2012.12.109Ï

8¹

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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

¶[üù

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

∫ ∞

0+a1e

∫ ∞

0+a2e

2

∫ ∞

0+ · · ·+ ane

n

∫ ∞

0

TLª©)XeüLªÚµ

P1 = a∫∞0 + a1e

∫ ∞

1+a2e

2

∫ ∞

2+ · · ·+ ane

n

∫ ∞

n,

P2 = a1e

∫ 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)

∗ÈgGesammelte Abhandlungen1ò11-4§KUber die Transzendenz der Zahlen e und π¶Ø

©Ñ?IKe"Èö5"

1

,¡§e©O^KÚkL«¼ê

z(z − 1)(z − 2) · · · (z − n)

Ú

(z − 1)(z − 2) · · · (z − n)e−z

3«mz = 0z = nS¤ýéµ

|∫ 1

0| < kKρ, |

∫ 2

0| < 2kKρ, . . . , |

∫ n

0| < nkKρ

¿P

κ = |a1e|+ 2|a2e2|+ · · ·+ n|anen|k,

Kkت

|P2| < κKρ. (2)

y3(½êρ§ÙÄkUêa · n!اÙg÷vκKρ

ρ! < 1"u´duÓ

ª(1)§P1ρ!´ØUρ+ 1Ø! 7ØÓu0ê§qÏlت(2) P2

ρ!

ýéu1§¤±ªP1

ρ!+P2

ρ!= 0

ØU¤á"

bπ´ê꧿êα1 = iπ÷vXêng§"u´·^α2, . . . , αnL

«T§Ù§Kdu1 + eiπ0§Lª

(1 + eα1)(1 + eα2) · · · (1 + eαn) = 1 + eβ1 + eβ2 + · · ·+ eβN

7,0¶ùp´§Nêβ1, . . . , βN´XêNg§"d§~

XMêβ1, . . . , βMØÓu0§ Ùu0§KùMêβ1, . . . , βM´Xe/

ªMg§µ

f(z) = bzM + b1zM−1 + · · ·+ bM = 0,

ÙXêÓ´knê§AO/§XêbMØÓu0"u´XþLªe/ª

a+ eβ1 + eβ2 + · · ·+ eβM ,

ùpa´ê"

òTLª¦±È© ∫ ∞

0=

∫ ∞

0zρ[g(z)]ρ+1e−zdz,

2

Ù¥ρ2gL«ê§¿Pg(z) = bMf(z)¶K

a

∫ ∞

0+eβ1

∫ ∞

0+eβ2

∫ ∞

0+ · · ·+ eβM

∫ ∞

0

TLª©)XeüLªÚµ

P1 = a∫∞0 + eβ1

∫ ∞

β1

+eβ2∫ ∞

β2

+ · · ·+ eβM∫ ∞

βM

,

P2 = eβ1∫ β1

0+eβ2

∫ β2

0+ · · ·+ eβM

∫ βM

0,

Ù¥§/§È©∫∞βi3zE²¡¥l:z = βi÷^²1u¢ê¶òz =

+∞§È©∫ βi0 l:z = 0÷ëü:ãò:z = βi"

È©∫∞0 quρ!Økn꧿¤áρ+ 1Óª

1

ρ!

∫ ∞

0≡ bρM+Mbρ+1

M , (ρ+ 1).

/Ïz = z′ + βi¿dug(βi) = 0§?Ú

eβi∫ ∞

βi

=

∫ ∞

0(z′ + βi)

ρ[g(z′ + βi)]ρ+1e−z

′dz′ = (ρ+ 1)!G(βi),

Ù¥G(βi)L«βiXêõª¼ê§Ù'uβigê3ρM +Me§ÙXê

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)

,¡§e©O^KÚkL«¼êzg(z)Úg(z)e−z30uz = 0z = βim@

È©ãþ¤ýéµ

|∫ βi

0| < |βi|kKρ (i = 1, 2, . . . ,M)

¿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

ρ!ýé

u1§¤±ªP1

ρ!+P2

ρ!= 0

ØU¤á"

N´wѧXÛUìXþÀå»Óü/y²'uê¼ê

Lindemann½n1"

Konigsberg i. Pr.§1893c15F

£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¤"

4

Èa

g,éê.êeÚ±Çπ5§Q´(6êÆ.õcJK"ÙÄgy²©O

dIêÆ[C. Hermiteu1873cÚIêÆ[F. Lindemannu1882cѧù´19­V

­êƤÒ"¯¢þ§e´1¿y²ê/<E0ê£1<¤

êu1844cdIêÆ[J. LiouvilleEÑ5¤§ π5F1n

êÆJK/z0¯KªÄ½)"

Hermite©Ø©Sur la fonction exponentielle©,§uLuComptes Rendus

de l’Academie des Sciences, 77, (1873), Paris, pp. 18-24, 226-233, 285-293¶5êÆû

))¤©z°À61408-416¹Ù1Ü©£=©Ñ1«y²¤¥È

©"Lindemann÷^Hermite Ñπ5y²£þ5¤©z¤§

Óé"ùp[Ñè´Ð«§ùü¯K3¤þQ²´õo(J§±9c

<§)ûGÑõoãå"

Hilbertu1893cÑ#y²§¾U3áá4SòùüJKZÀ|¢/)

û§|©§Lã'§-<7'",/#y²03êƤþ/ جp

uÐy²§ùØ©ú@#§Ï§Ð«êÆrº¯K

ÛÉå"XHermann Weyl3David Hilbert and his mathematical work

¥¤µ/A characteristic feature of Hilbert’s method is a peculiarly direct attack on

problems, unfettered by algorithms; he always goes back to the questions in their original

simplicity.0

,·;êØÃ'§ùͶêƯK°ç)§3·w5´é

Ö"duóþ(J§&ÖL©<¿Øõ¶ 8cS!£123F¤

qT½David Hilbert150±c§yòd©Èѧ±"Öö§¿Vg",§

Hilbert2êÆó9ÙéêÆ.K§Ø©¤UL"+½ÿ§

D²;©Ø©§¦c¬±õ<¤!ü§½ØVg«

ª"

Èö

2012c1uZurich

5

Riemann-Roch½n²;y²

Á²Ê∗

Úó

©8´0Riemann­¡Ä£¿Ñ;Riemann­¡þ

Riemann-Roch½n²;y²"y²ì[1]"©.´ö3nÀ?Ü

l7PRiemann­¡§¤Ï"w"RiemannïÄ)Û¼êN

5§Ú\Riemann­¡ùAÛé"^yâ§Riemann­¡Ò´E

6/"-<¯Û´§;ùÿÀ^¬éRiemann­¡()4"

;Riemann­¡þäk`êAÛ(§ Riemann-Roch½n´n)ù«(

Ñu:"~XdRiemann-Roch½n·±y²z;Riemann­¡ÑÓ

uu^1wê­"

­V¥m§H.Cartan§ Serre§GrothendieckòØ(Sheaf the-

ory)ÚþÓNØ(cohomology theory)XÚ/Ú\)ÛAÛ(Analytic Geome-

try)ÚêAÛ(Algebraic Geometry)" Serreó«ùüöméX§

Grothendieck3êAÛ¥éRiemann-Roch½n?1í2"^ùy

ó䧷±Ñ;Riemann­¡þRiemann-Roch½nØÓu²;

y²yy²"ù«éRiemann­¡nØ?n[2]"

3©¥§·k0Riemann­¡Ä£§,QãRiemann-Roch½n§

¿y²§­íا·^Ðóä(Ä¢©Û§4­¡ÿ

À§5ê)ÑRiemann-Roch½ny²"duöY²k§kØ?

"ÖöØ"

1 Riemann­¡Ä£

1.1 Riemann­¡gVg

ÄüCþX¼ê§ÙÛÜ5(XTaylorÐm§Iz 7→ zk, k ∈Z≥0)±3E²¡Cm8SéÐn)"1e·ïÄX¼ên5

§Ò¬õ5ù(J"~õ¼êX√z§log z",·±C

∗Äê82§y3MÃÆ

6

m8§XC×−]−∞, 0[§¿Ùþü©|§ù*þw84¼ê

Ü©&E§·¤rº,´¼êÛÜ5§ N5"3C×−] −∞, 0[

þ§√zküü©|f+(z)Úf−(z)§¦f+(1) = 1§f−(1) = −1"Riemann

´§·Äü°C×−] − ∞, 0[§©OPX+§X−"3X+þ½Â¼êf+§

X−þ½Â¼êf−§òX+¥]−∞, 0[þ>X−¥]−∞, 0[e>ÑÊÜ^ú

]−∞, 0[þ§2òX+¥]−∞, 0[e>X−¥]−∞, 0[þ>ÑÊÜ,^ú

] −∞, 0[þ§K·mX,f+f− 3XþÊÜëY¼êf"(X

eã)

3mX¥§ÊÜ/¿0þ:kg,/ÓuC¥m8§ f3ù

m8þ´X"

l ·mX,§3ÛÜþÑÓuCm8§Ïd±!ØÙþ

X§æX¼ê§ÙNÿÀØÓuC?Ûm8"·f´XþX¼ê§

§¹√zÜ&E"¡(X, f)

√z3C×þû½NX¼ê"ddRiemannòï

ÄX¼êN5¯K§8(ïÄùa#m(§ïÄÙþX!æ

X¼ê§X!æX©/ª"ùam¡Riemann­¡.

½ Â1. Riemann­ ¡ ´ ë Ï E 6 /§ = ë Ï ¢ 1 w 6

/X(C2, Hausdorff)\E("Ù¥E(´X1wãþ§ÙØÓ

ãm=£¼êþCm8mX¼ê"éXm8U§½ÂUþE1wk

©/ª(k = 0, 1, 2)§X¼ê§æX¼ê§X©/ª§æX©/ª§ÙN©

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

!0;Riemann­¡ê­'X§ÙSNòج3e©¥^"ê­

´êAÛ¥ü!Äé"oÑ/`§ê­Ò´^õª":½Â

AÛé§XF1êÆïÄI­ax2+bxy+cy2+dx+ex+f = 0§½3Lü

­V3êÆõ¡Óâ¥%/ ý­y2 +a1xy+a3y = x3 +a2x2 +a4x+a6"

ù§Xê±3?Û¥§XQ,Q(i),R,C,Fq"ùXê(½.)Ø´

ê4§·±Òê­¯êدK§Xxn + yn = 1þ´Äk¦xy 6= 0kn

:(x, y)(Fermat’s Last Theorem)ºÐ/ïÄ­AÛ§·kò.À

,ê4§~XQ,C,Fq"yâAÛïÄL²§n)­3ê4þ

AÛ§¬én)ÙþâkÏ"3e©¥·.Ü´C"d§òïÄ

§àgz¬5éõB§X§x2 +y+1 = 0C¤x2 +yz+z2 = 0,§y2 = x3−xC¤y2z = x3 − xz2"3AÛþùéAu\\á:(Kz)"3©¥·

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)

Ù¥L«Ñ"Pn¥^1wKê­(±e¡­)´Pnf8X ⊂ Pn§

÷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¥­"

~5. £ý­¤X = [x : y : z] ∈ P2|y2z = 4x3 − g2xz2 − g3z

3,Ù¥g2, g3 ∈ C ÷vg3

2 − 27g23 6= 0"XP2¥­"

·½Â­mXeµ

½Â3. X ⊂ Pm, Y ⊂ Pn­§XY´Nϕ : X → Y ,÷v

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

2E6/þ§·wX´Pn4i\Ef6/§AO/§Ù;Riemann­

¡"5¿eϕ : X → Y­m§KeòX,Yw;Riemann­¡, ϕ ´

XN§Ïd·l­Æ;Riemann­¡Æ¼fµ

F : ­ → ;Riemann­¡

~6. ~3~4¥­;Riemann­¡ÓuP1"

~5¥­;Riemann­¡,g2, g3CħÙÓaH~2¥;

Riemann­¡Óa"

·¯§F´Ä´essentially surjectiveº=§éu?¿;Riemann­¡X,´Ä

o3­X ′,¦X ′;Riemann­¡ÓuXº

Y´½"

Äk§·kXe½nµ

·K1. £±èû¤X ⊂ PnPn4i\Ef6/§KXPn¥­"

10

Ïd§yF´essentially surjective,Iy²§z;Riemann­¡ÑUXi

\,Pn¥"·ò3e©¥^Riemann-Roch½ny²ù:"l ·kµ

·K2. F´essentially surjective,=éu?¿;Riemann­¡X,3­X ′¦X ′

;Riemann­¡ÓuX"

(J´µ

·K3. F´Æmd"

Ïd§ïÄ;Riemann­¡mXN§ïÄ­m´£¯"

2 Riemann-Roch½nQãíØ

3!¥§·kQã;Riemann­¡þRiemann-Roch½nAÏ/§2

^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"

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21

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al

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∫

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bl

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(k)pj = −

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= degD − g + i(D).

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22

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[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.

£¤9!<m!ܵ!oÛ!oWÔ n¤

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Lars Hormander£1931.1-2012.11¤´a;êÆ["¦”5 ©§y

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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

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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["

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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"

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61William Thomson, 1st Baron Kelvin (Lord Kelvin)§m©§1824-1907§O=ÔnÆ[!êÆ[Ú

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77

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[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

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