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1 II :00~10:30? a b a

2 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

3 = = rosário

4 Euclid Maya-glyph quipe 9

6 number digits = 10 = 0,1,2,3,4,5,6,7,8, m 43 = 2-1 0' = 1, 0'' = 1' = 2, 2' = 3, 3' = 4, 0 = +, 1 =, 2 = ^, a 0 0 = a a 0 b'= a 0 b ' a 0' 0 = 0 a c'' 0 = 0' = 1 a c' b'= a c a c' b = 64, = 2 = 65536, = 3 = , -,, 0 field, Körper, corps 1. axiom system a+b=b+a ab=ba a+ b+c = a+b +c a bc = ab c a b+c = ab+ac a+0=a a1=a a+ -a = 0 a 0 aa -1 = 1 432/1001, rational number Q 619 algebtraic number A, π = real number R i = -1 C = a+b i : a,b R complex number C 2 i = j = k =-1, ij =- ji = k, jk =- kj = i, ki =- ik = j H = a+b i+c j+dkquaternion H ij = - ji = k ab = ba O = p+q l :p,q H *1 Cayley octanion O O H q = a+b i+c j+dk conjugate q = a-bi-cj-dk p+ql r+s l = pr -sq + ps+qr l p,q H p q l = pq l, pl q = pq l, pl q l = - qp *1

7 l ij = lk = - kl, li j = - il j = ij l = kl a bc = ab c 0 division algebra, skew field O n finite field p 2 3 2,3,4=2,5,6=2 3,7,8=2,9=3,10=2 5,11,12=2 3,13, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 25, 29, 31, p= p= n=4=2 2 F 2 = 2 = 0,1 irreducible 2 x +x+1 0,1,x,x = x = x+1 x+1 = x +2x+1 = x +1+ x +x+1 = x = 2 x+1 x+x+1 = x x=2

8 p= =7=2,3 4=12=2 5 p= =9=2,3 6=18=4 p=7 3 n=8=2 3 F 2 = 2 = 0,1 irreducible 3 x +x+1 8 = 0,1,2,3,4,5,6,7 = 0, 1, x, x+1, x, x +1, x +x, x +x+1 2 x = x+1 + x+1 =x+x+2=x+x= = x+1 x +1 = x +x +x+1+ x +x+1 = 2x +x +2x+2 = x = 4 2. xy 3 e a -1

9 -1-1 a bc = ab c, ae = ea = a, aa = a a = e ab = ba commutaive = abelian group x x -1 +y y -1 = x+y x -yx+y -1 = 0 2x x -1 +y y -1 = 0 3 C: 2x x -1 +y y -1 = 0 9 1,-1, 1,0, 1,1, 0,-1, 0,0, 0,1, -1,-1, -1,0, -1,1 O -1,1 + P, Q C P Q C P,Q PQ O PQ O PQ P Q P+Q P+Q = O PQ 3 C O Poincaré-Mordell P -1,0, Q 0,0, R 0,-1 P+Q +R = P+ Q+R PQ y=0pq= 1,0 O -1,1 x= 1-2y 2x x -1 +y y -1 = -3y y-1 5y-3 0,1 y =3/5,x=-1/5 P+Q= -1/5,3/5 R P+Q 0,-1-1/5,3/5y+8x+1 = 0 y = -8x-1 2x x-1 +y y-1 = 6x 5x+1 17x+3

10 x = -3/17, y = 7/17 R P+Q = -3/17, 7/17-1,1-3/17, 7/17-7y-5x+2 = 0 x = -7/5 y+2/5 2x x -1 +y y -1 = -3/125 y-1 17y-7 11y+4 y=-4/11,x=10/11 R+ P+Q = 10/11, -4/11 QR = 0,1 Q+R = 1,1 P Q+R x -2y +1 = 0 x = 2y-1 2x x-1 +y y-1 = y y-1 17y-7 y = 7/17, x = -3/17 P Q+R = P+Q R = -3/17, 7/17 P+ Q+R = P+Q +R = 10/11, -4/ x-2y+1 = 0 x = 0 x+2y-1 = 0 line Q+R 1,1 QR 0,1 O = -1,1 y = 1 P -1,0 Q 0,0 PQ 1,0 y = 0 P Q+R, P+Q R R 0,-1 P+Q -1/5,3/5 y+8x+1 = 0 3 A, B A, B 9 x x-2y+1 x+2y-1 = 0 y y-1 y+8x+1 = 0 C: 2x x -1 +y y -1 = 0 9 O, P, Q, R, PQ, QR, Q+R, P+Q, P Q+R = P+Q R 3 3 2:1 2x x-1 +y y-1 = 2x x-2y+1 x+2y-1 +y y-1 y+8x y = x+ax+b Weierstrass normal form 2x x -1 +y y -1 = 0 s = 1/2 4x-y-1 / 2x+1+y, t = -3/4 4x+3y +2y-1 / 2x+1+y 2 3 t = s-3/4 s+5/16

11 3. top, toupie solid body xy 3 = J. F. Lagrange, f x 2 y = f x hyper-elliptic curve Sof'ya, V. Kovalevskaya, Kovalevskaya gyroscope Riemann's ζ-function ζ 1/2+ it t = -5 ~ 40 critical-line ζ 1/2+t i :t=-5~40 1 violin Riemann's hypothesis, RH

12 0 1/2+ it t = RH counter example self-conjugate operator unitary operator quest ζ 1/2+ it t = -5 ~ 40 congruence ζ 2 5 y = x+5x+5 = f x f1 u = f x x x+u, f2 u,v =f x x x+ux+v, a p = x p f1 x /p, b p = x, y p f1 x,y /p p plot 4 3 x+ax+bx+pax+p p p p =0 p = 3 ~ sin θ + sin 2θ -sin θ Sato-Tait sin -conjecture

13 2 sin θ - C: 2 3 y = x +ax+b order ±1/4 a12 z = z P p/6 z 2 cyclic involution = = = = 0 identity E A 2 A = E

14 p = 47 = a12 x = x 18x -11x +5x-11 5 n n 23 a12 x = n p-1a12 5 x x -1-3,5,-9,-15,-14,-14,17,2,6,3,-8,15,4,-13,9,2,-7,-18,7,-8,-2,-9,3 4. a x+y+10xy+1=0 O P, Q, R b. 9 = F 3 =3= 0,1,2 2 x+1 F 9 = 0, 1, 2, x, x+1, x+2, 2x, 2x+1, 2x+2 x=3 4+5 = x+1 + x+2 = 2x+3 = 2x = = x+1 x+2 = x +3x+2- x +1 = 3x+1 = 1

15 g g g g =

all.dvi

all.dvi 5,, Euclid.,..,... Euclid,.,.,, e i (i =,, ). 6 x a x e e e x.:,,. a,,. a a = a e + a e + a e = {e, e, e } a (.) = a i e i = a i e i (.) i= {a,a,a } T ( T ),.,,,,. (.),.,...,,. a 0 0 a = a 0 + a + a 0