H = H 1 (Jac(R); Z) Sp 1 H (Jac(R); Z) = Λ Z H, H (Jac(R); Z) = Λ Z H = Λ Z H Poincaré duality canonical ( ) canonical symplectic form foliation (2) F

Size: px
Start display at page:

Download "H = H 1 (Jac(R); Z) Sp 1 H (Jac(R); Z) = Λ Z H, H (Jac(R); Z) = Λ Z H = Λ Z H Poincaré duality canonical ( ) canonical symplectic form foliation (2) F"

Transcription

1 Sp-modules symplectic Sp-module low dimensional., 0 1, 2, 3, 4 (n),. foliation n. Sp-modules, intersection form H = H 1 (Σ; Z), µ : H H Z H rank 2g free module Q C Q foliation R Q, R H Q = H Q, H R = H R g n Sp Sp(2g, Q), Sp(2g, R). Sp(2g, Z)- Sp(2g, Q) 0 (0) Lie sp(2g, Q) S 2 H Q. Lie Lie 4 g = 1 sp(2, Q) sl(2, Q) sl(2, Q) S 2 H g=1 Q Q x 2, xy, y 2. R 1 (1) Sp Riemann R Jacobian 1

2 H = H 1 (Jac(R); Z) Sp 1 H (Jac(R); Z) = Λ Z H, H (Jac(R); Z) = Λ Z H = Λ Z H Poincaré duality canonical ( ) canonical symplectic form foliation (2) Foliation standard symplectic vector space H R 2n symplectic x 1,..., x n, y 1,..., y n coordinate standard symplectic form ω 0 = n dx i dy i i=1 H R = (R 2n, ω 0 ) standard symplectic vector space Hamilton foliation 1972 Gel fand-kalinin-fuchs 40 transverse symplectic foliation foliation. exotic Gel fand Kalinin arxiv first name Fuchs transversely symplectic foliation symplectic manifold Definition 1.1. (M 2n, ω) symplectic manifold, ω A 2 (M) 2-form dω = 0 ω n volume form. 2n symplectic form standard standard symplectic vector space symplectic manifold 2

3 symplectic ω n Darboux Theorem 1.2 (Darboux). symplectic manifold(m, ω) (R 2n, ω 0 ) symplectical symplectically diffeomorphism symplectomorphism contact structure contactomorphism symplectic symplectic Darboux local global transversely symplectic foliation F C - W codimension 2n transversely symplectic foliation 2n, leaf local normal product local global Gel fand-fuchs Chern-Simons Thurston W = α U α, ϕ α : U α R 2n standard symplectic space (R 2n, ω 0 ) submersion submersion submersion 1 leaf R 2n symplectic W f αβ = ϕ β ϕ 1 α : ϕ α (U α U β ) ϕ β (U α U β ) symplectomorphism U α ϕ 1 α (p) p ϕ α f αβ : symplectomorphism U β ϕ β Gel fand-fuchs-kalinin 3

4 n = 1, R Gel fand-kalinin-fuchs H 7 (W; R). GKF class exotic class. exotic, primary 1999 Metoki class H 9 (W; R) Bott Perchik 1974,5 Topology euler 100 Perchik transversely symplectic, n = 1 2 symplectic structure area preserving holonomy foliation Gel fand-kalinin-fuchs class Metoki class 0 Gel fand-kalinin-fuchs 40 transversely symplectic foliation Gel fand-fuchs theory, Chern-Simons theory GKF-class GF-theory Gel fand-fuchs 1969 Chern-Simons 1974 Annals Chern-Simons Thurston hyperbolization hyperbolic geometry 1982 Bulletin Haken hyperbolic structure 1976 pseudo-anosov Bulletin Bulletin Bulletin hyperbolic geometry Fathi-Poénaru-Laudenbach, 1988 Bulletin measured foliation measured lamination foliation lamination lamination , 100 Nielsen Nielsen Nielsen-Thurston 76 Thurston Nielsen 4

5 Nielsen ,3 Nielsen pseudo-anosov Thurston Nielsen Thurston Nielsen Riemann quasi-conformal map Teichmülar Gel fand-fuchs 1969 Chern-Simons 1974 Chern-Simons introduction Q Pontrjagin class Novikov Pontrjagin class piecewise linear formula 3 rational Pontrjagin class smooth tangent bundle connection Chern-Weil Pontrjagin form R Pontrjagin class formula Chern-Simons 2 Pontrjagin form exact form Chern-Simons p 1, first Pontrjagin class Gel fand-fuchs 5 Chern-Siomns foliation Godbillon-Vey foliation Gel fand-fuchs, Chern-Simons Godbillon Vey 70 1 de Rham well-defined de Rham cohomology 1971 Godbillon-Vey class, Roussarie PSL(2, R) locally homogeneous foliation Thurston Bulletin Noncobordant foliations of S foliation, Godbillon-Vey 1970 Gel fand-kalinin-fuchs class Thurston Thurston. Thurston monster Perelmann Thurston

6 cover S 1 Thurston, foliation, Gel fand-fuchs, Sp-module transverse foliation standard symplectic vector space H R symmetric algebra, S H R = R[x 1,..., x n, y 1,..., y n ] R x 1,..., y n derivation Der poly (S H R ) = {X = n ( i=1 f i ) + g i ; f i, g i R[x 1,..., y n ]} x i y i h S H R X Der poly (S H R ) X(h) = n i=1 ( ) h h f i + g i x i y i symplectic form symplectic form infinitesimal Der poly (S H R, ω 0 ) = {X Der poly(s H R ); L X ω 0 = 0} symplectic form Lie symplectic form infinitesimal symplectic form Lie symplectic symplectic Hamilton Proposition 1.3. L X ω 0 = 0 n ( F X = F ) x i y i y i x i F S H R /R = R[x 1,..., x n, y 1,..., y n ]/R i=1 modulo R F Hamiltonian function n = 1 4 derivation Corollary 1.4. Lie Der poly (S H R, ω 0 ) S 0 H R 6

7 Lie bracket Poisson bracket Der poly (S H R ) S 0 H R Der poly (S H R ) X, X F = X F = F X S 0 H R F S 0 H R X F = X X Der poly (S H R ) Y Der poly (S H R ), G = G Y S 0 H R [X, Y] Der poly (S H R ) S 0 H R, {F, G} Poisson bracket n ( F G {F, G} = F ) G x i y i y i x i i=1 Lie transversely symplectic foliation, x i y i linear derivation 1 2 subalgebra S 2 H R ( sp(2n, R)). Gel fand-fuchs foliation Gel fand-fuchs R 1 Gel fand-fuchs jet 3 H 3 2 foliation 3 (3) mapping class group. symplectic D. Johnson 1970 Torelli abel Johnson kernel Johnson genus g M g = π 0 Diff + (Σ g ) Sp g = 1 g 2 kernel 1 I g M g Sp(2g, Z) 1 Thurston Nielsen 1 Nielsen Nielsen pseudo-anosov Nielsen genus 2, Torelli pseudo-anosov Thurston 1 Thurston 76 genus 2 7

8 ( ) Thurston fill up transverse cut contractible separating separating Dehn-twist separating curve separating curve 2 separating curves Dehn-twist, Dehn-twist pseudo-anosov separating curves Dehn-twsit Torelli Nielsen Dehn-twist, Dehn-twist 2 GL(2, Z) Torelli Dennis Johnson ,8 1 Torelli abel torsion Q H 1 (I g, Q) = Λ 3 H Q /H Q, Sp H Q H Q u u ω 0 Λ 3 H Q Sp-map Sp-submodule Young GL 3 Sp Torelli Dennis Johnson Torelli 5, abel Johnson homomorphism τ 1 : I g Λ 3 H/H Z τ Johnson τ 1 abel Rohlin Birman-Craggs 2-torsion, homology cylinder 8

9 Johnson Malcév completion lower central series abel 2-step nilpotent group Dehn-Nielsen automorphims lower central series Johnson kernel ker τ 1, ker τ 1 = K g Johnson kernel g = 2 g 3 abel modulo torsion abel torsion rank Johnson filtration lower central series M g M g (1) = I g M g (2) = K g M g (3)... Malcév completion M g I g Sp(2g, Z), relative Malcév completion Deligne Hain Sp semi-simple part nilpotent part nipotent part + k=1 M g(k)/m g (k + 1) Lie Lie + k=1 M g(k)/m g (k + 1) Der ( L(π(Σ g ) ) Malcév completion derivation Lie disc completion Lie symplectic form relation symplectic form derivation Johnson τ 1 : M g ( ) 1 2 Λ3 H/H Sp(2g, Z) Johnson H Hain Λ 3 H/H π 1 PH 3 (Jac(R)) P primitive part primitive Λ 3 H/H Hain Deligne relative Malcév completion nilpotent part Hain 9

10 H (Λ 3 H/H) Sp-module Sp-module Sp part symplectic Sp- H (Λ 3 H/H) Sp H (M g ; Q), H (Λ 3 H/H) Sp image tautological algebra., Teichmüller moduli abel Λ 3 H/H Sp-module H, 1 Grassmann Sp-module (4) Kontsevich graph (co)homology. 1992,3. Kontsevich graph cohomology Lie commutative, associative, Lie c g, a g, l g c g Kontsevich graph cohomology stable, g g = 1, 2 unstable cohomology transversely symplectic foliation Gel fand-kalinin-fuchs a g unstable stable Kontsevich Riemann moduli. l g Kontsevich Kontsevich Lie l g, Riemann cohomology 2 5 (5) Gal( Q/Q) Johnson Lie ,6 Johnson 1994 expectation, τ image, Sp Hain Johnson 10

11 Johnson Hain Grothendieck, Deligne, Drinfeld, Grothendieck anabelian geometry, Soule element Johnson image genus 0 S 2 CP 1 3 rank 2 derivation Soule trace genus 2 genus 1 trace genus 2 genus genus 0, 1, 2 3 genus 0 Newton genus 1 genus 1 parabolic Euclidean, genus 1 parabolic hyperbolic elliptic genus 1 6 (6) Vassiliev finite type invariants. symplectic Kontsevich knot Morsification knot Morse critical point critical point configuration chord diagram linear chord diagram Vassiliev chord diagram circular chord diagram. S 1, 2k pairing chord diagram relation 4T-relation frame independence relation. relations Vassiliev finite type invariatnt generating function free algebra Bar-Natan, Vassiliev invariant HP generators 11

12 . k = 1, 2, 3, 4... google sequence, mathematics, number, chord diagram k linear chord diagram (2k 1)!! cyclic symmetry cyclic action plot cyclic invariant chord diagrams cyclic invariant linear chord diagrams cyclic invariant symmetric alternating representation, Young 90 Young symmetric product. alternating product. symmetry cyclic invariants chord diagrams 1,2,5,18,105,902,9749,... cyclic 0,1,2,17,88,897,9562,... chord diagrams 2000 Vassiliev knot cyclic invariant tensors unknown Vassiliev 3 M link L, (M 3, L) LMO (Le-Murakami J.-Ohtsuki) 3 finite type invariants LMO Kontsevich commutative algebra c g symmetric algebra S H Q, derivation 3 Kontsevich 3 2n + 1 M 2n+1 1 tangent bundle symplectic knot Vassiliev invariant generating function. 3 generating function, 3 relation 12

13 generating function Galfalidis commutative version S 3 H Q, abel S 4 H Q 1 bracket relation 3 generating function Galoufalidis-Nakamura Sp-module Stoimenow linear chord diagram parallel regular regular chord diagram Vassiliev invariant (2k 1)!! Zagier Zagier Stoimenow Dedekind η- function, modular form Zagier Vassiliev invariants and a strange identity related to the Dedekind eta-function Zagier : 1 parabolic symplectic form ideal Σ 1 Σ 1,1 symplectic form : parabolic hyperbolic Goldman Σ 1,1 1 H Lie Σ 1 ideal parabaolic? genus 2 genus 0 genus 1 tangential bese point, parabolic 13

(5 19 ) 10,,, 2 3 Mathematica,, 1 Gauss( ), ,8,, 1827 Gauss 50, Gauss 200 xyz R 3 Σ Σ, surface, S Σ Σ, Σ g 2

(5 19 ) 10,,, 2 3 Mathematica,, 1 Gauss( ), ,8,, 1827 Gauss 50, Gauss 200 xyz R 3 Σ Σ, surface, S Σ Σ, Σ g 2 2010 5-2011 3 10 (2011 7 ) 1 2 2 20 3 (Mal cev completion) 37 4 52 5 Sp-modules 75 6 mapping class group 115 1 (5 19 ) 10,,, 2 3 Mathematica,, 1 Gauss(1777-1855), 1777 1855 77,8,, 1827 Gauss 50, 50 1827

More information

1 10 9 1 1.1 1 6 1,2 6 4 3 2 6 (1) Characteristic classes of foliations F foliation. breakthrough Thurston (1946-2012) 2 Gauß, Gauß, Riemann, Poincaré

1 10 9 1 1.1 1 6 1,2 6 4 3 2 6 (1) Characteristic classes of foliations F foliation. breakthrough Thurston (1946-2012) 2 Gauß, Gauß, Riemann, Poincaré 2013 10 9-2013 10, 1 2 1.1....... 2 1.2 C vs............................... 16 1.3........................... 28 2 3 41 2.1 commutative graph homology 3....................... 41 2.2 homology 3 homology...........................

More information

0. I II I II (1) linear type: GL( ), Sp( ), O( ), (2) loop type: loop current Kac-Moody affine, hyperbolic (3) diffeo t

0. I II I II (1) linear type: GL( ), Sp( ), O( ), (2) loop type: loop current Kac-Moody affine, hyperbolic (3) diffeo t e-mail: koyama@math.keio.ac.jp 0. I II I II (1) linear type: GL( ), Sp( ), O( ), (2) loop type: loop current Kac-Moody affine, hyperbolic (3) diffeo type: diffeo universal Teichmuller modular I. I-. Weyl

More information

1 = = = (set) (element) a A a A a A a A a A {2, 5, (0, 1)}, [ 1, 1] = {x; 1 x 1}. (proposition) A = {x; P (x)} P (x) x x a A a A Remark. (i) {2, 0, 0,

1 = = = (set) (element) a A a A a A a A a A {2, 5, (0, 1)}, [ 1, 1] = {x; 1 x 1}. (proposition) A = {x; P (x)} P (x) x x a A a A Remark. (i) {2, 0, 0, 2005 4 1 1 2 2 6 3 8 4 11 5 14 6 18 7 20 8 22 9 24 10 26 11 27 http://matcmadison.edu/alehnen/weblogic/logset.htm 1 1 = = = (set) (element) a A a A a A a A a A {2, 5, (0, 1)}, [ 1, 1] = {x; 1 x 1}. (proposition)

More information

E1 (4/12)., ( )., 3,4 ( ). ( ) Allen Hatcher, Vector bundle and K-theory ( HP ) 1

E1 (4/12)., ( )., 3,4 ( ). ( ) Allen Hatcher, Vector bundle and K-theory ( HP ) 1 E1 (4/12)., ( )., 3,4 ( ). ( ) Allen Hatcher, Vector bundle and K-theory ( HP ) 1 (4/12) 1 1.. 2. F R C H P n F E n := {((x 0,..., x n ), [v 0 : : v n ]) F n+1 P n F n x i v i = 0 }. i=0 E n P n F P n

More information

' , 24 :,,,,, ( ) Cech Index theorem 22 5 Stability 44 6 compact 49 7 Donaldson 58 8 Symplectic structure 63 9 Wall crossing 66 1

' , 24 :,,,,, ( ) Cech Index theorem 22 5 Stability 44 6 compact 49 7 Donaldson 58 8 Symplectic structure 63 9 Wall crossing 66 1 1998 1998 7 20 26, 44. 400,,., (KEK), ( ) ( )..,.,,,. 1998 1 '98 7 23, 24 :,,,,, ( ) 1 3 2 Cech 6 3 13 4 Index theorem 22 5 Stability 44 6 compact 49 7 Donaldson 58 8 Symplectic structure 63 9 Wall crossing

More information

Akbulut-Karakurt diagram L = K 1 K 2 in S 3 Stein corks W = W (L). W the Mazur manifold K 1, K 2 are unknotted lk(k 1, K 2 ) = ±1 Involution τ : K 1 K

Akbulut-Karakurt diagram L = K 1 K 2 in S 3 Stein corks W = W (L). W the Mazur manifold K 1, K 2 are unknotted lk(k 1, K 2 ) = ±1 Involution τ : K 1 K Akbulut-Karakurt diagram L = K 1 K 2 in S 3 Stein corks W = W (L) W the Mazur manifold K 1, K 2 are unknotted lk(k 1, K 2 ) = ±1 Involution τ : K 1 K 2 admits a Stein diagram h 0 h 1 h 2 h 2 is attached

More information

Chern-Simons Jones 3 Chern-Simons 1 - Chern-Simons - Jones J(K; q) [1] Jones q 1 J (K + ; q) qj (K ; q) = (q 1/2 q

Chern-Simons   Jones 3 Chern-Simons 1 - Chern-Simons - Jones J(K; q) [1] Jones q 1 J (K + ; q) qj (K ; q) = (q 1/2 q Chern-Simons E-mail: fuji@th.phys.nagoya-u.ac.jp Jones 3 Chern-Simons - Chern-Simons - Jones J(K; q) []Jones q J (K + ; q) qj (K ; q) = (q /2 q /2 )J (K 0 ; q), () J( ; q) =. (2) K Figure : K +, K, K 0

More information

Euler, Yang-Mills Clebsch variable Helicity ( Tosiaki Kori ) School of Sciences and Technology, Waseda Uiversity (i) Yang-Mills 3 A T (T A) Poisson Ha

Euler, Yang-Mills Clebsch variable Helicity ( Tosiaki Kori ) School of Sciences and Technology, Waseda Uiversity (i) Yang-Mills 3 A T (T A) Poisson Ha Euler, Yang-ills Clebsch variable Helicity Tosiaki Kori ) School of Sciences and Technology, Waseda Uiversity i) Yang-ills 3 A T T A) Poisson Hamilton ii) Clebsch parametrization iii) Y- Y-iv) Euler,v)

More information

Twist knot orbifold Chern-Simons

Twist knot orbifold Chern-Simons Twist knot orbifold Chern-Simons 1 3 M π F : F (M) M ω = {ω ij }, Ω = {Ω ij }, cs := 1 4π 2 (ω 12 ω 13 ω 23 + ω 12 Ω 12 + ω 13 Ω 13 + ω 23 Ω 23 ) M Chern-Simons., S. Chern J. Simons, F (M) Pontrjagin 2.,

More information

1 M = (M, g) m Riemann N = (N, h) n Riemann M N C f : M N f df : T M T N M T M f N T N M f 1 T N T M f 1 T N C X, Y Γ(T M) M C T M f 1 T N M Levi-Civi

1 M = (M, g) m Riemann N = (N, h) n Riemann M N C f : M N f df : T M T N M T M f N T N M f 1 T N T M f 1 T N C X, Y Γ(T M) M C T M f 1 T N M Levi-Civi 1 Surveys in Geometry 1980 2 6, 7 Harmonic Map Plateau Eells-Sampson [5] Siu [19, 20] Kähler 6 Reports on Global Analysis [15] Sacks- Uhlenbeck [18] Siu-Yau [21] Frankel Siu Yau Frankel [13] 1 Surveys

More information

SUSY DWs

SUSY DWs @ 2013 1 25 Supersymmetric Domain Walls Eric A. Bergshoeff, Axel Kleinschmidt, and Fabio Riccioni Phys. Rev. D86 (2012) 085043 (arxiv:1206.5697) ( ) Contents 1 2 SUSY Domain Walls Wess-Zumino Embedding

More information

Exercise in Mathematics IIB IIB (Seiji HIRABA) 0.1, =,,,. n R n, B(a; δ) = B δ (a) or U δ (a) = U(a;, δ) δ-. R n,,,, ;,,, ;,,. (S, O),,,,,,,, 1 C I 2

Exercise in Mathematics IIB IIB (Seiji HIRABA) 0.1, =,,,. n R n, B(a; δ) = B δ (a) or U δ (a) = U(a;, δ) δ-. R n,,,, ;,,, ;,,. (S, O),,,,,,,, 1 C I 2 Exercise in Mathematics IIB IIB (Seiji HIRABA) 0.1, =,,,. n R n, B(a; δ) = B δ (a) or U δ (a) = U(a;, δ) δ-. R n,,,, ;,,, ;,,. (S, O),,,,,,,, 1 C I 2 C II,,,,,,,,,,, 0.2. 1 (Connectivity) 3 2 (Compactness)

More information

1 1, 2016 D B. 1.1,.,,. (1). (2). (3) Milnor., (1) (2)., (3). 1.2,.,, ( )..,.,,. 1.3, webpage,.,,.

1 1, 2016 D B. 1.1,.,,. (1). (2). (3) Milnor., (1) (2)., (3). 1.2,.,, ( )..,.,,. 1.3, webpage,.,,. 1 1, 2016 D B. 1.1,.,,. (1). (2). (3) Milnor., (1) (2)., (3). 1.2,.,, ( )..,.,,. 1.3, 2015. webpage,.,,. 2 1 (1),, ( ). (2),,. (3),.,, : Hashinaga, T., Tamaru, H.: Three-dimensional solvsolitons and the

More information

Milnor 1 ( ), IX,. [KN].,. 2 : (1),. (2). 1 ; 1950, Milnor[M1, M2]. Milnor,,. ([Hil, HM, IO, St] ).,.,,, ( 2 5 )., Milnor ( 4.1)..,,., [CEGS],. Ω m, P

Milnor 1 ( ), IX,. [KN].,. 2 : (1),. (2). 1 ; 1950, Milnor[M1, M2]. Milnor,,. ([Hil, HM, IO, St] ).,.,,, ( 2 5 )., Milnor ( 4.1)..,,., [CEGS],. Ω m, P Milnor 1 ( ), IX,. [KN].,. 2 : (1),. (2). 1 ; 1950, Milnor[M1, M2]. Milnor,,. ([Hil, HM, IO, St] ).,.,,, ( 2 5 )., Milnor ( 4.1)..,,., [CEGS],. Ω m, PC ( 4 5 )., 5, Milnor Milnor., ( 6 )., (I) Z modulo

More information

Date Wed, 20 Jun (JST) From Kuroki Gen Message-Id Subject Part 4

Date Wed, 20 Jun (JST) From Kuroki Gen Message-Id Subject Part 4 Part 4 2001 6 20 1 2 2 generator 3 3 L 7 4 Manin triple 8 5 KP Hamiltonian 10 6 n-component KP 12 7 nonlinear Schrödinger Hamiltonian 13 http//wwwmathtohokuacjp/ kuroki/hyogen/soliton-4txt TEX 2002 1 17

More information

R R P N (C) 7 C Riemann R K ( ) C R C K 8 (R ) R C K 9 Riemann /C /C Riemann 10 C k 11 k C/k 12 Riemann k Riemann C/k k(c)/k R k F q Riemann 15

R R P N (C) 7 C Riemann R K ( ) C R C K 8 (R ) R C K 9 Riemann /C /C Riemann 10 C k 11 k C/k 12 Riemann k Riemann C/k k(c)/k R k F q Riemann 15 (Gen KUROKI) 1 1 : Riemann Spec Z 2? 3 : 4 2 Riemann Riemann Riemann 1 C 5 Riemann Riemann R compact R K C ( C(x) ) K C(R) Riemann R 6 (E-mail address: kuroki@math.tohoku.ac.jp) 1 1 ( 5 ) 2 ( Q ) Spec

More information

2 2.1 d q dt i(t = d p dt i(t = H p i (q(t, p(t H q i (q(t, p(t 1 i n (1 (1 X H = ( H H p k q k q k p k (2 ϕ H (t = (q 1 (t,, q n (t, p 1 (t,, p n (t

2 2.1 d q dt i(t = d p dt i(t = H p i (q(t, p(t H q i (q(t, p(t 1 i n (1 (1 X H = ( H H p k q k q k p k (2 ϕ H (t = (q 1 (t,, q n (t, p 1 (t,, p n (t (Clebsch parametrization, Helicity 1 Langer-Perline rylinski vortex Chern-Simons Atiyah-ott symplectic symplectic reduction Jackiw Jackiw Marsden- Weinstein ( Marsden-Weinstein Poisson Poisson Clebch parametrization

More information

compact compact Hermann compact Hermite ( - ) Hermann Hermann ( ) compact Hermite Lagrange compact Hermite ( ) a, Σ a {0} a 3 1

compact compact Hermann compact Hermite ( - ) Hermann Hermann ( ) compact Hermite Lagrange compact Hermite ( ) a, Σ a {0} a 3 1 014 5 4 compact compact Hermann compact Hermite ( - ) Hermann Hermann ( ) compact Hermite Lagrange compact Hermite ( ) 1 1.1. a, Σ a {0} a 3 1 (1) a = span(σ). () α, β Σ s α β := β α,β α α Σ. (3) α, β

More information

/ 2 n n n n x 1,..., x n 1 n 2 n R n n ndimensional Euclidean space R n vector point R n set space R n R n x = x 1 x n y = y 1 y n distance dx,

/ 2 n n n n x 1,..., x n 1 n 2 n R n n ndimensional Euclidean space R n vector point R n set space R n R n x = x 1 x n y = y 1 y n distance dx, 1 1.1 R n 1.1.1 3 xyz xyz 3 x, y, z R 3 := x y : x, y, z R z 1 3. n n x 1,..., x n x 1. x n x 1 x n 1 / 2 n n n n x 1,..., x n 1 n 2 n R n n ndimensional Euclidean space R n vector point 1.1.2 R n set

More information

0. Intro ( K CohFT etc CohFT 5.IKKT 6.

0. Intro ( K CohFT etc CohFT 5.IKKT 6. E-mail: sako@math.keio.ac.jp 0. Intro ( K 1. 2. CohFT etc 3. 4. CohFT 5.IKKT 6. 1 µ, ν : d (x 0,x 1,,x d 1 ) t = x 0 ( t τ ) x i i, j, :, α, β, SO(D) ( x µ g µν x µ µ g µν x ν (1) g µν g µν vector x µ,y

More information

3 exotica

3 exotica ( / ) 2013 2 23 embedding tensors (non)geometric fluxes exotic branes + D U-duality G 0 R-symmetry H dim(g 0 /H) T-duality 11 1 1 0 1 IIA R + 1 1 1 IIB SL(2, R) SO(2) 2 1 9 GL(2, R) SO(2) 3 SO(1, 1) 8

More information

G (n) (x 1, x 2,..., x n ) = 1 Dφe is φ(x 1 )φ(x 2 ) φ(x n ) (5) N N = Dφe is (6) G (n) (generating functional) 1 Z[J] d 4 x 1 d 4 x n G (n) (x 1, x 2

G (n) (x 1, x 2,..., x n ) = 1 Dφe is φ(x 1 )φ(x 2 ) φ(x n ) (5) N N = Dφe is (6) G (n) (generating functional) 1 Z[J] d 4 x 1 d 4 x n G (n) (x 1, x 2 6 Feynman (Green ) Feynman 6.1 Green generating functional Z[J] φ 4 L = 1 2 µφ µ φ m 2 φ2 λ 4! φ4 (1) ( 1 S[φ] = d 4 x 2 φkφ λ ) 4! φ4 (2) K = ( 2 + m 2 ) (3) n G (n) (x 1, x 2,..., x n ) = φ(x 1 )φ(x

More information

i Version 1.1, (2012/02/22 24),.,..,.,,. R-space,, ( R- space),, Kahler (Kähler C-space)., R-space,., R-space, Hermite,.

i Version 1.1, (2012/02/22 24),.,..,.,,. R-space,, ( R- space),, Kahler (Kähler C-space)., R-space,., R-space, Hermite,. R-space ( ) Version 1.1 (2012/02/29) i Version 1.1, (2012/02/22 24),.,..,.,,. R-space,, ( R- space),, Kahler (Kähler C-space)., R-space,., R-space, Hermite,. ii 1 Lie 1 1.1 Killing................................

More information

2/14 2 () (O O) O O (O O) id γ γ id O O γ O O O γ η id id η I O O O O I γ O. O(n) n *5 γ η γ S M, N M N (M N)(n) ( ) M(k) Sk Ind S n S i1 S ik N(i 1 )

2/14 2 () (O O) O O (O O) id γ γ id O O γ O O O γ η id id η I O O O O I γ O. O(n) n *5 γ η γ S M, N M N (M N)(n) ( ) M(k) Sk Ind S n S i1 S ik N(i 1 ) 1/14 * 1. Vassiliev Hopf P = k P k Kontsevich Bar-Natan P k (g,n) k=g 1+n, n>0, g 0 H 1 g ( S 1 H F(Com) ) ((g, n)) Sn. Com F Feynman ()S 1 H S n ()Kontsevich ( - - Lie ) 1 *2 () [LV12] Koszul 1.1 S F

More information

. Mac Lane [ML98]. 1 2 (strict monoidal category) S 1 R 3 A S 1 [0, 1] C 2 C End C (1) C 4 1 U q (sl 2 ) Drinfeld double. 6 2

. Mac Lane [ML98]. 1 2 (strict monoidal category) S 1 R 3 A S 1 [0, 1] C 2 C End C (1) C 4 1 U q (sl 2 ) Drinfeld double. 6 2 2014 6 30. 2014 3 1 6 (Hopf algebra) (group) Andruskiewitsch-Santos [AFS09] 1980 Drinfeld (quantum group) Lie Lie (ribbon Hopf algebra) (ribbon category) Turaev [Tur94] Kassel [Kas95] (PD) x12005i@math.nagoya-u.ac.jp

More information

eng10june10.dvi

eng10june10.dvi The Gauss-Bonnet type formulas for surfaces with singular points Masaaki Umehara Osaka University 1 2 1. Gaussian curvature K Figure 1. (Surfaces of K0 L p (r) =the length of the geod. circle of

More information

QCD 1 QCD GeV 2014 QCD 2015 QCD SU(3) QCD A µ g µν QCD 1

QCD 1 QCD GeV 2014 QCD 2015 QCD SU(3) QCD A µ g µν QCD 1 QCD 1 QCD GeV 2014 QCD 2015 QCD SU(3) QCD A µ g µν QCD 1 (vierbein) QCD QCD 1 1: QCD QCD Γ ρ µν A µ R σ µνρ F µν g µν A µ Lagrangian gr TrFµν F µν No. Yes. Yes. No. No! Yes! [1] Nash & Sen [2] Riemann

More information

I A A441 : April 15, 2013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida )

I A A441 : April 15, 2013 Version : 1.1 I   Kawahira, Tomoki TA (Shigehiro, Yoshida ) I013 00-1 : April 15, 013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida) http://www.math.nagoya-u.ac.jp/~kawahira/courses/13s-tenbou.html pdf * 4 15 4 5 13 e πi = 1 5 0 5 7 3 4 6 3 6 10 6 17

More information

I. (CREMONA ) : Cremona [C],., modular form f E f. 1., modular X H 1 (X, Q). modular symbol M-symbol, ( ) modular symbol., notation. H = { z = x

I. (CREMONA ) : Cremona [C],., modular form f E f. 1., modular X H 1 (X, Q). modular symbol M-symbol, ( ) modular symbol., notation. H = { z = x I. (CREMONA ) : Cremona [C],., modular form f E f. 1., modular X H 1 (X, Q). modular symbol M-symbol, ( ). 1.1. modular symbol., notation. H = z = x iy C y > 0, cusp H = H Q., Γ = PSL 2 (Z), G Γ [Γ : G]

More information

n ( (

n ( ( 1 2 27 6 1 1 m-mat@mathscihiroshima-uacjp 2 http://wwwmathscihiroshima-uacjp/~m-mat/teach/teachhtml 2 1 3 11 3 111 3 112 4 113 n 4 114 5 115 5 12 7 121 7 122 9 123 11 124 11 125 12 126 2 2 13 127 15 128

More information

main.dvi

main.dvi SGC - 70 2, 3 23 ɛ-δ 2.12.8 3 2.92.13 4 2 3 1 2.1 2.102.12 [8][14] [1],[2] [4][7] 2 [4] 1 2009 8 1 1 1.1... 1 1.2... 4 1.3 1... 8 1.4 2... 9 1.5... 12 1.6 1... 16 1.7... 18 1.8... 21 1.9... 23 2 27 2.1

More information

1

1 1 Borel1956 Groupes linéaire algébriques, Ann. of Math. 64 (1956), 20 82. Chevalley1956/58 Sur la classification des groupes de Lie algébriques, Sém. Chevalley 1956/58, E.N.S., Paris. Tits1959 Sur la classification

More information

SAMA- SUKU-RU Contents p-adic families of Eisenstein series (modular form) Hecke Eisenstein Eisenstein p T

SAMA- SUKU-RU Contents p-adic families of Eisenstein series (modular form) Hecke Eisenstein Eisenstein p T SAMA- SUKU-RU Contents 1. 1 2. 7.1. p-adic families of Eisenstein series 3 2.1. modular form Hecke 3 2.2. Eisenstein 5 2.3. Eisenstein p 7 3. 7.2. The projection to the ordinary part 9 3.1. The ordinary

More information

2017 : msjmeeting-2017sep-00f003 ( ) 1. 1 = A 1.1. / R T { } Λ R = a i T λ i a i R, λ i R, lim λ i = + i i=0 R Λ R v T ( a i T λ i ) = inf λ i, v T (0

2017 : msjmeeting-2017sep-00f003 ( ) 1. 1 = A 1.1. / R T { } Λ R = a i T λ i a i R, λ i R, lim λ i = + i i=0 R Λ R v T ( a i T λ i ) = inf λ i, v T (0 2017 : msjmeeting-2017sep-00f003 ( ) 1. 1 = A 1.1. / R T { } Λ R = a i T λ i a i R, λ i R, lim λ i = + i i=0 R Λ R v T ( a i T λ i ) = inf λ i, v T (0) = + a i 0 i=0 v T Λ R 0 = {x Λ R v T (x) 0} R Remark

More information

Basic Math. 1 0 [ N Z Q Q c R C] 1, 2, 3,... natural numbers, N Def.(Definition) N (1) 1 N, (2) n N = n +1 N, (3) N (1), (2), n N n N (element). n/ N.

Basic Math. 1 0 [ N Z Q Q c R C] 1, 2, 3,... natural numbers, N Def.(Definition) N (1) 1 N, (2) n N = n +1 N, (3) N (1), (2), n N n N (element). n/ N. Basic Mathematics 16 4 16 3-4 (10:40-12:10) 0 1 1 2 2 2 3 (mapping) 5 4 ε-δ (ε-δ Logic) 6 5 (Potency) 9 6 (Equivalence Relation and Order) 13 7 Zorn (Axiom of Choice, Zorn s Lemma) 14 8 (Set and Topology)

More information

1 nakayama/print/ Def (Definition ) Thm (Theorem ) Prop (Proposition ) Lem (Lemma ) Cor (Corollary ) 1. (1) A, B (2) ABC

1   nakayama/print/ Def (Definition ) Thm (Theorem ) Prop (Proposition ) Lem (Lemma ) Cor (Corollary ) 1. (1) A, B (2) ABC 1 http://www.gem.aoyama.ac.jp/ nakayama/print/ Def (Definition ) Thm (Theorem ) Prop (Proposition ) Lem (Lemma ) Cor (Corollary ) 1. (1) A, B (2) ABC r 1 A B B C C A (1),(2),, (8) A, B, C A,B,C 2 1 ABC

More information

1 1.1 [ ]., D R m, f : D R n C -. f p D (df) p : (df) p : R m R n f(p + vt) f(p) : v lim. t 0 t, (df) p., R m {x 1,..., x m }, (df) p (x i ) =

1 1.1 [ ]., D R m, f : D R n C -. f p D (df) p : (df) p : R m R n f(p + vt) f(p) : v lim. t 0 t, (df) p., R m {x 1,..., x m }, (df) p (x i ) = 2004 / D : 0,.,., :,.,.,,.,,,.,.,,.. :,,,,,,,., web page.,,. C-613 e-mail tamaru math.sci.hiroshima-u.ac.jp url http://www.math.sci.hiroshima-u.ac.jp/ tamaru/index-j.html 2004 D - 1 - 1 1.1 [ ].,. 1.1.1

More information

( Symplectic ) A B ( )

( Symplectic ) A B ( ) ( Symplectic A B (2002 9 2003 6 1. (a i. ii. iii. (b Poisson i. Poisson ii. ( so(3 iii. ( (c i. Lie Lie ii. Lie Lie iii. (d Moment maps i. ii. (e 2. (a Poisson (b Poisson Lie i. Hamilton Hamilton ii. infinitesimal

More information

D-brane K 1, 2 ( ) 1 K D-brane K K D-brane Witten [1] D-brane K K K K D-brane D-brane K RR BPS D-brane

D-brane K 1, 2   ( ) 1 K D-brane K K D-brane Witten [1] D-brane K K K K D-brane D-brane K RR BPS D-brane D-brane K 1, 2 E-mail: sugimoto@yukawa.kyoto-u.ac.jp (2004 12 16 ) 1 K D-brane K K D-brane Witten [1] D-brane K K K K D-brane D-brane K RR BPS D-brane D-brane RR D-brane K D-brane K D-brane K K [2, 3]

More information

( ) (, ) arxiv: hgm OpenXM search. d n A = (a ij ). A i a i Z d, Z d. i a ij > 0. β N 0 A = N 0 a N 0 a n Z A (β; p) = Au=β,u N n 0 A

( ) (, ) arxiv: hgm OpenXM search. d n A = (a ij ). A i a i Z d, Z d. i a ij > 0. β N 0 A = N 0 a N 0 a n Z A (β; p) = Au=β,u N n 0 A ( ) (, ) arxiv: 1510.02269 hgm OpenXM search. d n A = (a ij ). A i a i Z d, Z d. i a ij > 0. β N 0 A = N 0 a 1 + + N 0 a n Z A (β; p) = Au=β,u N n 0 A-. u! = n i=1 u i!, p u = n i=1 pu i i. Z = Z A Au

More information

2016 Course Description of Undergraduate Seminars (2015 12 16 ) 2016 12 16 ( ) 13:00 15:00 12 16 ( ) 1 21 ( ) 1 13 ( ) 17:00 1 14 ( ) 12:00 1 21 ( ) 15:00 1 27 ( ) 13:00 14:00 2 1 ( ) 17:00 2 3 ( ) 12

More information

( ) (, ) ( )

( ) (, ) ( ) ( ) (, ) ( ) 1 2 2 2 2.1......................... 2 2.2.............................. 3 2.3............................... 4 2.4.............................. 5 2.5.............................. 6 2.6..........................

More information

第73回 微分同相群のトポロジー Smale予想を巡って 2019 年 3 月 16 日 (土) 10:00 於 東京都 文京区 春日 中央大学理工学部 5 号館 5534 教室 3 月 16 日 (土) 10:00 11:00 11:20 12:20 13:50 14:50 15:

第73回 微分同相群のトポロジー Smale予想を巡って 2019 年 3 月 16 日 (土) 10:00 於 東京都 文京区 春日 中央大学理工学部 5 号館 5534 教室 3 月 16 日 (土) 10:00 11:00 11:20 12:20 13:50 14:50 15: 第73回 微分同相群のトポロジー Smale予想を巡って 2019 年 3 月 16 日 (土) 10:00 於 東京都 文京区 春日 1-13-27 中央大学理工学部 5 号館 5534 教室 3 月 16 日 (土) 10:00 11:00 11:20 12:20 13:50 14:50 15:10 16:10 16:30 17:30 17:40 微分同相群は Lie 群 だろうか 4 次元球面の微分同相群

More information

3 de Sitter CMC 1 (Shoichi Fujimori) Department of Mathematics, Kobe University 3 de Sitter S (CMC 1), 1 ( [AA]). 3 H 3 CMC 1 Bryant ([B, UY1]).

3 de Sitter CMC 1 (Shoichi Fujimori) Department of Mathematics, Kobe University 3 de Sitter S (CMC 1), 1 ( [AA]). 3 H 3 CMC 1 Bryant ([B, UY1]). 3 de Sitter CMC 1 (Shoichi Fujimori) Department of Mathematics, Kobe University 3 de Sitter S 3 1 1 (CMC 1), 1 ( [AA]) 3 H 3 CMC 1 Bryant ([B, UY1]) H 3 CMC 1, Bryant ([CHR, RUY1, RUY2, UY1, UY2, UY3,

More information

k + (1/2) S k+(1/2) (Γ 0 (N)) N p Hecke T k+(1/2) (p 2 ) S k+1/2 (Γ 0 (N)) M > 0 2k, M S 2k (Γ 0 (M)) Hecke T 2k (p) (p M) 1.1 ( ). k 2 M N M N f S k+

k + (1/2) S k+(1/2) (Γ 0 (N)) N p Hecke T k+(1/2) (p 2 ) S k+1/2 (Γ 0 (N)) M > 0 2k, M S 2k (Γ 0 (M)) Hecke T 2k (p) (p M) 1.1 ( ). k 2 M N M N f S k+ 1 SL 2 (R) γ(z) = az + b cz + d ( ) a b z h, γ = SL c d 2 (R) h 4 N Γ 0 (N) {( ) } a b Γ 0 (N) = SL c d 2 (Z) c 0 mod N θ(z) θ(z) = q n2 q = e 2π 1z, z h n Z Γ 0 (4) j(γ, z) ( ) a b θ(γ(z)) = j(γ, z)θ(z)

More information

R C Gunning, Lectures on Riemann Surfaces, Princeton Math Notes, Princeton Univ Press 1966,, (4),,, Gunning, Schwarz Schwarz Schwarz, {z; x}, [z; x] =

R C Gunning, Lectures on Riemann Surfaces, Princeton Math Notes, Princeton Univ Press 1966,, (4),,, Gunning, Schwarz Schwarz Schwarz, {z; x}, [z; x] = Schwarz 1, x z = z(x) {z; x} {z; x} = z z 1 2 z z, = d/dx (1) a 0, b {az; x} = {z; x}, {z + b; x} = {z; x} {1/z; x} = {z; x} (2) ad bc 0 a, b, c, d 2 { az + b cz + d ; x } = {z; x} (3) z(x) = (ax + b)/(cx

More information

12 2 e S,T S s S T t T (map) α α : S T s t = α(s) (2.1) S (domain) T (codomain) (target set), {α(s)} T (range) (image) s, s S t T s S

12 2 e S,T S s S T t T (map) α α : S T s t = α(s) (2.1) S (domain) T (codomain) (target set), {α(s)} T (range) (image) s, s S t T s S 12 2 e 2.1 2.1.1 S,T S s S T t T (map α α : S T s t = α(s (2.1 S (domain T (codomain (target set, {α(s} T (range (image 2.1.2 s, s S t T s S t T, α s, s S s s, α(s α(s (2.2 α (injection 4 T t T (coimage

More information

/ n (M1) M (M2) n Λ A = {ϕ λ : U λ R n } λ Λ M (atlas) A (a) {U λ } λ Λ M (open covering) U λ M λ Λ U λ = M (b) λ Λ ϕ λ : U λ ϕ λ (U λ ) R n ϕ

/ n (M1) M (M2) n Λ A = {ϕ λ : U λ R n } λ Λ M (atlas) A (a) {U λ } λ Λ M (open covering) U λ M λ Λ U λ = M (b) λ Λ ϕ λ : U λ ϕ λ (U λ ) R n ϕ 4 4.1 1 2 1 4 2 1 / 2 4.1.1 n (M1) M (M2) n Λ A = {ϕ λ : U λ R n } λ Λ M (atlas) A (a) {U λ } λ Λ M (open covering) U λ M λ Λ U λ = M (b) λ Λ ϕ λ : U λ ϕ λ (U λ ) R n ϕ λ U λ (local chart, local coordinate)

More information

2.3. p(n)x n = n=0 i= x = i x x 2 x 3 x..,?. p(n)x n = + x + 2 x x 3 + x + 7 x + x + n=0, n p(n) x n, ( ). p(n) (mother function)., x i = + xi +

2.3. p(n)x n = n=0 i= x = i x x 2 x 3 x..,?. p(n)x n = + x + 2 x x 3 + x + 7 x + x + n=0, n p(n) x n, ( ). p(n) (mother function)., x i = + xi + ( ) : ( ) n, n., = 2+2+,, = 2 + 2 + = 2 + + 2 = + 2 + 2,,,. ( composition.), λ = (2, 2, )... n (partition), λ = (λ, λ 2,..., λ r ), λ λ 2 λ r > 0, r λ i = n i=. r λ, l(λ)., r λ i = n i=, λ, λ., n P n,

More information

Armstrong culture Web

Armstrong culture Web 2004 5 10 M.A. Armstrong, Groups and Symmetry, Springer-Verlag, NewYork, 1988 (2000) (1989) (2001) (2002) 1 Armstrong culture Web 1 3 1.1................................. 3 1.2.................................

More information

Broadhurst-Kreimer Brown ( D3) 1 Broadhurst-Kreimer Zagier Gangl- -Zagi

Broadhurst-Kreimer Brown ( D3) 1 Broadhurst-Kreimer Zagier Gangl- -Zagi Broadhurst-Kreimer Brown ( D3) 1 Broadhurst-Kreimer 2 2 - -Zagier 5 2.1............................. 5 2.2........................... 8 3 Gangl- -Zagier 11 3.1.................................. 11 3.2

More information

第5章 偏微分方程式の境界値問題

第5章 偏微分方程式の境界値問題 October 5, 2018 1 / 113 4 ( ) 2 / 113 Poisson 5.1 Poisson ( A.7.1) Poisson Poisson 1 (A.6 ) Γ p p N u D Γ D b 5.1.1: = Γ D Γ N 3 / 113 Poisson 5.1.1 d {2, 3} Lipschitz (A.5 ) Γ D Γ N = \ Γ D Γ p Γ N Γ

More information

x 3 a (mod p) ( ). a, b, m Z a b m a b (mod m) a b m 2.2 (Z/mZ). a = {x x a (mod m)} a Z m 0, 1... m 1 Z/mZ = {0, 1... m 1} a + b = a +

x 3 a (mod p) ( ). a, b, m Z a b m a b (mod m) a b m 2.2 (Z/mZ). a = {x x a (mod m)} a Z m 0, 1... m 1 Z/mZ = {0, 1... m 1} a + b = a + 1 1 22 1 x 3 (mod ) 2 2.1 ( )., b, m Z b m b (mod m) b m 2.2 (Z/mZ). = {x x (mod m)} Z m 0, 1... m 1 Z/mZ = {0, 1... m 1} + b = + b, b = b Z/mZ 1 1 Z Q R Z/Z 2.3 ( ). m {x 0, x 1,..., x m 1 } modm 2.4

More information

C:/yokoyama/book/fubook/tft/tft5h/tft5h.dvi

C:/yokoyama/book/fubook/tft/tft5h/tft5h.dvi 1997.10 2000 10 15 ver. 2.72 i 1 1 1.1 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 1.2 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 1.3 :

More information

, CH n. CH n, CP n,,,., CH n,,. RH n ( Cartan )., CH n., RH n CH n,,., RH n, CH n., RH n ( ), CH n ( 1.1 (v), (vi) )., RH n,, CH n,., CH n,. 1.2, CH n

, CH n. CH n, CP n,,,., CH n,,. RH n ( Cartan )., CH n., RH n CH n,,., RH n, CH n., RH n ( ), CH n ( 1.1 (v), (vi) )., RH n,, CH n,., CH n,. 1.2, CH n ( ), Jürgen Berndt,.,. 1, CH n.,,. 1.1 ([6]). CH n (n 2), : (i) CH k (k = 0,..., n 1) tube. (ii) RH n tube. (iii). (iv) ruled minimal, equidistant. (v) normally homogeneous submanifold F k tube. (vi) normally

More information

(check matrices and minimum distances) H : a check matrix of C the minimum distance d = (the minimum # of column vectors of H which are linearly depen

(check matrices and minimum distances) H : a check matrix of C the minimum distance d = (the minimum # of column vectors of H which are linearly depen Hamming (Hamming codes) c 1 # of the lines in F q c through the origin n = qc 1 q 1 Choose a direction vector h i for each line. No two vectors are colinear. A linearly dependent system of h i s consists

More information

July 28, H H 0 H int = H H 0 H int = H int (x)d 3 x Schrödinger Picture Ψ(t) S =e iht Ψ H O S Heisenberg Picture Ψ H O H (t) =e iht O S e i

July 28, H H 0 H int = H H 0 H int = H int (x)d 3 x Schrödinger Picture Ψ(t) S =e iht Ψ H O S Heisenberg Picture Ψ H O H (t) =e iht O S e i July 8, 4. H H H int H H H int H int (x)d 3 x Schrödinger Picture Ψ(t) S e iht Ψ H O S Heisenberg Picture Ψ H O H (t) e iht O S e iht Interaction Picture Ψ(t) D e iht Ψ(t) S O D (t) e iht O S e ih t (Dirac

More information

2.1 H f 3, SL(2, Z) Γ k (1) f H (2) γ Γ f k γ = f (3) f Γ \H cusp γ SL(2, Z) f k γ Fourier f k γ = a γ (n)e 2πinz/N n=0 (3) γ SL(2, Z) a γ (0) = 0 f c

2.1 H f 3, SL(2, Z) Γ k (1) f H (2) γ Γ f k γ = f (3) f Γ \H cusp γ SL(2, Z) f k γ Fourier f k γ = a γ (n)e 2πinz/N n=0 (3) γ SL(2, Z) a γ (0) = 0 f c GL 2 1 Lie SL(2, R) GL(2, A) Gelbart [Ge] 1 3 [Ge] Jacquet-Langlands [JL] Bump [Bu] Borel([Bo]) ([Ko]) ([Mo]) [Mo] 2 2.1 H = {z C Im(z) > 0} Γ SL(2, Z) Γ N N Γ (N) = {γ SL(2, Z) γ = 1 2 mod N} g SL(2,

More information

tomocci ,. :,,,, Lie,,,, Einstein, Newton. 1 M n C. s, M p. M f, p d ds f = dxµ p ds µ f p, X p = X µ µ p = dxµ ds µ p. µ, X µ.,. p,. T M p.

tomocci ,. :,,,, Lie,,,, Einstein, Newton. 1 M n C. s, M p. M f, p d ds f = dxµ p ds µ f p, X p = X µ µ p = dxµ ds µ p. µ, X µ.,. p,. T M p. tomocci 18 7 5...,. :,,,, Lie,,,, Einstein, Newton. 1 M n C. s, M p. M f, p d ds f = dxµ p ds µ f p, X p = X µ µ p = dxµ ds µ p. µ, X µ.,. p,. T M p. M F (M), X(F (M)).. T M p e i = e µ i µ. a a = a i

More information

O x y z O ( O ) O (O ) 3 x y z O O x v t = t = 0 ( 1 ) O t = 0 c t r = ct P (x, y, z) r 2 = x 2 + y 2 + z 2 (t, x, y, z) (ct) 2 x 2 y 2 z 2 = 0

O x y z O ( O ) O (O ) 3 x y z O O x v t = t = 0 ( 1 ) O t = 0 c t r = ct P (x, y, z) r 2 = x 2 + y 2 + z 2 (t, x, y, z) (ct) 2 x 2 y 2 z 2 = 0 9 O y O ( O ) O (O ) 3 y O O v t = t = 0 ( ) O t = 0 t r = t P (, y, ) r = + y + (t,, y, ) (t) y = 0 () ( )O O t (t ) y = 0 () (t) y = (t ) y = 0 (3) O O v O O v O O O y y O O v P(, y,, t) t (, y,, t )

More information

( ),.,,., C A (2008, ). 1,, (M, g) (Riemannian symmetric space), : p M, s p : M M :.,.,.,, (, ).,, (M, g) p M, s p : M M p, : (1) p s p, (

( ),.,,., C A (2008, ). 1,, (M, g) (Riemannian symmetric space), : p M, s p : M M :.,.,.,, (, ).,, (M, g) p M, s p : M M p, : (1) p s p, ( ( ),.,,., C A (2008, ). 1,,. 1.1. (M, g) (Riemannian symmetric space), : p M, s p : M M :.,.,.,, (, ).,,. 1.2. (M, g) p M, s p : M M p, : (1) p s p, (2) s 2 p = id ( id ), (3) s p ( )., p ( s p (p) = p),,

More information

xia2.dvi

xia2.dvi Journal of Differential Equations 96 (992), 70-84 Melnikov method and transversal homoclinic points in the restricted three-body problem Zhihong Xia Department of Mathematics, Harvard University Cambridge,

More information

Dynkin Serre Weyl

Dynkin Serre Weyl Dynkin Naoya Enomoto 2003.3. paper Dynkin Introduction Dynkin Lie Lie paper 1 0 Introduction 3 I ( ) Lie Dynkin 4 1 ( ) Lie 4 1.1 Lie ( )................................ 4 1.2 Killing form...........................................

More information

Mazur [Ma1] Schlessinger [Sch] [SL] [Ma1] [Ma1] [Ma2] Galois [] 17 R m R R R M End R M) M R ut R M) M R R G R[G] R G Sets 1 Λ Noether Λ k Λ m Λ k C Λ

Mazur [Ma1] Schlessinger [Sch] [SL] [Ma1] [Ma1] [Ma2] Galois [] 17 R m R R R M End R M) M R ut R M) M R R G R[G] R G Sets 1 Λ Noether Λ k Λ m Λ k C Λ Galois ) 0 1 1 2 2 4 3 10 4 12 5 14 16 0 Galois Galois Galois TaylorWiles Fermat [W][TW] Galois Galois Galois 1 Noether 2 1 Mazur [Ma1] Schlessinger [Sch] [SL] [Ma1] [Ma1] [Ma2] Galois [] 17 R m R R R

More information

Macdonald, ,,, Macdonald. Macdonald,,,,,.,, Gauss,,.,, Lauricella A, B, C, D, Gelfand, A,., Heckman Opdam.,,,.,,., intersection,. Macdona

Macdonald, ,,, Macdonald. Macdonald,,,,,.,, Gauss,,.,, Lauricella A, B, C, D, Gelfand, A,., Heckman Opdam.,,,.,,., intersection,. Macdona Macdonald, 2015.9.1 9.2.,,, Macdonald. Macdonald,,,,,.,, Gauss,,.,, Lauricella A, B, C, D, Gelfand, A,., Heckman Opdam.,,,.,,., intersection,. Macdonald,, q., Heckman Opdam q,, Macdonald., 1 ,,. Macdonald,

More information

sakigake1.dvi

sakigake1.dvi (Zin ARAI) arai@cris.hokudai.ac.jp http://www.cris.hokudai.ac.jp/arai/ 1 dynamical systems ( mechanics ) dynamical systems 3 G X Ψ:G X X, (g, x) Ψ(g, x) =:Ψ g (x) Ψ id (x) =x, Ψ gh (x) =Ψ h (Ψ g (x)) (

More information

コホモロジー的AGT対応とK群類似

コホモロジー的AGT対応とK群類似 AGT K ( ) Encounter with Mathematics October 29, 2016 AGT L. F. Alday, D. Gaiotto, Y. Tachikawa, Liouville Correlation Functions from Four-dimensional Gauge Theories, Lett. Math. Phys. 91 (2010), arxiv:0906.3219.

More information

( ) Lemma 2.2. X ultra filter (1) X = X 1 X 2 X 1 X 2 (2) X = X 1 X 2 X 3... X N X 1, X 2,..., X N (3) disjoint union X j Definition 2.3. X ultra filt

( ) Lemma 2.2. X ultra filter (1) X = X 1 X 2 X 1 X 2 (2) X = X 1 X 2 X 3... X N X 1, X 2,..., X N (3) disjoint union X j Definition 2.3. X ultra filt NON COMMTATIVE ALGEBRAIC SPACE OF FINITE ARITHMETIC TYPE ( ) 1. Introduction (1) (2) universality C ( ) R (1) (2) ultra filter 0 (1) (1) ( ) (2) (2) (3) 2. ultra filter Definition 2.1. X F filter (1) F

More information

Donaldson Seiberg-Witten [GNY] f U U C 1 f(z)dz = Res f(a) 2πi C a U U α = f(z)dz dα = 0 U f U U P 1 α 0 a P 1 Res a α = 0. P 1 Donaldson Seib

Donaldson Seiberg-Witten [GNY] f U U C 1 f(z)dz = Res f(a) 2πi C a U U α = f(z)dz dα = 0 U f U U P 1 α 0 a P 1 Res a α = 0. P 1 Donaldson Seib ( ) Donaldson Seiberg-Witten Witten Göttsche [GNY] L. Göttsche, H. Nakajima and K. Yoshioka, Donaldson = Seiberg-Witten from Mochizuki s formula and instanton counting, Publ. of RIMS, to appear Donaldson

More information

四変数基本対称式の解放

四変数基本対称式の解放 Solving the simultaneous equation of the symmetric tetravariate polynomials and The roots of a quartic equation Oomori, Yasuhiro in Himeji City, Japan Dec.1, 2011 Abstract 1. S 4 2. 1. {α, β, γ, δ} (1)

More information

( Goo Ishikawa, Go-o Ishikawa) Department of Mathematics, Hokkaido University, Sapporo , JAPAN. 1 (?)

( Goo Ishikawa, Go-o Ishikawa) Department of Mathematics, Hokkaido University, Sapporo , JAPAN.   1 (?) ( Goo Ishikawa, Go-o Ishikawa) Department of Mathematics, Hokkaido University, Sapporo 060-0810, JAPAN. E-mail: ishikawa@math.sci.hokudai.ac.jp 1 (?) (ray) (wave front) ray (reflection) (refraction) (diffraction)

More information

1 Affine Lie 1.1 Affine Lie g Lie, 2h A B = tr g ad A ad B A, B g Killig form., h g daul Coxeter number., g = sl n C h = n., g long root 2 2., ρ half

1 Affine Lie 1.1 Affine Lie g Lie, 2h A B = tr g ad A ad B A, B g Killig form., h g daul Coxeter number., g = sl n C h = n., g long root 2 2., ρ half Wess-Zumino-Witten 1999 3 18 Wess-Zumino-Witten., Knizhnik-Zamolodchikov-Bernard,,. 1 Affine Lie 2 1.1 Affine Lie.............................. 2 1.2..................................... 3 2 WZW 4 3 Knizhnik-Zamolodchikov-Bernard

More information

第1章 微分方程式と近似解法

第1章 微分方程式と近似解法 April 12, 2018 1 / 52 1.1 ( ) 2 / 52 1.2 1.1 1.1: 3 / 52 1.3 Poisson Poisson Poisson 1 d {2, 3} 4 / 52 1 1.3.1 1 u,b b(t,x) u(t,x) x=0 1.1: 1 a x=l 1.1 1 (0, t T ) (0, l) 1 a b : (0, t T ) (0, l) R, u

More information

JKR Point loading of an elastic half-space 2 3 Pressure applied to a circular region Boussinesq, n =

JKR Point loading of an elastic half-space 2 3 Pressure applied to a circular region Boussinesq, n = JKR 17 9 15 1 Point loading of an elastic half-space Pressure applied to a circular region 4.1 Boussinesq, n = 1.............................. 4. Hertz, n = 1.................................. 6 4 Hertz

More information

1. A0 A B A0 A : A1,...,A5 B : B1,...,B

1. A0 A B A0 A : A1,...,A5 B : B1,...,B 1. A0 A B A0 A : A1,...,A5 B : B1,...,B12 2. 3. 4. 5. A0 A, B Z Z m, n Z m n m, n A m, n B m=n (1) A, B (2) A B = A B = Z/ π : Z Z/ (3) A B Z/ (4) Z/ A, B (5) f : Z Z f(n) = n f = g π g : Z/ Z A, B (6)

More information

1 1. R 2n non-kähler complex structure n = 2 n = 1 complex curve Kähler No n 3 Calabi Eckmann Yes 1953 2 complex structure 2 Hopf h p : S 2p+1 CP p, h

1 1. R 2n non-kähler complex structure n = 2 n = 1 complex curve Kähler No n 3 Calabi Eckmann Yes 1953 2 complex structure 2 Hopf h p : S 2p+1 CP p, h Non-Kähler complex structures on R 4 ( ) 1. Antonio J. Di Scala (Politecnico di Torino), Daniele Zuddas (KIAS) [1] R 4 non-kähler complex surfaces Kähler 1. (M, J) complex manifold M complex structure

More information

March 4, R R R- R R

March 4, R R R- R R March 4, 2016 1. R- 2 1.1. R- 2 1.2. R- R- 4 1.3. R- 5 2. 6 2.1. 6 2.2. 6 2.3. 6 2.4. 7 3. 8 3.1. 8 3.2. 8 4. 10 4.1. 10 4.2. 10 4.3. 10 5. 12 5.1. 12 5.2. 14 6. Hom 14 6.1. Hom 14 6.2. Hom 15 6.3. Hom

More information

2

2 III ( Dirac ) ( ) ( ) 2001. 9.22 2 1 2 1.1... 3 1.2... 3 1.3 G P... 5 2 5 2.1... 6 2.2... 6 2.3 G P... 7 2.4... 7 3 8 3.1... 8 3.2... 9 3.3... 10 3.4... 11 3.5... 12 4 Dirac 13 4.1 Spin... 13 4.2 Spin

More information

2005 2006.2.22-1 - 1 Fig. 1 2005 2006.2.22-2 - Element-Free Galerkin Method (EFGM) Meshless Local Petrov-Galerkin Method (MLPGM) 2005 2006.2.22-3 - 2 MLS u h (x) 1 p T (x) = [1, x, y]. (1) φ(x) 0.5 φ(x)

More information

2 R U, U Hausdorff, R. R. S R = (S, A) (closed), (open). (complete projective smooth algebraic curve) (cf. 2). 1., ( ).,. countable ( 2 ) ,,.,,

2 R U, U Hausdorff, R. R. S R = (S, A) (closed), (open). (complete projective smooth algebraic curve) (cf. 2). 1., ( ).,. countable ( 2 ) ,,.,, 15, pp.1-13 1 1.1,. 1.1. C ( ) f = u + iv, (, u, v f ). 1 1. f f x = i f x u x = v y, u y = v x.., u, v u = v = 0 (, f = 2 f x + 2 f )., 2 y2 u = 0. u, u. 1,. 1.2. S, A S. (i) A φ S U φ C. (ii) φ A U φ

More information

2018 : msjmeeting-2018sep-11i001 WKB ( ) Eynard-Orantin WKB.,, Schrödinger WKB Voros, Painlevé (τ- ). 1. WKB,, WKB Voros WKB, Painlevé WKB. WKB, [ ],.

2018 : msjmeeting-2018sep-11i001 WKB ( ) Eynard-Orantin WKB.,, Schrödinger WKB Voros, Painlevé (τ- ). 1. WKB,, WKB Voros WKB, Painlevé WKB. WKB, [ ],. 018 : msjmeeting-018sep-11i001 WKB ( Eynard-Orantin WKB.,, Schrödinger WKB Voros, Painlevé (τ-. 1. WKB,, WKB Voros WKB, Painlevé WKB. WKB, [ ],. 1.1. WKB Voros Voros ([V] WKB (exact WKB analysis, ( 1 Schrödinger

More information

IA 2013 : :10722 : 2 : :2 :761 :1 (23-27) : : ( / ) (1 /, ) / e.g. (Taylar ) e x = 1 + x + x xn n! +... sin x = x x3 6 + x5 x2n+1 + (

IA 2013 : :10722 : 2 : :2 :761 :1 (23-27) : : ( / ) (1 /, ) / e.g. (Taylar ) e x = 1 + x + x xn n! +... sin x = x x3 6 + x5 x2n+1 + ( IA 2013 : :10722 : 2 : :2 :761 :1 23-27) : : 1 1.1 / ) 1 /, ) / e.g. Taylar ) e x = 1 + x + x2 2 +... + xn n! +... sin x = x x3 6 + x5 x2n+1 + 1)n 5! 2n + 1)! 2 2.1 = 1 e.g. 0 = 0.00..., π = 3.14..., 1

More information

平成 15 年度 ( 第 25 回 ) 数学入門公開講座テキスト ( 京都大学数理解析研究所, 平成 ~8 15 月年 78 日開催月 4 日 ) X 2 = 1 ( ) f 1 (X 1,..., X n ) = 0,..., f r (X 1,..., X n ) = 0 X = (

平成 15 年度 ( 第 25 回 ) 数学入門公開講座テキスト ( 京都大学数理解析研究所, 平成 ~8 15 月年 78 日開催月 4 日 ) X 2 = 1 ( ) f 1 (X 1,..., X n ) = 0,..., f r (X 1,..., X n ) = 0 X = ( 1 1.1 X 2 = 1 ( ) f 1 (X 1,..., X n ) = 0,..., f r (X 1,..., X n ) = 0 X = (X 1,..., X n ) ( ) X 1,..., X n f 1,..., f r A T X + XA XBR 1 B T X + C T QC = O X 1.2 X 1,..., X n X i X j X j X i = 0, P i

More information

untitled

untitled 18 18 8 17 18 8 19 3. II 3-8 18 9:00~10:30? 3 30 3 a b a x n nx n-1 x n n+1 x / n+1 log log = logos + arithmos n+1 x / n+1 incompleteness theorem log b = = rosário Euclid Maya-glyph quipe 9 number digits

More information

Siegel modular forms of middle parahoric subgroups and Ihara lift ( Tomoyoshi Ibukiyama Osaka University 1. Introduction [8] Ihara Sp(2, R) p

Siegel modular forms of middle parahoric subgroups and Ihara lift ( Tomoyoshi Ibukiyama Osaka University 1. Introduction [8] Ihara Sp(2, R) p Siegel modular forms of middle parahoric subgroups and Ihara lift ( Tomoyoshi Ibukiyama Osaka University 1. Introduction [8] Ihara 80 1963 Sp(2, R) p L holomorphic discrete series Eichler Brandt Eichler

More information

On a branched Zp-cover of Q-homology 3-spheres

On a branched Zp-cover of Q-homology 3-spheres Zp 拡大と分岐 Zp 被覆 GL1 表現の変形理論としての岩澤理論 SL2 表現の変形理論 On a branched Zp -cover of Q-homology 3-spheres 植木 潤 九州大学大学院数理学府 D2 December 23, 2014 植木 潤 九州大学大学院数理学府 D2 On a branched Zp -cover of Q-homology 3-spheres

More information

第 61 回トポロジーシンポジウム講演集 2014 年 7 月於東北大学 ( ) 1 ( ) [6],[7] J.W. Alexander 3 1 : t 2 t +1=0 4 1 : t 2 3t +1=0 8 2 : 1 3t +3t 2 3t 3 +3t 4 3t 5 + t

第 61 回トポロジーシンポジウム講演集 2014 年 7 月於東北大学 ( ) 1 ( ) [6],[7] J.W. Alexander 3 1 : t 2 t +1=0 4 1 : t 2 3t +1=0 8 2 : 1 3t +3t 2 3t 3 +3t 4 3t 5 + t ( ) 1 ( ) [6],[7] 1. 1928 J.W. Alexander 3 1 : t 2 t +1=0 4 1 : t 2 3t +1=0 8 2 : 1 3t +3t 2 3t 3 +3t 4 3t 5 + t 6 7 7 : 1 5t +9t 2 5t 3 + t 4 ( :25400086) 2010 Mathematics Subject Classification: 57M25,

More information

研修コーナー

研修コーナー l l l l l l l l l l l α α β l µ l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l

More information

D 24 D D D

D 24 D D D 5 Paper I.R. 2001 5 Paper HP Paper 5 3 5.1................................................... 3 5.2.................................................... 4 5.3.......................................... 6

More information

2 CAD : CAD 7

2 CAD : CAD 7 1 CAD 2017.6.25 2 CAD 2 3 1998 1 0 6 : CAD 7 3 CAD 2017 6 4 0 7 0.1 1............................. 7 0.2 2............................. 8 0.3 3............................ 9 0.4 4............................

More information

211 kotaro@math.titech.ac.jp 1 R *1 n n R n *2 R n = {(x 1,..., x n ) x 1,..., x n R}. R R 2 R 3 R n R n R n D D R n *3 ) (x 1,..., x n ) f(x 1,..., x n ) f D *4 n 2 n = 1 ( ) 1 f D R n f : D R 1.1. (x,

More information

¿ô³Ø³Ø½øÏÀ¥Î¡¼¥È

¿ô³Ø³Ø½øÏÀ¥Î¡¼¥È 2011 i N Z Q R C A def B, A B. ii..,.,.. (, ), ( ),.?????????,. iii 04-13 04-20 04-27 05-04 [ ] 05-11 05-18 05-25 06-01 06-08 06-15 06-22 06-29 07-06 07-13 07-20 07-27 08-03 10-05 10-12 10-19 [ ] 10-26

More information

A11 (1993,1994) 29 A12 (1994) 29 A13 Trefethen and Bau Numerical Linear Algebra (1997) 29 A14 (1999) 30 A15 (2003) 30 A16 (2004) 30 A17 (2007) 30 A18

A11 (1993,1994) 29 A12 (1994) 29 A13 Trefethen and Bau Numerical Linear Algebra (1997) 29 A14 (1999) 30 A15 (2003) 30 A16 (2004) 30 A17 (2007) 30 A18 2013 8 29y, 2016 10 29 1 2 2 Jordan 3 21 3 3 Jordan (1) 3 31 Jordan 4 32 Jordan 4 33 Jordan 6 34 Jordan 8 35 9 4 Jordan (2) 10 41 x 11 42 x 12 43 16 44 19 441 19 442 20 443 25 45 25 5 Jordan 26 A 26 A1

More information

[2, 3, 4, 5] * C s (a m k (symmetry operation E m[ 1(a ] σ m σ (symmetry element E σ {E, σ} C s 32 ( ( =, 2 =, (3 0 1 v = x 1 1 +

[2, 3, 4, 5] * C s (a m k (symmetry operation E m[ 1(a ] σ m σ (symmetry element E σ {E, σ} C s 32 ( ( =, 2 =, (3 0 1 v = x 1 1 + 2016 12 16 1 1 2 2 2.1 C s................. 2 2.2 C 3v................ 9 3 11 3.1.............. 11 3.2 32............... 12 3.3.............. 13 4 14 4.1........... 14 4.2................ 15 4.3................

More information

日本数学会・2011年度年会(早稲田大学)・総合講演

日本数学会・2011年度年会(早稲田大学)・総合講演 日本数学会 2011 年度年会 ( 早稲田大学 ) 総合講演 2011 年度日本数学会春季賞受賞記念講演 MSJMEETING-2011-0 ( ) 1. p>0 p C ( ) p p 0 smooth l (l p ) p p André, Christol, Mebkhout, Kedlaya K 0 O K K k O K k p>0 K K : K R 0 p = p 1 Γ := K k

More information

1.2 (Kleppe, cf. [6]). C S 3 P 3 3 S 3. χ(p 3, I C (3)) 1 C, C P 3 ( ) 3 S 3( S 3 S 3 ). V 3 del Pezzo (cf. 2.1), S V, del Pezzo 1.1, V 3 del Pe

1.2 (Kleppe, cf. [6]). C S 3 P 3 3 S 3. χ(p 3, I C (3)) 1 C, C P 3 ( ) 3 S 3( S 3 S 3 ). V 3 del Pezzo (cf. 2.1), S V, del Pezzo 1.1, V 3 del Pe 3 del Pezzo (Hirokazu Nasu) 1 [10]. 3 V C C, V Hilbert scheme Hilb V [C]. C V C S V S. C S S V, C V. Hilbert schemes Hilb V Hilb S [S] [C] ( χ(s, N S/V ) χ(c, N C/S )), Hilb V [C] (generically non-reduced)

More information