(D1)− degD2" (3)
edegD ≥ g§dت(2)§kr(D1) ≥ 1 + degD2"
l d(3)§r(D) ≥ 1"d1Ú§½néD¤á"
edegD ≤ g − 2§Kdeg(K −D) = 2g − 2− deg(D) ≥ g"l dþ½néK −D¤á§l d1Ú§éD¤á"Ïdeu/´degD = g − 1§r(D) = i(D) = 0"
ù·k0− 0 = 1− g + g − 1§½n¤á"
½n2A^§·y²Xe½n§Ù531.2¥®`²"
½n3. (;Riemann¡Ki\) X;Riemann¡§K∃N ∈ N9X
i\F : X → PN"=§FüëYN§÷vµ∀0 ≤ i ≤ N§-Di =
[z0 :
· · · : zN ] ∈ PN |zi 6= 0§ϕi : Di → CN , [z0 : · · · : zN ] 7→ ( z0zi , · · · ,
zizi, · · · , zNzi )§
14
K∃F−1(Di)þX¼êfi1, · · · , fiN§s.t. Fi := ϕi F∣∣F−1(Di)
: F−1(Di) → CN÷
vFi(z) = (fi1(z), · · · , fiN (z)),∀z ∈ F−1(Di)"∀x ∈ X,∃0 ≤ i ≤ N, 1 ≤ j ≤ N§s.t.
x ∈ F−1(Di)f ′ij(x) 6= 0"
Ún1. X´;Riemann¡§ºg"∀D ∈ Div(X)§edegD > 2g− 2§Ki(D) =
0"
y²µKX;KØf§-D′ = K − D§degD′ < 0"i(D) = r(K − D) =
dimC Γ(X,OX(D′))"
ÏdIy²Γ(X,OX(D′)) = 0"eØ,§@o∀f ∈ Γ(X,OX(D′))−0§(f)+D′ ≥0§deg(f) +D′ = degD′ ≥ 0§gñ"
·Ú\Vgµ
½Â4. X´;Riemann¡§D =∑
p∈Xnp[p] ∈ Div(X)"¡OX(D)N)¤§
e∀p ∈ X,∃f ∈ Γ(X,OX(D))− 0 s.t. vp(f) = −np"
Ún2. X´;Riemann¡§ºg"∀D ∈ Div(X)§edegD ≥ 2g§KOX(D)
N)¤"
y²µPD =∑
p∈Xnp[p]"
∀p ∈ X§-D′ = D− [p]§KdegD > degD′ > 2g− 2"dÚn1§i(D) = i(D′) = 0"
l r(D) = 1− g + degD§r(D′) = 1− g + degD′§r(D) > r(D′)"
Γ(X,OX(D′)) ⊂ Γ(X,OX(D))§Γ(X,OX(D′)) $ Γ(X,OX(D))"
f ∈ Γ(X,OX(D′))− Γ(X,OX(D))§Kvp(f) = −np"e¡·5y²½n3"
½n3y²µ£â[2]§Chapter 17¤"
Xºg§D ∈ Div(X) s.t. degD ≥ 2g + 1"-f0, · · · , fNΓ(X,OX(D))
|ħ·½ÂF : X → PN"
∀x0 ∈ X§zx0NCÛÜI¦z(x0) = 0§K3z = 0NCfi(z) = znigi(z)§é
,ni ∈ Z 9,3z = 0?X§ØX¼êgi¤á§∀0 ≤ i ≤ N"-n = min
0≤i≤Nni§½ÂF (x0) = [zn0−ng0(z) : · · · : znN−ngN (z)]
∣∣z=0
(½00 = 1)"
w,F½ÂØ6uÛÜIzÀ"·y²F´Xi\"dIy
²µ
1)F´ü"
2)∀x0 ∈ X§x0?ÛÜIz s.t. z(x0) = 0"
PÒÓþ§-hi(z) = zni−ngi(z), 0 ≤ i ≤ N"
15
Øn0 = n§Kh0(0) 6= 0"éx0NCy§
F (y) = [h0(z) : · · · : hN (z)]∣∣z=z(y)
=
[1 :
h1(z)
h0(z): · · · : hN (z)
h0(z)
]∣∣∣∣z=z(y)
·y²§∃1 ≤ i ≤ N§s.t.d
dz
∣∣∣∣z=0
hi(z)
h0(z)6= 0"
1)yµx1, x2 ∈ X§x1 6= x2"PD =∑
p∈Xnp[p]"
PD′ = D − [x2]§KdegD′ ≥ 2g§OX(D′)N)¤"
∃f ∈ Γ(X,OX(D′))− 0 s.t. vx1(f) = −nx1 , vx2(f) ≥ −nx2 + 1"
Γ(X,OX(D′)) ⊂ Γ(X,OX(D))"f =
N∑
i=0
λifi§é,λi ∈ C"
-z1, z2©O´x1, x2NCÛÜI§z1(x1) = z2(x2) = 0"
ÏOX(D)´N)¤§
kµ := min0≤i≤N
vxµ(fi) = −nxµ , µ = 1, 2"
-fi = zkµµ gµ,i, f = z
kµµ gµ3xµNC¤á"KF (xµ) = [gµ,0(xµ) : · · · : gµ,N (xµ)], µ =
1, 2§N∑
i=0
λigµ,i(xµ) = gµ(xµ)"dvx1(f) = −nx1 = k1, vx2(f) ≥ −nx2 + 1 = k2 + 1§
g1(x1) 6= 0§g2(x2) = 0"ÏdF (x1) 6= F (x2)"
2)yµPD =∑
p∈Xnp[p], n = min
0≤i≤Nvx0(fi)"3x0NCfi = znhi, ∀0 ≤ i ≤ N"
duOX(D)N)¤§n = −nx0"-D′ = D − [x0],KdegD′ ≥ 2g"OX(D′)N)¤"∃f ∈ Γ(X,OX(D′)) − 0
s.t. vx0(f) = 1− nx0 = 1 + n"-f(z) = znh(z), h(z)3z = 0NCX"Kh3z = 0Tk
":"
dΓ(X,OX(D′)) ⊂ Γ(X,OX(D))§f =N∑
i=0
λifi é,λi ∈ C"
znh =N∑
i=0
λiznhi, h =
N∑
i=0
λihi"
h
h0− λ0 =
N∑
i=1
λi(hih0
)§3z = 0NC¤á"
duh3z = 0Tk":
0 6= d
dz
∣∣∣∣z=0
(h
h0) =
N∑
i=1
λid
dz
∣∣∣∣z=0
(hih0
)"
∃1 ≤ i ≤ N s.t. ddz
∣∣z=0
( hih0 ) 6= 0"
16
3 Riemann-Roch½ny²
3.1 ĽÂÚPÒ
XRiemann¡§K1w6/§ÙþûнÂÿ¼êVg"
9Γ(X,FX)½Â3Xþÿ¼êN§KÙΓ(X,EX)−ê"-
Γ(X,F(1)X ) := Γ(X,FX)⊗Γ(X,EX) Γ(X,E
(1)X )
Ù¥¡Xþÿ1-/ª"∀ω ∈ Γ(X,F(1)X )§3ÛÜIz = x + iyeω = fdx +
gdyé,ÿ¼êf, g"
½Âf∗ : Γ(X,F(1)X )→ Γ(X,F
(1)X )Xeµ3ÛÜIz = x+ iye§∗(fdz + gdz) =
−ifdz + igdz§∗(fdx+ gdy) = −gdx+ fdy"
¡ω ∈ Γ(X,E(1)X )NÚ§edω = 0, d ∗ ω = 0"XþNÚ/ªNPH(X)"
¡f ∈ Γ(X,EX)NÚ§edf ∈ H(X)"Kf ∈ Γ(X,EX) NÚ⇔3?ÛÜIz =
x+ iye§∂2f
∂x2+∂2f
∂y2= 4
∂
∂z
∂
∂zf = 0
½ÂL2(X) = ωXþÿ1-/ª|∫X ω ∧ ∗ω < ∞"L2(X)3SÈ< ω1, ω2 >:=
∫X ω1 ∧ ∗ω2eHilbertm§ù´î¼m¥²È¼êN¤Hilbertmù¯
¢üíØ"
αX¥1w4§dPoincareéó§3Xþ1¢;|4/ªηᦵé?
¿Xþ14/ªω §¤á∫α ω =
∫X ω ∧ ηα"T^(½ηαde RhamþÓNa"
α, βX¥1w4§½Âêα·β =∫X ηα∧ηβ"Ù÷vµα·β ∈ Z, α·β = −β·
α§α · β=6uα, βÓNa§dd½ÂH1(X,Z)¥mê(1w4
H1(X,Z)ħ2U©Æ[α]·([β]+[γ]) = α·β+α·γ§([β]+[γ])·[α] = β ·α+γ ·α½Â)
X;Riemann¡,Kºf4¡Ùkõ>/L«µ
aa−1(º0)
½
∏gj=1 bjajb
−1j a−1
j (ºg ≥ 1)
òaj , bjwH1(X,Z)¥§P(N1, · · · ,N2g) = (a1, ..., ag, b1, ..., bg),Kݵ
(Nj ·Nk) =
(0 I
−I 0
)
2g×2g
(∗)
/§eH1(X,Z)Ä(Nj)÷v(∗)§K¡Ù;KÄ"
17
3.2 L2(X)©)§±9;Riemann Surface þ1aAbel©
ïÄRiemann¡XþAbel©(XæX©/ª)9æX¼ê§Bé
´NÚ©/ª"RiemannÄuDirichletn5/y²0Riemann¡þ,NÚ©
/ª35§¦ØyXeµ/ΩX¥«§éu3ΩþëY§Ωþ1w§3∂Ωþ
÷v½>^¼êu§ÄDirichletÈ©D(u) =∫
Ω du ∧ ∗du =< du,du >"´
u´D(u):⇔u3ΩþNÚ" Dok§NÚuo3"0¢Sþ§
D(u)ÛU´Ø²w(,D(u) ≥ 0)"WeierstrassÚHadamard©OE
,aq4¯K§Ù¥©OÑyD(u)ØÚD(u)o+∞y"ÖÍ´ò¦)m°L2,2|^±eóä5lL2£C∞"
½n4. (WeylÚn) ϕ´ü Dþ²Èÿ¼ê§KϕNÚ=éz
Dþ1w;|¼ê秤á∫D ϕ∆η = 0 §Ù¥∆η := d ∗ dη"
y²[1]"
|^StokesúªN´y²Xe·Kµ
·K4. : XRiemann¡,α ∈ L2(X)C1a§Kµ
1)α4⇔< α, ∗df >= 0,ézXþ;|1w¼êf"
2)αT⇔< α, ∗ω >= 0,ézXþ;|4/ªω"
PEc(X)Xþ1w;|¼êN"PEdEc(X)3L2(X)¥4§KE ⊥ ∗E§d·K4¥1)§eα ∈ L2(X)C1a§Kα ∈ H := H(X) ∩ L2(X)⇔ α ∈ (E
⊕ ∗E)⊥"
¯¢þ§|^WeylÚn±y²µ
½n5. H ⊂ L2(X)4L2(X)kXe©)µ
L2(X) = E⊕ ∗E⊕H
T½n´ïÄRiemann¡þAbel©ÚæX¼ê35Ä"dT½nÚ·
K4±Xeíص
íØ5. X;Riemann¡§KXþ4/ªmZ(1)(X)kXeÚ©)µ
Z(1)(X) = dE (X)⊕H(X)
yX;Riemann¡§ºg§½(Nj)H1(X,Z)|;KħKdde
Rham½n§íØ5§±9'XH(X) = Γ(X,ΩX)⊕
Γ(X,ΩX) Xe½nµ
½n6. dimCH(X) = 2g,3H(X)|Ä(ω1, ..., ω2g)¦
18
∫Njωk = δjk, 1 ≤ j, k ≤ 2g
dimC Γ(X,ΩX) = g,3Γ(X,ΩX)|Ä(ζ1, ..., ζg)¦
∫Njζk = δjk, 1 ≤ j ≤ g, 1 ≤ k ≤ g
dd·x;Riemann¡Xþ1aAbel©(X©/ª)"
3.3 üaAbel©35
·3Riemann¡XþEäk½4:5æX©/ª(1!n
aAbel©)"5¿euXþNÚ¼ê(Û:)Kdu + i ∗ du3uK:?K§
äkuAÛ:5"±eÚnÑEXþäkA½Û:5NÚ¼ê
"
Ún3. XRiemann¡,p0 ∈ X, (U, z)p0NCÛÜI§z(U) = D(ü )§
z(p0) = 0, 0 < a < 1"
hXþ¼ê§÷vµ
1)h3a2 ≤ |z| ≤ aþNÚ(=3Ù,mSNÚ)
2)∫|z|=a ∗dh = 0(d^¿©^´:3Iz = reiθe§ ∂
∂rh = 03|z| = aþ¤
á)
K3X\p0þ¼êu§¦u−h3|z| < aþNÚ§u3p ∈ U | |z(p)| > a2∪U cþ
NÚ"
TÚny²Äu½n4§·K4§½n5"
A^§©O-h(z) = z−n + zn
a2n, n ∈ N9h(z) = log
∣∣∣∣ z−z1z−z2 ·z−a2/z1z−a2/z2
∣∣∣∣, 0 < |z1|, |z2| <a2Xeµ
½n7. XRiemann¡, pj ∈ X, zjpjNCÛÜI§zj(pj) = 0, j = 1, 2K
1)éên ≥ 2§3XþæX©/ª§Ù±p14:§3z1e1zn1
dz1"
2)3XþæX©/ª§Ù±p1, p24:§3Iz1e1z1
dz1,3Iz2e
−1z2
dz2"
3.4 ª
X´;Riemann¡,ºg,(Nj) = (a1, ..., ag, b1, ..., bg)H1(X,Z);KÄ"
éXþ4/ªθ, θ,·í∫X θ ∧ ∗θ =< θ, θ >Oúª"ØÄ
∫X θ ∧ θ, 5
¿Ù=6uθ, θþÓNa"
19
(ωj)H(X)ħéóu(Nj),=∫Njωk = δjk§K
∫X θ ∧ θ =
∑µjµk
∫X ωi ∧
ωk§Ù¥∑
j µjωj ,∑
k µkωk ©Oθ, θû½Óde RhamþÓNa"-(αj) =
(ηb1 , ..., ηbg ,−ηa1 , ...,−ηag)§K·ωjαj 3H(X)¥ÝK§l ´µ
∫X ωj ∧ ωk = (Nj ·Nk ) =
(0 I
−I 0
)
µj =∫Njθ, µj =
∫Njθ"
ddµ
½n8 (RiemannV5'X).∫X θ ∧ θ =
∑gj=1
∫ajθ∫bjθ −
∫bjθ∫ajθ
AO/§eθ, θ ∈ Γ(X,ΩX),5¿θ ∧ θ = 0§µ
0 =∑g
j=1
∫ajθ∫bjθ −
∫bjθ∫ajθ
Ïdeθzaj±Ï∫ajθ = 0,θajéó§=
∫akθ = δjk,K
∫bjθ = 0,l θXþ
TX/ª§θ = 0"C-5NΓ(X,ΩX) −→ Cg, θ 7−→ (∫a1θ, ...,
∫agθ)ü§l
Ó(Ïü>þg-)"½n7¥1)Üå5ħҵ
í Ø6. X ; Riemann ¡§ p ∈ X, z 3p? Û Ü I,(Nj) =
(a1, ..., ag, b1, ..., bg)H1(X,Z);Kħn ≥ 2§K3τ(n)p ∈ Γ(X,M
(1)X )§Ù
3Ize dzzn ,3X\pþX§
∫ajτ
(n)p = 0, j = 1, ..., g.
d·ÐRiemann-Roch½ny²O"
3.5 Riemann-Roch½ny²
X;Riemann¡ºg§P
r(D) = dimC Γ(X,OX(D)), i(D) = dimC Γ(X,ΩX(−D))
Riemann-Roch½nQãXe:
½n9. r(D), i(D) <∞r(D)− i(D) = 1− g + degD (∗)
·3¥©1Ü©¥wIy²D ≥ 0kØf/"
y²µeD = 0,Kr(D) = dimC Γ(X,OX(D)) = 1§i(D) = dimC Γ(X,ΩX(−D)) = g§
(*)¤á"
D > 0, D = Σrj=1nj [pj ]"½(Nj) = (ai, · · · , ag, b1, · · · , bg)H1(X,Z);K
ħòai, biþwXþ4§ÙþØLpj , ∀1 ≤ j ≤ r"-D′
= Σrj=1(nj +
20
1)[pj ]§KdΓ(X,OX(D)) ⊂ Γ(X,ΩX(D′)) Ù¥Γ(X,ΩX(D
′)) := ω ∈ Γ(X,ΩX(D
′)) |
∫aiω = 0, ∀1 ≤ i ≤ g,ω3Xþ?¿:?3ê0"é1 ≤ j ≤ r§½pj?ÛÜ
Izj s.t.zj(pj) = 0"5¿τ(2)p1 , · · · , τ (n1+1)
p1 , τ(2)p2 , · · · , τ (n2+1)
p2 , · · · , τ (nr+1)pr ∈ Γ(X,ΩX(D
′))§
5Ã'"dimC Γ(X,ΩX(D′)) ≥ Σr
j=1nj = degD"
,¡§∀ω ∈ Γ(X,M(1)X ),(ω) + D
′ ≥ 0ω3Xþz:?3êþ0§Kω3
Izje
ω = (−2∑
l=−nj−1
λljzlj)dzj +/XÜ©0
eωai−±Ïþ0§díØ6/XÜ©0þ0"5¿ω3X − pj1≤j≤rþX§ÏdÙd3zpj ?ÌÜû½§l wÑdimC Γ(X,ΩX(D
′)) ≤ degD§
dimC Γ(X,ΩX(D′)) = degD"
PNΦ = d : Γ(X,OX(D)) → Γ(X,ΩX(D′))§w,ker Φ = C§ ∀ω ∈
Γ(X,ΩX(D′))§ω ∈ Im Φ ⇔ ωbi−±Ïþ0. l Im Φ´Γ(X,ΩX(D
′))þg5
§)m§
dimC Im Φ = dimC Γ(X,OX(D))− dimC ker Φ = r(D)− 1
≥ dimC Γ(X,ΩX(D′)) − g = degD − g
=r(D) ≥ 1− g + degD"
ùت¡Riemannت§@dRiemanny²"RiemannÆ)RochÑü
àØi(D)§Ïd·½n¡Riemann-Roch½n"
·e¡5y²þãتüàØi(D)§d·ÑdimC Im Φ"5
¿Γ(X,ΩX(D′)) = ⊕rj=1 ⊕
nj+1k=2 C · τ (k)
pj §Ïd∫biτ
(k)pj "Ø(ai, bi)5gX
õ>/L«Πgi=1biaib
−1i a−1
i (5¿g = 0§Φ÷§r(D) = 1 + degD§ i(D) = 0§
½n¤á"¤±ùp·bg ≥ 1)
KX′
= X − ai, · · · , ag, b1, · · · , bgüëÏ"ωi ∈ Γ(X,ΩX)aiéó§=∫ai′ ωi =
δi′ i,∫bi′ ωi = 0§Kωi3X
′þT§ωi = dfi"D±pj%X¥¿©m
§K
0 =
∫
X′−Dωi ∧ τ (k)
pj =
∫
X′−Dd(fiτ
(k)pj )
=
∫
∂(X′−D)fiτ
(k)pj = [
g∑
l=1
∫
al
[fi(x)− fi(x′)]τ (k)
pj +
∫
bl
[fi(y)− fi(y′)]τ (k)
pj ]−∫
∂Dfiτ
(k)pj (?)
21
Ù¥éx ∈ al, x′a−1
l þxéA:(al, a−1l õ>/þü^>§òÙÊÜXþ
al)§yy′'Xaq"K´
fi(x)− fi(x′) =
∫ x′
xωi =
∫
bl
ωi, ∀x ∈ al
fi(y)− fi(y′) =
∫ y′
yωi = −
∫
al
ωi, ∀y ∈ bl
d(?)
0 =
g∑
l=1
[
∫
al
τ (k)pj
∫
bl
ωi −∫
al
ωi
∫
bl
τ (k)pj ]−
∫
∂Dfiτ
(k)pj = −
∫
bi
τ (k)pj −
∫
∂Dfiτ
(k)pj
ωi3Izje
ωi = (+∞∑
m=0
µ(j)imzmj )dzj
Kd3ê½n ∫
∂Dfiτ
(k)pj = 2π
√−1
µ(j)i,k−2
k − 1
∫
bi
τ (k)pj = −
∫
∂Dfiτ
(k)pj = 2π
√−1
µ(j)i,k−2
1− k½Â
T :Γ(X,ΩX(D′)) → Cg
ω 7→ (
∫
b1
ω, · · · ,∫
bg
ω)
KT3Γ(X,ΩX(D′))Ä
1− k2π√−1
τ (k)pj | 1 ≤ j ≤ r, 2 ≤ k ≤ nj + 1
e§Ý(µ(j)i,k−2)i,(j,k)§kerT = Im Φ"
é1 ≤ i ≤ g§½ÂT′(ωi) = (µ
(j)i,k−2)1≤j≤r, 2≤k≤nj+1 ∈ CdegD"*нÂspanωi |
1 ≤ i ≤ g = Γ(X,ΩX)þ"´dimC ImT = dimC ImT′ kerT
′= Γ(X,ΩX(−D)) §Ï
d
dimC Im Φ = dimC kerT = dimC Γ(X,ΩX(D′)) − dimC ImT
= degD − dimC ImT′
= degD − (dimC Γ(X,ΩX)− dimC kerT′)
= degD − g + i(D).
qdimC Im Φ = r(D)− 1§r(D), i(D) <∞r(D)− i(D) = 1− g + degD"
22
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[1] Farkas and Kra, Riemann Surfaces, GTM 71
[2] Forster, Lectures on Riemann Surfaces, GTM 81
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23
A Short Introduction to Integrable Systems
Di Yang∗
Abstract
This monograph follows closely a ‘He-Si’ series talk, organized by undergraduate students
of DMS, THU. It is a short introduction to integrable systems (IS) and was supposed to give
to a wide class of audiences. The talk includes some fundamental and beautiful theories of IS,
such as the concept of Hamiltonian reduction, the inverse scattering method and bihamiltonian
recursions. Many materials come from Professor Youjin Zhang’s beautiful lectures “Theories
of IS” given at THU during the autumn semester of 2008-2009. Other related references are
listed inside this monograph. The hope is to let more people know “what is an integrable
system” and be interested in the beautiful notions of integrability and in this beautiful theory.
Calculus of several variables, some manifold concepts and theory of ordinary differen-
tial equations are pre-required. Second-year undergraduates of Tsinghua should be able to
understand parts of the talk.
Contents
1 Finite dimensional integrable systems 251.1 Hamiltonian systems and first integrals . . . . . . . . . . . . . . . . . . . . . . . 25
1.2 Symmetries and conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.3 Hamiltonian reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.4 Arnold-Liouville integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2 Integrable partial differential equations 322.1 The KdV equation : History, one-soliton solution and conservation laws . . . . . 32
2.2 Inverse scattering transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3 Bihamiltonian structures and the KdV hierarchy . . . . . . . . . . . . . . . . . . 42
2.4 Virasoro symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.5 Witten’s conjecture and the Virasoro constraints . . . . . . . . . . . . . . . . . . 47∗PhD student (2008), supervisor: Professor Youjin Zhang.
24
1 Finite dimensional integrable systems
1.1 Hamiltonian systems and first integrals
V. I. Arnold, “Mathematical methods in classical mechanics”.
R. Abraham and J. Marsden, “Foundations of mechanics”.
• Newtonian mechanics = Dynamical systems on an n-dimensional Riemannian manifold
(N, g) satisfying Newton’s equations. (N, g) is called the the configuration space, where g is
a Riemannian metric. Take local coordinates: (q1, ..., qn); Newton’s equations of motion are
written as
qi = −gi j ∂V∂q j (i = 1, ..., n) (Newton’s equations),
where V = V(q1, ..., qn) is the potential of the considered system. Note that here we assume
potential V does not contain t explicitly. Below, except otherwise stated, we always consider
the autonomous case.
• Lagrangian mechanics = Dynamical systems on the tangent bundle T N of the con-
figuration space N satisfying The Euler-Lagrangian equations. Local coordinates:
(q1, · · · , qn, q1, · · · , qn); Lagrangian: L = L(q1, · · · , qn, q1, · · · , qn) = T − V , where
T =∑n
i, j=1 gi jqiq j is the kinetic energy. The equations of motion are given by
ddt
(∂L∂qi
)=∂L∂qi (i = 1, ..., n) (The Euler-Lagrange equations).
• Hamiltonian mechanics = Dynamical systems on the cotangent bundle T ∗N of the configu-
ration space N satisfying Hamilton’s equations. Local coordinates: (q1, · · · , qn, p1, · · · , pn).
Let H = H(p1, · · · , pn, q1, · · · , qn) denote the Hamiltonian.
pi = −∂H∂qi
qi =∂H∂pi
(i = 1, ..., n) (Hamilton’s equations).
We consider in this talk the Hamiltonian formulation. Let (M2n, ω) be a symplectic manifold,
i.e.
M : 2n-dimensional smooth manifold,
ω : non-degenerate closed 2-form.
25
On M there be a Darboux coordinate chart (p1, ..., pn, q1, ..., qn), such that
ω = dp1 ∧ dq1 + · · · + dpn ∧ dqn.
Let H : M2n → R be any smooth function; locally, H = H(p1, ..., pn, q1, ..., qn).
Definition 1.1. The Hamiltonian system on (M, ω) associated with H is defined by
dpi
dt= −∂H
∂qidqi
dt=∂H∂pi
(i = 1, ..., n), (1.1)
where H is called the Hamiltonian. Equations (1.1) are called Hamilton’s equations. We denote
the system by the triple (M2n, ω,H).
Definition 1.2. The Hamiltonian vector field associated with H is defined by vH =] dH, i.e.
dH(X) = ω(X, vH), ∀X ∈ X(M). Locally, vH :=n∑
i=1
(−∂H∂qi
∂
∂pi+∂H∂pi
∂
∂qi
). The phase flow corre-
sponding to vH is called a Hamiltonian flow.
Remark 1.3. By definition, dH = ivHω. In usual coordinates (x1, ..., x2n),
ω =12ωi jdxi ∧ dx j,
H = H(x1, ..., x2n),
vH = ωi j ∂H∂xi
∂
∂x j ,
where (ωi j) = (ωi j)−1. And Hamilton’s equations are given by
dxi
dt= ωi j ∂H
∂x j , i = 1, · · · , 2n (1.2)
Definition 1.4. Let F ∈ C∞(M2n,R). F is called a conserved quantity or a first integral of (1.1), ifddt
F(pi(t), qi(t)) = 0,∀t ∈ (−ε, ε) (1.3)
for some ε > 0.
Geometrically, F is a first integral of (1.1) means along any ”stream line” of (1.1), F(pi(t), qi(t))
is a constant.
Note that
(1.3)⇔n∑
i=1
(∂F∂pi
pi +∂F∂qi
qi
)= 0
⇔n∑
i=1
(− ∂F∂pi
∂H∂qi
+∂F∂qi
∂H∂pi
)= 0.
This suggests the definition of a Poisson Bracket.
26
Definition 1.5. ∀ G1,G2 ∈ C∞(M,R), define the Poisson bracket by
G1,G2 =
n∑
i=1
(∂G1
∂pi
∂G2
∂qi− ∂G1
∂qi
∂G2
∂pi
)(1.4)
From the above definition, it is easy to see that F ∈ C∞(M) is a first integral of (1.1), i.f.f.
H, F = 0. (1.5)
• Properties of a Poisson bracket:
1. Anti-symmetric: F,G = −G, F, ∀ F,G ∈ C∞(M).
2. Bi-linearility.
3. The Jacobi identity: F1, F2, F3 + F2, F3, F1 + F3, F1, F2 = 0.
Remark 1.6. In usual coordinate (x1, · · · , x2n),
F,G = ωi j ∂F∂xi
∂G∂x j ,
Remark 1.7. The Jacobi identity⇔ dω = 0.
Remark 1.8. Three facts:
1. Anti-symmetric⇒ H,H = 0⇒ H is a first integral of (1.1).
2. If F,G are first integrals, so is F,G.
3. [vF , vG] = −vF,G, i.e. the map
C∞(M)→ X(M)
F 7→ vF
is a Lie algebra anti-homomorphism.
1.2 Symmetries and conservation laws
Let (M, ω) be a connected symplectic manifold and let G be a connected Lie group. Denote
by g its corresponding Lie algebra. Consider a G-action on (M, ω):
G × M −→ M
(g, x) 7−→ gx,
satisfying
27
1. ∀g ∈ G, g : M → M is a diffeomorphism;
2. ∀g ∈ G, g keep the symplectic structure, i.e. g∗ω = ω.
Such a G-action is called a symplectomorphism.
Definition 1.9. A G-action on (M, ω) is called a Hamiltonian action is if it is a symplectomorphism
satisfying the following two more assumptions:
A1. ∀X ∈ g, we require the flows on M generated by X are Hamiltonian flows, namely, the vector
field X on M induced by X is a Hamiltonian vector field.
To be precise, the one-parameter group gt = etX acts on M.
gt : M −→ M
x 7−→ gt(x) = etX(x).
X : the corresponding flow field on M, i.e., X ∈ X(M).
g∗tω = ω,∀t ∈ R⇒LXω = 0
⇒d iXω + iXdω = 0
⇒d iXω = 0
⇒locally, ∃FX , s.t. iXω = d FX .
As a result, assumption A1 means FX should exist globally.
Remark 1.10. If H1DR(M) = 0, then A1 is always true.
A2. Denote r = dim G = dim g. Take X1, ..., Xr basis of g.
Under assumption A1, we have X1, X2, ..., Xr Hamiltonian vector fields on M, and
FX1, FX2
, ..., FXr∈ C∞(M,R) the corresponding Hamiltonians.
∀X ∈ g, X =∑
ciXi,
X =∑
ciXi.
Define:FX =∑
ciFXi
F : g→ C∞(M,R)
X 7−→ FX .
28
Our assumption A2 is that F is a Lie algebraic homomorphism, i.e.
FX , FY = FX,Y. (1.6)
Remark: The topological obstruction here is H2(g,R).
Definition 1.11. Let the G-action on (M, ω) be a Hamiltonian action. Define
J : M → g∗
x 7−→ Jx
satisfying
Jx(X) = FX(x), ∀X ∈ g.Then J is called a momentum map.
Theorem 1.12. (Equivariant Theorem) The momentum map J : M → g∗ is equivariant w.r.t the
coadjoint action of G on g∗, namely, ∀g ∈ G, the following chart commutes.
Mg //
J
M
J
g∗Ad∗g // g∗
Proof
∀x ∈ M, to show Jgx = Ad∗g Jx
⇔∀X ∈ g, Jgx = Ad∗g Jx(X) = Jx(Adg−1 X)
⇔FX(gx) = F ˜Adg−1 X(x)
⇔FX(etY x) = F ˜Ade−tY X(x),∀Y ∈ g
⇔
ddt |t=0 FX(etY x) = d
dt |t=0 F ˜Ade−tY X(x)
FX(etY x) |t=0= F ˜Ade−tY X |t=0
⇔
dFX(Y)(x) = F ˜ad−Y X(x)
FX , FY = F−Y,X
Theorem 1.13. (Nother’s Theorem) Let G act on (M, ω) as a Hamiltonian action. And let H ∈C∞(M) be a G-invariant Hamiltonian. Then ∀ X ∈ g, Jx(X) is a first integral of (1.1).
29
ProofJx(X) = FX(x)
To show:
H, FX(x) = 0,∀x ∈ g.
H is G − invariant ⇔ H(gx) = H(x),∀x ∈ M,∀g ∈ G
⇔ H(etY x) = H(x),∀x ∈ M,∀Y ∈ g
⇔ ddt|t=0 H(etY x) = 0,∀x ∈ M,∀Y ∈ g
⇔ dH(Y) = 0,∀Y ∈ g⇔ H, FY = 0,∀Y ∈ g.
1.3 Hamiltonian reduction
Consider a connected Lie group G acting on (M, ω). Denote by g and lg∗ the Lie algebra and
dual Lie algebra of G respectively. Assume the G-action is a Hamiltonian action.
J : M // g∗ momentum map
∀ µ ∈ g∗, define J−1(µ) = x ∈ M|Jx = µ.Let µ be any regular value of J.
=⇒ J−1(µ) is a smooth submanifold of M.
Question: Is it possible to let G act on J−1(µ)?
x ∈ J−1(µ) ⇐⇒ Jx = µ
gx ∈ J−1(µ) ⇐⇒ Jgx = µ = Ad∗gµ
Write
Gµ = g ∈ G | Ad∗gµ = µ=⇒ Gµ acts on J−1(µ).
Mg //
J
M
J
g∗Ad∗g // g∗
30
Assume the Gµ-action is free and proper.
Definition 1.14. The orbit space of the Gµ-action on J−1(µ) is defined by
Mµ := J−1(µ)/Gµ.
Theorem 1.15. Mµ is a symplectic manifold with the symplectic structure ωµ, where ωµ is induced
by ω through
Π∗µωµ = i∗µω. (1.7)
J−1(µ)iµ //
Πµ
M
Mµ = J−1(µ)/Gµ
We call (Mµ, ωµ) the symplectic reduced space. Such a reduced space is usually referred to
as a “remaining symmetry”.
Theorem 1.16. Let H ∈ C∞(M) be a G-invariant Hamiltonian. Let ft denote the Hamiltonian
flow corresponding to the Hamiltonian vector field vH . Due to Nother’s theorem, ft leaves J−1(µ)
invariant and commutes with the G-action. Then we have
1. ft induces flow f µt on Mµ by Πµ ft iµ = f µt Πµ.
2. f µt on (Mµ, ωµ) is Hamiltonian with reduced Hamiltonian Hµ, specified by Hµ Πµ = H iµ.
J−1(µ) iM //
Πµ
M H // R
Mµ = J−1(µ)/Gµ
hµ
55
J−1(µ) iM //
Πµ
Mft // R
Mµ = J−1(µ)
/Gµ
f µt // Mµ
Proofs of the above two theorems could be found in the following references.
“momentum map and Hamiltonian reduction” by J. P. Ortega and T. S. Ratiu.
“Reduction of symplectic manifolds with symmetry” by J. Marsden and A. Weinstein.
31
1.4 Arnold-Liouville integrability
The notion of “reduction” suggests that sufficiently many symmetries or first integrals will
ensure “integrability”, for which we mean solvability or a full geometric picture. Such an idea
is natural and simple; It came form I. Newton, but it was Marsden and Weinstein who made it
mathematically and beautifully formulated.
There are also many other senses of integrability such as the existence of Lax pairs, lineariz-
ability, the Painleve property, ....
The word first integral is “good”, since solvability in our discussion relates to integrations by
quadratures.
Question: How many first integrals are enough?
In fact, we need n independent first integrals.
Theorem 1.17. (Arnold-Liouville criterion of integrability) Let (M2n, ω,H) be a Hamiltonian
system. On M, n independent first integrals I1, ..., In in involution ensures integrability. The inte-
grability in this criterion is usually called a complete integrability.
Furthermore, there exists “action-angle” coordinates (F1, ..., Fn, ϑ1, ..., ϑn) satisfying ω =∑
dFi ∧ dϑi and Hamilton’s equations (1.1) written in this coordinate chart are
dFidt = 0
dϑidt = ωi(F1, ..., Fn).
(1.8)
where Fi = Fi(I1, ..., In).
2 Integrable partial differential equations
2.1 The KdV equation : History, one-soliton solution and conservation laws
• In 1834, J. S. Russell observed a very interesting a phenomenon of a water wave in a narrow
channel. He wrote down the following sentences in his diary.
“I was observing the motion of a boat which was rapidly drawn along a narrow channel by a
pair of horses, when the boat suddenly stopped)not so the mass of water in the channel which it
had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then
suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary
elevation, a rounded, smooth and well-defined heap of water, which continued its course along the
channel apparently without change of form or diminution of speed. I followed it on horseback,
and overtook it still rolling on at a rate of some eight or nine miles an hour (14km/h), preserving
32
its original figure some thirty feet (9m) long and a foot to a foot and a half (300-450mm) in height.
Its height gradually diminished, and after a chase of one or two miles (2õ3 km) I lost it in the
windings of the channel. Such, in the month of August 1834, was my first chance interview with
that singular and beautiful phenomenon which I have called the Wave of Translation.”
——J. S. Russell
Russell’s observation puzzled many famous applied mathematicians at that time, such as G.
Stokes and G. Airy.
Lord Kelvin, J. Boussinesq and Lord Rayleigh et. al. once also tried to understand this
interesting phenomenon.
• In 1895, D. Korteweg and G. de Vries derived the following approximation of the water-wave
equation
ut + 6uux + uxxx = 0, (2.1)
which is now known as the KdV equation.
• Travelling wave solution (later called the 1-soliton solution) of the KdV equation:
Write u(x, t) = f (x − ct)
=⇒w′2 = 4w3 − g2w − g3, where w =12
f − c12.
=⇒w = ℘(φ + φ0) where φ = x − ct, ℘ is the Weierstrass ℘-function.
=⇒u = 2w +c6
= 2℘(φ + φ0) +c6
(periodic solution, “genus − 1 curves”)
Here, c, g2, g3 are certain constants such that g2 = 112 c2 + c1 and g3 = − c3
116 +cc16 − c2.
In particular, when c = k2, c1 = c2 = 0,
u(x, t) =
12 k2
cosh( k2 (x − k2t + φ0))
As x→ ±∞, this solution u goes to 0. This solution is called the 1-soliton solution; it exhibits the
behavior described by Russell. (The picture of 1−soliton solution.)
However, after Korteweg and de Vries’ work, for a long time, there were no effective methods
for solving nonlinear problems.
Two new challenges:• In 1965, Kruskal, Zabusky et. al. observed the phenomenon of “2-soliton” interaction by
numerical experiments. This gave a new challenge to mathematicians. Indeed, although there
were no effective methods for solving non-linear problems, it was possible to solve a particular
33
problem as long as it is ‘good’ enough. Now we know there do exists such nonlinear PDEs which
can be ‘solved’. This refers to integrability in the sense of solvability for PDEs.
• Around the same time, another problem challenged mathematicians. It refers to integrability
in the sense of first integrals in the PDE case. Let’s look at the KdV equation again
ut + 6uux + uxxx = 0 (t > 0, x ∈ R). (2.2)
Suppose u, ux, uxx, ...→ 0 as |x| → ∞.
We observe that (2.2) can be re-written in a divergence form:
ut + (3u2 + ux)x = 0.
This implies that∫ +∞−∞ u dx is a first integral of (2.2).
Before 1965, people found a list of first several conservation laws of (2.2):
Momentum conservation: ut + (3u2 + uxx)x = 0 (2.3)
Kinetic energy conservation: (12
u2)t + (2u3 + uux − 12
ux2)x = 0 (2.4)
A third order conservation law: (13
u3)t + (32
u4 + u2uxx − 2uux2 − 1
3uxuxx +
16
uxx2)x = 0 (2.5)
... (2.6)
Any PDE possesses infinitely many degrees of freedom, by analogy to the finite dimensional cases,
at least one needs at least infinitely many “first integrals” to ensure its “integrability”. After ob-
serving the first several conservation laws of KdV, some people believed there would be infinitely
many but others didn’t believe.
Positive evidences:
1. 1, 2, 3, 4, 5, 6, 7, (8, 9, 10, 11) : In math, if a problem with n is right for n = 1, ..., 11, it’s
better to conjecture it is right for all n.
2. Kruskal and Zabusky’s numerical experiments suggested certain ‘strong’ conservation in
KdV.
Negative evidences:
The higher order conservation laws involve too many terms without much regular patterns.
And it is even very hard to find the next one.
34
“Coincidence” or “truth”? In 1967, Gardner, Greene, Kruskal and Miura (GGKM) developed
a “non-linear Fourier transform” method, later called the “the inverse scattering method (IST)”,
which answered the following:
(a) The KdV equation is integrable in the sense of solvability, i.e. the initial value problem can
be “solved”.
(b) The KdV equation possesses indeed infinitely many conservation laws. (=⇒ integrability in
the sense of infinitely first integrals.)
(c) The KdV equation possesses multi-soliton solutions. (=⇒ answered K.Z.’s experiments.)
Remark 2.1. (a)(b)(c) suggest that KdV has a very strong integrability. In fact, later on Faddeev
et. al. found the completely integrability of KdV, namely, the action-angle coordinates were
discovered.
Tools of IST:
“classical analysis”: complex analysis + asymptotic analysis.
In 1930s, this “classical analysis” was called “modern analysis”, see Watson and Whittaker.
• In 1968, P. D. Lax made the IST method in a systematic way. Lax pairs:
Lφ = λφ
φt = Aφ.
2.2 Inverse scattering transform
Consider the initial value problem of KdV:
ut + 6uux + uxxx = 0 (t > 0, x ∈ R),
u|t=0 = u0(x), (x ∈ R).(2.7)
Here we suppose that u, ux, uxx, ..., u0, u0x, u0xx, ...→ 0 as |x| → 0, which is a physically reasonable
requirement.
To understand IST, let’s first simplify the problem and start with solving a linearized KdV:
ut + uxxx = 0 (t > 0, x ∈ R)
u|t=0 = u0(x) (x ∈ R).(2.8)
This simpler problem can be easily solved by the Fourier transform method, which gives
u(x, t) =1
2π
∫ +∞
−∞u0(k)eikx+ik3tdk,
35
where u0(k) is the Fourier transform of u0(x).
u(x, 0) = u0(x) Fourier transform // u(k, 0) = u0(k)
time evolution in the spectral world
u(x, t) u(k, t), t > 0Inverse Fourier transform
oo
(real world) (spectral world)
Now we go back to our problem and explain the G.G.K.M’s “non-linear Fourier transform”
method for solving the initial value problem of KdV.
Crucial observation: Equation (2.7) can be written as the compatibility of the following two
linear PDEs:
(∂x2 + u(x, t))φ(k, x, t) = −k2φ(k, x, t), (2.9a)
φt = (γ + ut)φ − (2u − 4k2)φx. (2.9b)
Here, γ ∈ C is a constant. The first equation can be considered as a spectral problem with k ∈ Cas the spectral parameter. One can check that
φxxt = φtxx ⇔ the KdV equation.
Such a pair of compatible equations (2.9a) and (2.9b) are called a Lax pair.
Note that the spectral problem (2.9a) is the famous linear Schrodinger equation. Physicists
at that time were already very familiar with its solutions. This finally leads to solving the KdV
equation.
Indeed, consider
(∂x2 + u)φ = −k2φ. (2.10)
The condition
u→ 0 as |x| → ∞implies
φ ∼ c · e±ikx, as |x| → ∞.Let ψ1, ψ2 be solutions of (2.10) specified by
ψ1 ∼ eikx (x→ +∞)
ψ2 ∼ e−ikx (x→ +∞)⇒ (ψ1, ψ2) : a fundamental solution vector of (2.10).
36
Let φ1, φ2 be solutions of (2.10) specified byφ1 ∼ eikx (x→ −∞)
φ2 ∼ e−ikx (x→ −∞)⇒ (φ1, φ2) : another fundamental solution vector of (2.10).
Symmetry of (2.10): The transformation k → −k leaves (2.10) invariant.
=⇒ ψ1(−k, x) = ψ2(k, x),
φ1(−k, x) = φ2(k, x).
LetMi = φie−ikx (i = 1, 2),
Ni = ψieikx (i = 1, 2).
=⇒
M1 ∼ 1
M2 ∼ e2ikx(x→ −∞) and
N1 ∼ e2ikx
N2 ∼ 1(x→ +∞).
As a result,
=⇒
M1xx − 2ikMx = −uM1 (k ∈ R),
M1 → 1 (x→ −∞).(2.11)
The Green-function method gives
M1(k, x) = 1 +1
2ik
∫ x
−∞dξ(1 − e2ik(x−ξ))u(ξ)M1(k, ξ), (k ∈ R) (2.12)
Similarly,
N2(k, x) = 1 − 12ik
∫ −∞
xdξ(1 − e2ik(x−ξ))u(ξ)N2(k, ξ). (k ∈ R) (2.13)
By using the analytic continuation and the theory of Volterra integral equations, we find
M1(k, x) analytic on Im(k) > 0 and M1 → 1 as k → ∞.
N2(k, x) analytic on Im(k) < 0 and N2 → 1 as k → ∞.
For k real,
φ1(k, x) = a(k)ψ2(k, x) + b(k)ψ1(k, x) =⇒M1(k, x)
a(k)= N2(k, x) +
b(k)a(k)
N2(−k, x)e2ikx (2.14)
Definition 2.2. ρ := ba is called the reflection coefficients; 1
a is called the transmission coefficients.
The reflection coefficients and the transmission coefficients satisfy
(ba
)2+
(1a
)2= 1.
37
Lemma 2.3. a(k) can be extended to Im(k) > 0 analytically.
Proofx→ +∞,N2(k, x)→ 1
=⇒ M1(k, x) ∼ a(k) + b(k)e2ikx (x→ +∞).
But
M1 ∼ 1 +1
2ik
∫ +∞
−∞(1 − e2ik(x−ξ))u(ξ)M1(k, ξ)dξ (k ∈ R),
=⇒
a(k) = 1 + 12ik
∫ +∞−∞ u(ξ)M1(k, ξ)dξ,
b(k) = − 12ik
∫ +∞∞ u(ξ)M1(k, ξ)dξ.
(2.15)
Since M1(k, x) is analytic on Im(k) > 0, we find finally
a(k) is analytic on Im(k) > 0 and a(k)→ 1 (as k → ∞).
More analysis =⇒ a(k) has finitely many zeros ik1, ..., ikN on the complex k−plane, where
ki ∈ R, ki > 0.
c.f. M. J. Ablowitz and P. A. Clarkson, “Solitons, Nonlinear Evolution Equations and Inverse
Scattering.”
u(x, 0) = u0(x)direct scattering // scattering data ρ(k, 0), a(k, 0) at time 0
time evolution in the spectral world
u(x, t) scattering data ρ(k, t), a(k, t) at time tinverse scattering
also called reconstructionoo
Now we need two more steps.
• Time dependence (i.e. time evolution in the spectral world, see the above picture):
By using φt = (γ + ux)φ + (4k2 − m)φ where we take γ = 4ik3, it’s easy to get the evolution
of the scattering data (ρ, a) with respect to t.
at = 0,
bt = 8ik3b.(2.16)
=⇒
a(k, t) = a(k, 0),
b(k, t) = b(k, 0)e8ik3t.(2.17)
38
• Inverse scattering: For simplicity, assume N = 0.
Consider (2.14) as a Riemann-Hilbert problem, one finds
N1(k, x) = e2ikx(1 +
12πi
∫ +∞
−∞ρ(ξ)N1(ξ, x)ξ + k + i0
dξ). (2.18)
Finally,
u(x) = −∂x(1π
∫ +∞
−∞ρ(ξ)N1(ξ, x)dξ
). (2.19)
Let’s now see some applications of IST.
Application 1. Infinitely many conservation laws.
a(k, t) ≡ a(k, 0)→ 1 (k → ∞).
Idea: The coefficients of the full asymptotic expansion of a(k) give a family of conserved quanti-
ties.
Note: a(k) = limx→+∞ M1(k, x). It remains to find the asymptotic for M1(k, x) as k goes to infinity.
M1 = φ1eikx → 1 (x→ −∞),
Mxx − 2ikMx = uM.
Let M1 = e f → 1 (x→ −∞), ⇒
f → 0 (x→ −∞) and
( fx)2 + fxx − 2ik fx = u.
Let ψ = fx ⇒ f =∫ x−∞ ψ(k, ξ) dξ and
ψ2 + ψx − 2ikψ = u (Riccati equation).
For large k,
ψ ∼ ψ0 +ψ1
2ik+
ψ2
(2ik)2 + ...
⇒ ψ0 = 0, ψ1 = −u, ψ2 = ux, ψ3 = u2 − uxx, ψ4 = 2(u2)x − uxxx, ...
⇒ M1 ∼ e∫ x−∞( ψ1
2ik +ψ2
(2ik)2+··· )dx
(k → ∞).
⇒ a(k) = limx→+∞M1(k, x) ∼ e
∫ +∞−∞ ( ψ1
2ik +ψ2
(2ik)2+...)dx
(k → ∞).
dadt
= 0 ⇒ ψ1, ψ2, ψ3, ψ4, ψ5, ... are conserved densities.
Note: ψ2m are trivial.
39
Application 2. N-soliton solutions.
N = 1 u = 2k21 sech2k1(x − 4k2
1 − x1);
N = 2 u =4(k2
2 − k21)((k2
2 − k21) + k2
1 cosh(2k2ξ2) + k + 22 cosh(2k1ξ1))
((k2 − k1) cosh(k1ξ1 + k2ξ2) + (k2 + k1) cosh(k2ξ2 − k1ξ1))2 (ξi = x − 4ki − xi);
... ...
(pictures of 2−soliton equations.)
Now we turn to studying another problem—the initial boundary value problem of the KdV
equation with the periodic boundary conditions, which is more challenging and is closely related
to algebraic geometry. ut + 6uux + uxxx = 0 (t > 0, x ∈ S 1),
ut=0 = u0(x) (x ∈ S 1).
Again, the IST method can be used to this problem. However, here we focus on some particular
solutions.
e.g. u = 2℘(φ + φ0) +c6, φ = x − ct.⇒ elliptic curve
∂2x log τ = u, τ : the tau-function.⇒ Riemann surface
higher genus solution of KdV d complex structure of R.S.
Indeed, let C be a compact Riemann surface of genus g ≥ 1.
Introduce the period matrix Πg×2g = (Ag×g, Bg×g). Recall
AtB = BtA Riemann bilinear relation√−1(AtB − BtA) positive definite, Hermitian.
A suitable choice of basis of the first homology group ai, bi ∈ H1(C), as well as a suitable
choice of basis of the holomorphic differentials ωi ∈ Ω1(C)
⇒ normalized period matrix: Πnor = (Ig,Z).
Here, Z satisfies
a) Z = ZT ,
b) ImZ is a positive definite matrix.
40
Theorem 2.4. (Torelli) C1,C2 : R. S ., C1 C2 (bi-holomorphic), i.f.f.
Π1nor = Π2nor modulo Sp(2g).
Remark 2.5. Recall the symplectic group Sp(2g) acts on the space Hg which is the space of n× n
complex matrices satisfying the conditions a) and b) above. e.g. g = 1, H1 is the upper half plane.
The appearance of this discrete group Sp(2g) is due to the freedom of choice of the basis ai, bi,the intersection numbers between which satisfy ai · b j = δi j.
Question:
# of parameters of Π(C)nor =
g(g+1)2 := n1.
# of parameters of complex structures of (C) =: n2 =
1 g=1,
3g − 3 g ≥ 2.g = 1 n1 = 1 n2 = 1
g = 2 n1 = 3 n2 = 3
g = 3 n1 = 6 n2 = 6
g = 4 n1 > n2
Πnor discriminates complex structures but it has more parameters. So the question is which
Πnor comes from periods of a R.S.?
This question is called the Riemann-Schottky problem.
Periodic solutions of the Kadomtsev-Petviashvili (KP) equation
3uyy = (4ut − 6uux + uxxx),
lead to the answer of this question.
Remark 2.6. The KdV equation is a reduction of the KP equation. So it also provides complex
structures as we’ve already stated in the beginning of this topic, but it only provides a non-generic
family of the complex structures.
The precise statement of the answer to the question is given by
Novikov conjecture. “Jacobians of smooth algebraic curves are precisely those indecomposable
principally polarized Abelian varieties whose theta-functions provide explicit solutions of the KP
equation.”
In history, the Novikov conjecture, proved by T. Shiota and I. M. Krichever, gave the first
effective answer to the Riemann-Schottky problem, where the whole KP hierarchy should be con-
sidered to achieve effectiveness.
cf. T. Shiota and I. M. Krichever. A series of nice papers.
41
2.3 Bihamiltonian structures and the KdV hierarchy
It was known long before that the KdV equation
ut = ∂x(−(3u2 + uxx)) (2.20)
possesses a Hamiltonian (Poisson) structure. For convenience, let t → −t and the KdV equation
becomes
ut = ∂x(3u2 + uxx) = ∂xδHδu(x)
(2.21)
where H =∫
(u3 − 12 u2
x)dx =∫
hdx is the Hamiltonian, h is called the Hamiltonian density and
δHδu(x)
:=∂h∂u− ∂x
∂h∂ux
+ ...
denotes the variational derivative.
Hamiltonian structures for PDEs (infinite dimensional dynamical systems) can be understood
in a very similar way as in the case of ODEs (finite dimensional dynamical systems). Let’s briefly
recall the definitions.
Hamiltonian structure. Let
F := space of “local functionals ” over S(R).
To be precise, F ∈ F is a functional means
F : S(R)→ Ru(x) 7−→ F[u].
And “local” means F can be written in the form F[u(x)] =∫ +∞−∞ f (u, ux, uxx, . . .)dx.
Definition 2.7. The Poisson bracket , is a map
F ×F → F
F , G 7−→ F,G
such that , satisfies a) bilinearility b) anti-symmetric and c) the Jacobi identity.
Definition 2.8. Let∫
:=∫ +∞−∞ and let P be a ‘local’ operator on S(R). Write
F[u] =
∫f (u, ux, uxx, . . .)dx, (2.22)
G[u] =
∫g(u, ux, uxx, . . .)dx, (2.23)
F[u],G[u] =
∫δFδu(x)
PδGδu(x)
dx. (2.24)
42
Operator P is called a Hamiltonian operator if , is a Poisson bracket.
E.g.
e.g.1 P = ∂x is a Hamiltonian operator.
e.g.2 P = ∂xu + u∂x is a Hamiltonian operator.
In the rest of this subsection, we go back to the KdV equation. Take P = ∂x, then the KdV
equation can be written as
ut = P(3u2 + uxx) = ∂xδHδu(x)
= u(x),H[u],
where
u(x),H[u(x)] =
∫δu(x)δu(y)
∂yδHδu(y)
dy =
∫δ(x − y)∂y(3u2 + uyy)dy = ∂x(3u2 + uxx).
Proposition 2.9. ∀F ∈ F , F[u] =∫
f (u, ux, uxx, ...)dx, the follow formula holds
dF[u]dt
= F[u],H[u].
Corollary 2.10. F is a first integral⇔ F,H = 0.
Remark 2.11. Physicists write the Poisson bracket with P = ∂x as u(x), u(y) = ∂xδ(x − y).
Remark 2.12. In general, P = Pi∂ix, Pi = Pi(u, ux, ...).
Bihamiltonian structure. It turns out that equation (2.21) possesses a second Hamiltonian struc-
ture
P2 = 2(u∂x + ∂xu) + ∂3x,
which is independent to but is compatible with P1 = ∂x. Namely,
ut = P1δH1
δu(x)= P2
δH0
δu(x), (2.25)
where H1 =∫
(u3 − 12 u2
x)dx, H0 =∫
12 u2dx. “Compatible” means ∀λ ∈ C, P1 + λP2 is a Hamilto-
nian operator.
• The Lenard-Magri scheme tells one can obtain a bihamiltonian recursion
P1δHn
δu(x)= P2
δHn−1
δu(x). (2.26)
43
The starting point is the Casimir of the first Hamiltonian structure , 1: H−1 = 12
∫u(x)dx.
Indeed, δH−1δu(x) = 1
2 , and ∀G ∈ F ,
G,H−11 = 0.
H−1 ⇒ H0, indeed,
P1δH0
δu(x)= P2
δH−1
δu(x)= (2u∂x + 2∂xu + ∂3
x)12
= ux ⇒ H0 =
∫12
u2dx.
H0 ⇒ H1, indeed,
P1δH1
δu(x)= p2
δH0
δu(x)= (2u∂x + 2∂xu + ∂3
x)u(x) = 6uux + uxxx ⇒ H1 =
∫(u3 − 1
2u2
x)dx.
H1 ⇒ H2, ....
The compatibility between the first and the second Hamiltonian structures ensures that such
a recursive procedure can be followed one by one.
Lemma 2.13. One remarkable fact of the above recursion is that H−1,H0,H1,H2,H3, . . . are all
conserved quantities of the KdV equation.
ProofH−1,H11 = 0, H0,H11 = 0.
For k ≥ 2 ,
Hk,H11 = −H1,Hk1 = −∫
δH1
δuP1δHk
δudx = −
∫δH1
δuP2δHk−1
δudx
= −H1,Hk−12 = Hk−1,H12 =
∫δHk−1
δuP2
δH1
δu(x)dx =
∫δHk−1
δuP1δH2
δudx
= Hk−1,H21 = −H2,Hk−22 = Hk−2,H31 = ... = 0.
• A more interesting fact: We can build a hierarchy of Hamiltonian PDEs, each member of which
is “integrable” in the sense of admitting infinitely many conservation laws and H−1,H0,H1,H2, ...
are the corresponding Hamiltonians.
∂u∂tn
= P1δHn
δu(x), n = −1, 0, 1, 2, .... (2.27)
This hierarchy of PDEs is the celebrated KdV hierarchy:
∂u∂t0
= ux∂u∂t1
= 6uux + uxxx∂u∂t2
= ...
...
44
• The most remarkable fact:
Hk,Hl1 = 0, ∀k, l ≥ −1. (2.28)
This is already proved in the Lemma 2.13 and it implies that each member of the KdV hierarchy is
integrable.
Definition 2.14. (Symmetry) For an evolutionary PDE
ut = f (u, ux, uxx, ...), (2.29)
the infinitesimal transformation
us = g(u, ux, uxx, ...) (2.30)
is called a symmetry of (2.29), if∂
∂s∂
∂tu =
∂
∂t∂
∂su.
Proposition 2.15. For the KdV hierarchy, each member is a symmetry of all others.
Proof [VHm ,VHn] = −VHm,Hn = 0.
References:
P. Oliver, “Applications of Lie Groups to Differential Equations”.
A. S. Fokas, “Symmetries and integrability”.
Prop. 2.15 means flows in (2.27) are mutually commutative. ⇒ The PDEs in the KdV
hierarchy can be solved together, i.e. u = u(t0, t1, t2, ...) (t0 = x).
2.4 Virasoro symmetries
It is well-known that the KdV equation
ut = 6uux + ε2uxxx
possesses the Galilean symmetry
us = 6tux + 1.
Here and below, a parameter ε is introduced for the KdV hierarchy by the re-scaling ti → εti and
x→ εx.
By applying the bihamiltonian recursion operator to the Galilean symmetry, we obtain anoth-
er family of infinitely many symmetries of the KdV equation
∂u∂sm
= Φ∂u
∂sm−1(m ≥ 0)
with ∂u∂s−1
= 1 + 6tux.
45
Definition 2.16.∂
∂s−1,∂
∂s0,∂
∂s1, ...
are called the additional symmetries of the KdV equation.
These additional symmetries are indeed common properties of the whole KdV hierarchy.
Namely, we start from the follow infinitesimal transformation:
∂u∂s−1
= 1 + t1∂u∂t0
+ t2∂u∂t1
+ ... (t0 = x).
It is easy to show this is a symmetry of the whole KdV hierarchy. We call it the Galilean symmetry.
Let∂u∂sm
= Φ( ∂u∂sm−1
), m ≥ 0.
⇒ ∂
∂tk
∂u∂sm
=∂
∂sm
∂u∂tk
.
The commutation relations between these additional symmetries are
[∂
∂si,∂
∂s j] = ( j − i)
∂
∂si+ j, i, j ≥ −1.
Since these commutation relations coincides with those of half of the Virasoro algebra, we
call ∂∂sm
(m ≥ −1) the Virasoro symmetries.
Now let’s introduce the tau-function of the KdV hierarchy. Let u = ∂2x log τ. The Virasoro
symmetries then act on τ by∂τ
∂sm= Lmτ (m ≥ −1),
where Lm (m ≥ −1) are some linear operators defined by
L−1 =∑
k≥1
tk∂
∂tk−1+
12ε2 t2
0, (2.31a)
L0 =∑
k≥0
(k +12
)tk∂
∂tk+
116, (2.31b)
L1 =∑
k≥0
(k +12
)(k +32
)tk∂
∂tk+1+ε2
8∂2
∂t20
, (2.31c)
...... (2.31d)
Again, they form half of a Virasoro algebra:
[Lm, Ln] = −(m − n)Lm+n (m, n ≥ −1).
46
2.5 Witten’s conjecture and the Virasoro constraints
Consider the following initial value problem of the KdV hierarchy. It specifies a particular
solution: ∂u∂tk
= ∂xδHuδu(x) (k ≥ 1)
u(x, 0, 0, ...) = x.
ti → εti (i ≥ 0)GGGGGGGGGGGGGGGGGGGA
u = ε2∂2x log τv
τv = exp( 1ε2 (
x3
6+ ...) + (
124
t1 + ...) + ε2(t32
1440+ ...) + ...
).
Here, the particular solution u is written in terms of τv, which is a tau−function of the KdV
hierarchy.
In 1991, E. Witten, when he was studying the intersection theory of the Deligne-Mumford
moduli space Mg,n and 2D quantum gravity, made the following conjecture, now also known as
the Witten-Kontsevich theorem.
Witten’s conjecture: (Witten-Kontsevich theorem) Let
Fg :=∑
n
1n!
tp1 . . . tpn
∫
Mg,n
Cp11 (L1) ∧ . . . ∧Cpn
1 (Ln).
Here, Mg,n is the moduli space of stable curves of genus g with n marked points and Ln is the
tautological line bundle over Mg,n through the i-th marked point. Write
log τM =
∞∑
g=0
ε2g−2Fg.
The following statements hold true.
¬ τM = τv.
τM satisfies the following Virasoro constraints.
L−1τM = ∂t0τM (string equation),
LmτM = Πm+1j=1 ( j + 1/2)∂m+1τM (m ≥ 0).
(2.32)
® The Virasoro constraints specify τM uniquely.
cf. B. Dubrovin and Y. Zhang, “Normal forms”.
47
Acknowledgements I’m grateful to Professor Youjin Zhang, Professor A. S. Fokas and Professor
B. Dubrovin for their advises. I would like to thank to Si-Qi Liu, who suggested to me several
topics for this talk; Many thanks also to Tao Su who invited me to give this talk. Finally, I owe a
lot to Chenglong Yu, Jinzhao Pan, Ming Zhang, Qirui Li and Jialun Li for their interests and for
typing the tex file; Without them, the present note would never be finished.
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48
What is algebraic topology about?
Marcel Bokstedt
June 28, 2012
Topology is a young subject. Algebra, number theory and geometry go back at least a
couple of thousand year. The basics of number theory and geometry were probably known
even before the beginning of written history. Fields like analysis, mathematical physics
and probability theory are over 300 years old. But topology was born only a little more
than a century ago.
In the last 50 years many papers have been published in the subject. Today it has
developed into one of the main branches of mathematics, with lots of connections to all
the other main branches. Each month there are posted about 20 new preprints classified
as algebraic topology at arxiv.org .
The soul of the subject is geometrical, but the strongest tools are algebraic. Since
this claim is rather abstract, I’ll give a few examples to explain the point. These examples
will deal with “topological spaces”. I’m not going to define formally what a topological
space is, but you can easily look up the definition. Examples are Euclidean spaces, and
subsets of Euclidean spaces. You can illustrate most of the ideas of algebraic topology by
considering such spaces, so let’s stick with those.
For instance, intervals on the line are topological spaces. There are open ones (like
(0,1)) and closed ones (like [0,1]). Graphs in the plane are also topological spaces. Here
is a graph:
The small dots at the end of edges are called vertices. They are really points (but in
the picture they are not so small, since they are enlarged to emphasize them).
To create some order in the menagerie of spaces, one introduces an equivalence rela-
tion. Two spaces X,Y are homeomorphic if there is a bijective map f : X → Y such that
both f and f−1 are continuous. For instance, the open interval (0, 1) is homeomorphic to
the line R. On the other hand, the following map from a half open interval to the circle
is not a homeomorphism. Even if f is bijective and continuous, the inverse f−1 is not
49
continuous.
f : [0, 1)→ S1 = (x, y) ∈ R2|x2 + y2 = 1f(t) = (cos(2πt), sin(2πt))
If there exists a homeomorphism between two spaces X and Y , we say that they are
homeomorphic, and they are considered to be indistinguishable from a topological point
of view. The trouble with this is that it’s not so easy to check whether two spaces are
homeomorphic. There are very many continuous maps between them, and it’s often not
so easy to figure out if any of them has a continuous inverse.
We need some ways of telling that two spaces are different. This strategy comes with
the nice bonus that the properties that can be used to distinguish spaces are usually also
interesting properties in themselves.
For instance, one easy way to distinguish spaces is to see how many pieces they have.
For instance, the graph pictured above is all in one piece. Any two point of it can be
connected by a curve on the graph. We say that it is connected. It isn’t homeomorphic
to any graph which consists of two or more distinct connected pieces.
Exercise 1. Prove that there is no homeomorphism between an open and a closed interval.
There’s a hint at the end of the article.
Example 2. A plane with a finite number of points removed.
It is not so hard to see that this space only depends on the number of points. That
is, if P = p1, . . . , pn and Q = q1, . . . , qn are two subsets of n distinct points in the
50
plane R2, then the spaces we get by removing P , that is R2 \P , and by removing Q, that
is R2 \Q, are homeomorphic.
To see that the spaces are not the same for different n, we introduce a new concept.
When we investigate a space like space X = Xn = R2 \ p1, . . . , pn, we immediately
see that there are different types of closed curves in X. A curve in X is a curve in the
plane that keep away from the points pi. Think of the curves as rubber bands. Some of
these rubber bands can be contracted to a point, but some can’t because they will get
stuck at the points which are removed from the plane.
Figure 1: The curve at the bottom can be contracted to a point. The curve at the top
can’t be contracted, or can it??
We say that two curves are homotopic if they can be deformed to each other.
It turns out that it’s clever to fix a point x ∈ X. The homotopy classes of curves
starting and ending at x form a group, the fundamental group π1(X) of X. The composition
of two such paths consists of firsts walking along one curve and then along the other. To
know the fundamental group is usually basic to an understanding of the geometry of X.
In our case, we can prove, using algebra,
Theorem 3. If n 6= m, then the group π1(Xm) is not isomorphic to the group π1(Xn).
This directly shows
Corollary 4. If m 6= n, then the space Xm is not homeomorphic to the space Xn
51
Proof. From the definition of the fundamental group, it’s totally obvious that homeomor-
phic spaces have isomorphic fundamental groups.
This is a very typical example of how the “algebra” gets into the “algebraic topology.”
We want to prove that Xm is not homeomorphic to Xn. It’s not realistic to consider all
continuous maps from Xm to Xn, since there are too many of them. So instead we look at
shadows of the spaces. The spaces themselves live in the world of geometry. The shadow
of X is π1(X), which lives in the realm of algebra. To show that two spaces are different,
it’s enough to show that their shadows are different. The punchline is that it’s usually
far easier to get a hold of the algebraic homomorphisms between the fundamental groups
than to get a good overview of all continuous maps.
Example 5. The Euclidean spaces Rn.
These spaces are all connected. Just one innocent question about them: Could it
be that R2 is homeomorphic to R3? Obviously, this is false. Our intuition cries “NO”.
But it turns out that it’s not so easy to prove that. The problem here is not that our
intuition is wrong, it’s rather about whether the technical definition of “homeomorphisms”
is reasonable. And yes, it is true that there are no homeomorphisms between Rn and Rm
for n 6= m. The proof is not easy. The most common approach to this uses something
called “homology groups”. Homology groups are not so easy to define, but they carry a
lot of subtle information about a space. These groups and the many possible variations
of them are the favorite tools of every hard working topologist.
Example 6. Triangulated spheres.
Topologists don’t distinguish between the surface of a cube and a sphere. And there’s
not just the cube. There are lots of two-dimensional surfaces that are homeomorphic to a
sphere. As examples, take the “regular polytopes”.
These guys are very symmetrical, and there are exactly 5 of them. The great ancient
Greek, people like Plato the philosopher and Euclid the geometer, knew about them and
loved them. A surprising amount of mathematics is related to them, but let’s not go in
that direction today. Remember that for the moment we are only interested in topological
properties.
The surface of each of these regular polytopes contains a certain graph. We can see
vertices, edges and faces. Since we are topologists today, we see all of these as lying on
a sphere. Perhaps we are getting a little bored now, so to pass the time we count the
52
number of vertices, the number of edges and the number of faces. We get the following
table:Polytope Vertices Edges Faces V+F-E
Tetrahedron 4 6 4 2
Octahedron 6 12 8 2
Cube 8 12 6 2
Icosahedron 12 30 20 2
Dodecahedron 20 30 12 2
It is no big surprise that the number of vertices is different in each case. But the strange
thing is that if we sum the number of vertices and the number of faces, and subtract the
number of edges, in each case we get 2.
We might draw other similar, but less symmetrical graphs on a sphere. If only the
graph is connected, every time we do this calculation, we get the number 2. That’s kind
of neat, and it seems to be a topological property of the sphere.
Example 7. Surfaces.
There are other surfaces than spheres. For instance, there is the torus, the surface of
a ring. Then there are more complicated shapes that you can get for instance by taking
several metal rings, welding them together, and consider the surface of the body we have
obtained. There are many ways to weld, how many different surfaces do we actually get
in this way?
53
Here’s the interesting part. Given one of these surfaces S, we can generalize the
example and draw graphs on S. We have to be a little careful about exactly what graphs
we allow, but if we do it right, we find that the number χ(S) = V +F −E does not depend
on the graph, only on the surface. But the number is not necessarily two, as for the sphere.
For instance, if S is a torus, then χ(S) = 0. If we weld a set of rings pairwise together n
times to get a connected ring-sculpture, its surface will have Euler characteristic −2n.
The number χ(S) is called the Euler characteristic of S. There are several ways to
understand this, the best way is to use the homology groups we mentioned before. This
description gives a generalization of the Euler characteristic to a large class of spaces.
Theorem 8. For any surface S in R3, the Euler characteristic is an even integer, smaller
or equal to 2. Two surfaces are homeomorphic if and only if their Euler characteristics
agree.
Objection 9. The previous examples might are fun, but are they serious mathematics?
Mathematics is about numbers and integrals and equations, right? It should not be about
making drawings on a balls and rings.
The first answer to this objection touches on the question: What are the basic ob-
jects we study in geometry? The answer to that question will always be subjective, and
historically the answer has changed from time to time. Even today, to some extent it
will vary from university to university. However, one answer you would frequently get is
that the basic objects are “generalizations of surfaces to higher dimensions”. These are
known as n-dimensional manifolds, so that in particular surfaces are 2-dimensional mani-
folds. The defining property of them is that for every point p on a manifold M , there is a
neighborhood of the point that looks just like (is homeomorphic to) a Euclidean space of
dimension n.
These are the most basic objects. However, many people do something to the man-
ifolds before they study them. Some are interested in studying additional structures on
manifolds (differential geometry, symplectic geometry and lots of others), some people
54
like manifolds with singularities. These are strictly speaking not manifolds, but have a
few exceptional points p where the manifold property in not satisfied. As an example of
this, think of a space consisting of two intersecting lines. The intersection point is the
singularity.
If we are interested in manifolds of high dimensions, possibly with singularities, argu-
ments using drawings are usually not very convincing. But the general theory developed
to answer the questions we posed above works fine. The homology groups and the fun-
damental group still make sense. In this way, algebraic topology gives some of the most
fundamental properties of these spaces.
To give a second answer, let’s first do some real mathematics related to curves and
surfaces in our 3-dimensional space. At any point p on a (differentiable) curve, we can
approximate it by a line, the tangent to the curve at p. We can usually approximate the
curve even better with a circle, touching the curve at p. The inverse of the radius of the
circle which approximates best is called the curvature of the curve at p. If there is a line
that approximates better than every circle, we say that the curvature is zero.
There is a similar concept for a surface M . At each point p on M there is a real
number R which codifies how much the surface bends close to the point p. The precise
definition of R is a little more complicated for surfaces than for curves. This story is
interesting, and if you want to know more about it, look it up in any book on differential
geometry. For now, let’s not worry about details.
Anyhow, for each point on M we have a real number R. Another way of saying this
is that we have a function R : M → R. Given a surface, we want to find the ”‘average”’
curvature of M . Well, this is given by integrating the curvature over M , and dividing by
the area of M . We focus on the numerator of this quotient, and integrate the curvature
over the whole surface M .
Now we notice something interesting. If we scale the surface to make it smaller, the
area will shrink and the curvature will get bigger. When the area shrinks, the integral
gets smaller. On the other hand, when the curvature grows, the integral gets bigger. If
we change the surface in a more general way than just scaling, the same two effects will
still oppose each other. After doing a lot of analysis, we are delighted to find that they
exactly balance each other out. The integral does not change at all when we change the
surface, as long as we don’t change the homeomorphism class! In fact, we get the following
fundamental and surprising theorem.
55
Theorem 10 (Gauss-Bonnet). The integral of the curvature over a surface equals 2πχ(M).
This is not a theorem of topology. It’s a theorem about the interaction of topolo-
gy with analysis, or more precisely, with differential geometry. Historically, it predates
topology. It is typical of such theorems. The topological invariants, in this case the Euler
characteristic, set the general stage, and the finer and usually more complicated analysis
plays out on this stage, subject to the general conditions given by topology.
One should not assume that the two answers above are the only ways in which topol-
ogy can interact with the rest of mathematics, but one should be aware of that various
generalization of the Gauss-Bonnet theorem have had an enormous impact all over math-
ematics (Chern classes, the index theorem etc.).
Hint 11. In Exercise 1, try to remove one point from the intervals, and see what happens.
The upper curve in figure 1 is not contractible to a point. You have to be pretty
clever to find an elementary proof.
56
n/Zc
Instructor: Shou-Wu Zhang£ÜÆɤ[Princeton University]
2012.9.8
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S"24´Óê"ù´Ï:
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59
ln/©´N´§½n/>a, b, c§-
γ =c
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60
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61
ß. n´vk²Ïfê.
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2ab, a2 + b2 = c2.
a =411340519227716149383203
21666555693714761309610
b =6803298487826435051217540
411340519227716149383203
c =224403517704336969924557513090674863160948472041
8912332268928859588025535178967163570016480830
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62
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x =p
q⇔ (a, b, c) = (2pq, p2 − q2, p2 + q2)⇔ γ =
p2 + q2
2
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63
ã´kVg))½ü:PÚQ§·ÒUÑS§2'u¢¶§R,·
ÑP +Q = R,ù´g¤vk"·Ò\^§gÒÑü:§v"
¤±§ng'gk¿gõ"\p¥vkÆng´£¤"
e¡Äýkn:"e¡´\41901cß:
½n. (Modell 1922)C´Q þ^ý.u´
C(Q) ' Zr ⊕ C(Q)tor.
Ù¥r > 0, C(Q)tor´k+.
·'%´§ùoõkn:§kvkĺ)¤"·e¡a'¯§
ØL§Ø^ú%§·y3®²Ø2é¥Æ)"Ø´Æ)§´ïÄ)ù"
,§ýéAng§·v)§´§·±k2§3)"ézêp, -
Np = #y2 = x3 + ax+ b mod p)ê.
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ap = p−Np.
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L(C, s) =∏
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L-¼êùå5éOÛ§´ÐmÒ´p-?êOÑy?êaqé"ù?
ês¢Üu3/2Âñ§¢Üu3/2ÿ§§Ù¢´k)Ûòÿ"ù´Wilesy
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"y30100ø7¯K"ù¯K´`§fâ@¼ê3s = 1?kTaylorÐmµ
ß. (Birch and Swinnerton-Dyer)
L(C, s)3s = 1VЪäk/ª
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64
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eπ√163´êíº
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eπ√163 = 262537412640768743.99999999999925 . . .
= 6403203 + 744− 74× 10−14 . . .
Xw·NoP§µdlI£5§I[A¦<L§¦ÒJÆ\ýú
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William Thurston£1946.10-2012.8¤´IêÆ["¦´$ÿÀÆ++E<
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67
Joseph Liouville )
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37Gottfried Wilhelm Leibniz§4ÙZ[§1646-1716§IêÆ[ÚóÆ[§È©Ú?uyö"38Johann Bernoulli§¿#Ëã|§1667-1748§I-a¬êÆ[§Bernoulli[x¥Ãõ#ÑêÆ[¥
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IêÆ[Borchardt50ÚJoachimsthal51Liouville[¦ó¹§ 1850-
46Gaspard Monge, Comte de Peluse§F§1746-1818§IêÆ[§xAÛMáö"47Johann Carl Friedrich Gauss§pd§1777-1855§IêÆ[ÚÆ[§éêÆÚÆuÐ
ÑÃÔ'z§<¡“êÆf”"48Niels Henrik Abel§C§1802-1829§Uâé%êÆ[§y²Êg§Øª)5§27k
u¡ö"49Pierre-Simon, marquis de Laplace§.Ê.d§1749-1827§IêÆ[ÚU©Æ[§éUNåÆ¥ê
ÆÚVÇÚORkz"50Carl Wilhelm Borchardt §1817-1880§IêÆ["51Ferdinand J. Joachimsthal§1818-1861§IêÆ["
75
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52Charles Auguste Briot§1817-1882§IêÆ[ÚÔnÆ["53Jean-Claude Bouquet§1819-1885§IêÆ["54Friedrich Wilhelm Bessel §l§1784-1846§IêÆ[ÚU©Æ["55Adrien-Marie Legendre§V4§1752-1833§#ÑIêÆ[§éÚOÆ!©Û!êØ!Äê
uÐÑkz"56Sir William Rowan Hamilton§Çî§1805-1865§O=ÔnÆ[!U©Æ[ÚêÆ[§é²;å
Æ!1ÆÚêÆk#Ñz"57Vladimir Igorevich Arnold§Cì§1937-2010§IêÆ[§2001cWolføÌ"58Joseph Alfred Serret§1819-1885§IêÆ["59Joseph Louis Francois Bertrand§ËA=§1822-1900§IêÆ[§ó+)êØ!©AÛ!V
ÇØ!²LÆÚ9åÆ"60Charles Hermite§DA§1822-1901§IêÆ[§3êØ!êÆ+ÑkÑÚó"
76
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61William Thomson, 1st Baron Kelvin (Lord Kelvin)§m©§1824-1907§O=ÔnÆ[!êÆ[Ú
ó§§9åƧIu²ö"62Thomas Archer Hirst FRS§1830-1892§=IêÆ[§Ìl¯AÛÆïÄ"
77
ë©z
[1] B Belhoste and J Lutzen, Joseph Liouville et le College de France, Rev. Histoire Sci.
37 (3-4) (1984), 255-304.
[2] WIKIPEDIA, the Free Encyclopedia.
[3] http://en.wikipedia.org/wiki/Joseph Liouville
[4] http://zh.wikipedia.org/zh-cn/%E7%BA%A6%E7%91%9F%E5%A4%AB%C2%B7%
E5%88%98%E7%BB%B4%E5%B0%94
[5] http://www.chinavib.com/wiki/doc-view-394.html
[6] http://media.openedu.com.cn/media file/netcourse/asx/cwffc/ktsy/htm/lwe.htm
[7] http://wlifang.zgz.cn/huixing.htm
[8] http://thediaryjunction.blogspot.com/2009 03 01 archive.html
[9] O’Connor, John J.; Robertson, Edmund F., “Joseph Liouville”, MacTutor Histo-
ry of Mathematics archive, University of St Andrews, http://www-history.mcs.st-
andrews.ac.uk/Biographies/Liouville.html.
[10] Joseph Liouville at the Mathematics Genealogy Project, http://genealogy.math. nd-
su.nodak.edu/id.php?id=55185
[11] Hirst’s Comments at Mac Tutor, http://www-history.mcs.st-andrews.ac.uk/HistTopi-
cs/Hirst comments.html
78
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