... \h;,VI ~S-t~ f"'~

... i.. ( Ivv\:.e-n(Y() th...<..Yi o/>2-t~ pt'~~

Jo ( 11') = ~ \<;711'1 ~----'> wiA-V>

D

~ '1Y 4 "f CWl Jl (th..:. vi ~-&~)

1r= '} ().'¥'! o D ( ~ v~)

r \-v) =- 0,A. "(~) \\~I'{_,, ~-ee.. ~'I

~p.ec;t-e.J h k o{- eaJJ_W1..12M.-~~ ,;;_

.. Eukx-L~~:

) l:::."IJ"=O l.V\ Dj:

L '1Y41.f) 0J>'+1Y + ~d.,,-1)" ~ 0) (-v--\{) (-a)l .. -v- -+-~d)l-v') =. 0 o-n A

(Nd to k ~s-6-oJ ~~) ~~iy ~.

•• 1 ::: ,0 ~~:o 1 0r-~fd,,y :o O¥ ~ Sv~~n...: ~.

Nole.,: Wb .)..{ ~+. ~ u:- Cs .vv.e..v. ""· r. -t A t~

i...V1-t-vrc:o-Y ~~ <>td:. pra.U...wi C:s ~v~-t -to >t.~~ pr~

Arshak PetrosyanLecture notes for a mini-course given at IMPA July 6–17, 2015

·t3.E:-")(~ -to keep_~~~~-

..

. '

. '

. '

. J..t :=. \')<., =-o'J / 'R = 0 / ~ (,c.'J->C~) = ~(')(~ >C.,)

c.t. E. C 1 ' 112

( s: U IS~ )

\kt u. E: ( 0 'i(B1J 1

O-lh.vr ?~~Oo.fil.s:

~ = Re. ( "'"-1 r ~ IJ<..,\) ic.

K= 3h,)a.)-=i-/2)1../) ... / ~wi-:-±» .'.2..W\.J:·· "'wiEN

. , 4) Aei&~+~O'Y\ : S~--~ ~-es

D • D {\,~c.et.l_ uft

• 1 • J-L s~-pvr~ ~

• ~-ts >wvdivr ~ tQ ~ (s~-tt)

• ~s. ~~(skies] • f

• t Att. =- 0 ~VI D ~ ~ l..111 ~J;_'l'~ONI ('~~tr.~~ ~J

• 1 · ~ '-6-ps wb pr~ycuy-~

lvlct,-t h.e.wt,c:vh> .... PD · • CW\ A •I V-<--V"~

~ -W> ~ ~ !~ =o (n-o ~~) l !.\, ~ \I( ==> rdl(.. ,: - J_ (1.\,-10)

. ' ·~)) f "\

5_ R.~~±t ± ~~~UK"S.. ... 5, \ PrO:f?%-ti0'11 Le,t ~ &- o... s~cwi ~ ~~~ ~

w\,.\:;h S~h vll. ~ ~. lb 1.1EC 1'at(DUA) ¥s~ ol>o

J.{,={x.,.:o~f\1?:> 1 =i,' J D-= B1 >'f=O 1 ~(><1J-'1<.,)-3:(x~~..) .A1-t,..= 0 ~VI B :-

. )

I

(..(... >,.. 0) - I{><.., ~ 0 ) l(,.l(_ )( ... = 0 °"" B.~

\ \ -q i.t,'lt~\ 2- J ')( .:S. .~ \. LL~ ) \ ')4-)(0\"-:t f' .) "\'.. B~\'4•) 13~.si ()(o) '\ %_,,(')(.")

• L~k~ ~~ t-~-h-/

• B~ cwi o-e,~e-Y"V~CJ.Y\ ~ u;1e et.-'\"~ sL<1,~c.

• ~~CUt-S ~ ~ t ~l;,"=f .~ ·- I~'t;l.:t€.- ~ ~ €::: W 2 ' :i.. ( B~.,)

.. Pro-o+- ~ Pr-o--p s. i . w~ ....,~a sk..o-w I.\.. E. C11 .. ( E>~2 u ~v:i.) .1.) ~ ~h.- '"'7o sh...ow' IL)I" E Co(. ( B',,~)

2) £~h.. +o skow

I 0 r/ J 0 < S < ""7; ) X E- D112

Le..-t. 1(\c.)( ') :=: ~ \(J~~\:t

~ \ ')(\ n-:i.

~s )

5.> l~ 3 ~E (0,1)

1. (~)Cs>) ~ e 1Y'\ :if)

1.(j>(~) ~e1.Cj>(1 9) ~oJl, j=-iJlr·:.i~-1

.. Pr~ tl ~ ?.~ _y.Je., w~a s-h-~ 4-~e. +ke_ ~~L.~ I

lvl41\- =- 0 OM ___ B.\ I t

Ca._.se. i.. P~-b:.D-n o1r- t.t..= o t.,"' B>:ty" Bs ~s k~

\\u.=-01; n ('?>~9" B~)j ~ { \B~g'\B~\

W~-\:h, L~ S-.1 -t:~s ~v.e.-$

1 c3) ( ~) :s C,, ( 1°) (.;i_s-) - lm(§')

IC)\~)~ C.,, 1v)(:i..§') c. + 1

.,I.\:~~ w-e- o{,~V\ ~ ~ o<y < ~

~~

1..(1'1)(~) ~ c 3 J. o-r

10)(~) ~ C~"' 4o-r ovll. J ~i.) ... ,, k-1

.. 6. Oe-b.:~ ~~~±y

._, "6.i. Th.wr..e-wi... Le-1: u- -6..e. ~ s~+-~ ct- Si~~~ ~

.,w~-t\.i s~h A. ~'f. Ike.vi t.(.,t ({' 112 (DU...4)

~ CA..Se.:

.;. 5 AIA..-=O ~"' B~

• +

..

• +

l IA-"?-- o ) - U..x~ ~ 0 1 U..!.t.;<., ::::::- o Cl-YI .. B/

)

I

1 ~N(<')= .L + J2i9 -cir I l)(l")

( t.t.s<. R-e.lL:c-h's l~-tlty ~

( x · -q u.) Au..=- o )

_ J_ + n-Q. _ n-\ -+ r r ('

. . i..) o~~-€..Y'V-€..

Y- H ' ( r) = vt.- \ ~ 'J.., N (I() H (r)

.!_ 1 ~~ \ilr) ~ n-t-~

.av--~ r

.. 2-> NONI/) Lt.± o.-v--e. $~ ~c. ( u., .±. ~ 0 / Au..~= 0 i :t <.t ::io > 0 )

~ - ~ e.-s-b: wt.d~

sup I"-\ " C ( ~ .,_, )'" "' Cr 'h

;'72 ~9'"('

i.. i.

:; NO'f'~z.~~O'Y\ : ~Lt-,..\\~(-a~.) =i.

.~5~~ ~:bi ¥ N:

.. ,

. ;

. •· 'j

~\"7et.r\2.. = N(11u..r) =-N(<')IA.-) ~ N(~_,l-1.) s~

U..<'j ~ Llo ' I.."'

Ltrj ~ Llo ~VI

l.Lrj ~ Llo L YI

y,j\,a. ( B4) lto s~e.s

R~ btt,fO lri

0('0i6~) l<.o:,... 0 I -(U.o)11., '?- 0 /

C 1 )"(1~:- U B~) Uo Cc.to) 'I<,. :=. 0

Ott IR.lt-1

lto~ {) / lt 0 ~S ~ ~~ IC= N(o-'".)'-9

Lto \.Ax) =- ). \.o (><) / A~ o

., ~.[f. \ Pro.-pos-\--\:-i'o.-vi_ Xo \-----' K(~o) i~ ~ s~~-ti~s

f'~ N(Jtc.'~- -t-k....-t

Xo ~ N )(. (r)14) c:s UH-t.4,:~ Q11 r

~°"~ 011 ~ -tJ.uz.

~.eL~y

Xo E ll' ~ U.OWIA."(>> ll 0 ~-{;- X. G\.-t'"'e, ~ «{- ky~ K.

Q~-hOM WLvt- 'lH'-e,.. p-os<;-\;~ v~ e{-- 1<.?

... 1\ V\: 2 J ~

K-=- "!>/:i..) ~) +;')._) ~) ... / Q..M- <;,_) LWI.) .. . WIEN

5k-ko-1A.s ~(:x.);::. c R~ ( }(., -r~ \){.._\) ic. ( ~~s-. wka...t t=-2-"")

• Fo-ir ~~~?

tt. .. E:C''cl ="> K~'+at

-\.b l~ It LLc ~ a-f- ~..e..c K. ~ 1<ic<<

T~ K. = ?>h_ ~ Llo()() :::: c" R~ ( )(H-1 + ilx111) S/2. ( cv.{-k re+ L~ IR._"'-1)

,.,i.+ L~ (AU-~ ... f-n'e.d~. ~~ec.,'{y. ~)

L~-t- ~-=- c;c:v~s{-y ('it) . 1l-uw.

• t

~Cr) = Cr '2. <~ic.-1./) /'

., \kS:_.~-~-P-S.~Ue_ .. ~ • !- C =-o 1 ~~a.. IC.< Q

=:!;> «~ ('C3e_ttof .= 0 oY" (d,e_Lt-) - a. 0 i:.~ JR."

=> U.a ~s ~ .<>¥L O+tJL. ~-h.hl vo..n~ ::!) f~ ~ t\'.) 2-D ~ K.=-.! • ' . . .:i..

., 1 lks/ w.e, ~+a., ~~~0-1'\

; , r ~ '~,~ u u r IC.

IC.~~

L~ \~l:t t:s r.ela.Av~ ~ ~Y\.. r .. Pr~ ~~K()(o) ~<; ~ ~~Uni,~

\

.. 1...0. \N~c,.s-~p-e- ~~c.:.t-y ~

\_~ T~ )<.o E \Ko ~

w~ (r) = \ ~ 2 -ri~~- \"~1.t.\ - ~ ~ ~ ('ttt~ ,Lt. /' ~ V\ re-(01 i) ~r\"'o) 'O'C.,.t~ ... )

=- .J_ D(r) - K H(r) '::.

\-\(r) ( N (r) - 1<.) y111+:2.ic.-:t rl'l"'"21c.-1 '(' l'i. '.2...C:~1

Pr°'°4- \.J ~ ~ ~ s~ cL: « ~~~-es. o.-~ °fo"r. N (r) / )<., = '°

\-\ 1 lr) = V!::_! H (.r) ~ a.. D(r) r

'D1 l-") ::; ~-2 D(.r) 1- <. ~ u.; r

~Br

E r \-\ .. Cr) !'!E C r"-H~1e, x.. "" ~ ii:> c.Jic.) ~~s+

;lu,~)" ;S c \-low'.eNM J ~--\; ~ S ~ o.fw.uou.S

·3?,, -l:kt Lt}"') .:f 0

.~. )· A~::::: o__ tn B/

• .l. i Lt.~ 0 ) .. - Ll.y.rl. ~ 0)

.. I -e.6 .::: { 'f.. : _"'!> .. = o" 'Y-11-1 > 5 \1-'' I J

• t

3> . y~ 1ae.1.t.o('I'-)= :rC..(~.-e. .. -•) R~('lC.,-1-t-L)( 11 )

= t C,. (~·-t.i-f) JJ'1o:_,-r1'v-.2 +Xl\-1

. ;

... Nov1, l' LLv-j ~ Lto t111 C i,ot. ( i: UB>~) ~ ~ k'}A-)

..

•I

0-c. Lt..,-_ ~ - f l.\<\_ i 1 )

\=--a..r (,,. : ~,Ji;' •

W{y.) == 'Jeit(x) + ol..o 1\-1

S~se.. 3 'I<. E: { \v. .. \ < ~ 1 f'i ~v:i. : o-c. it(~o) < '° \)t1->'~\i - .J. 0 r.; C:n \) ::(B11., './\) 0 {~~\<C..)

i.) \9'Vl /\ : itT ~ 0

2.. 'l ) h..e.ure-wi ( L l. f> r~~ +y <>4 r~,~~

. --· }.-:\-_ 0-E:- f 3f2. ~ .. ...q 0 > 0 . ] 'f'b > o ... s. t I

'd~1..t- ~ o .. t-vi E>rs; ~ .~ -e. E f.s II

:::9. ~ ·• .:\:. : \3~b _., lR. s' i. . .. l \/ '!-" ~ \ ~ b

{LL=-01; f) ~~5 = l )<M-t~ ::\ (){'')1; fl B1crr

r n .?.:>.~~ = L ')("_I =- ~ c)( 11)J n B~

See Lecture 3, Section 4

.-···"

__ ,,_N_ote ___ '!':!_~/:J. ('.':1 I.lo) = 0

_ I~ u...,.J __..,. u... _ -l::b

3.i:_T~~ (.~(?YiVV\Jl;tnL l~±y) -~-e. ~··~t h>O ~ SE(o_,1)

i;.-t. ~ ~ 1Y ~s '3h__ - ~.s __ ,,.- ~x) = J.~1"v-("-)

V" '>,.- () QA'\ 8 • I

3.~. D~1{<.r~-l r-tU~~~

A~-evi.-.

l ch 4,0 ('I() C,.zt.. 0

Ci> 0

E.1 ~~ to U.rj .

.JW~

">,;,/l.._-~s

lt (1/z) -::.. I.(..( rj x)

rJ r· 3/~ J

~ ~·~ to ~ p-o-w'-k've.. ~~

£<.,~:/ .. )I j ~

.. \J~~ Er:p-e,r-T~ .

. , 3>, 3. ~s1:-bt'O'Vl Tks<-r-e. ~~sts )' > o s.-t

~ o<'f"<Y"o

• W'l>/:a.(v-,~)::. H(r) .(N(c)-ic) ~ o rt\•ih:.-~

· U ppex- ~ "frowi ET

fr w~,,. (r)~) • · ~ l_W11~ ( .t, 1-V,.) -W":>,#o ( 1, I{. :~a>)] ~ -; ) ( ~ .. {,(_~lb) - ~ Lt~l/ .. } ) ~

11'a>1

HtW-e.. v..}~t.v(14>:::.. k;r:i ··------~(><):. \'Jl.\°"1"" ~t~1&>(")(1t...c1) 3/_,,-~s

Bit Er-~~rv1 l~

. ~ \ lt~'M - Ll~~a>\ ~ ( t '11/2.

'dS~

) \ ~1/i) - ~(.3/2) \

~?>i

)

c , ..... ,, (J/.J 3/ ~. tJ{~) \ = ) CJ')) ""' r - I~ "'-r

'O~, t"

~

~(): ~ar )~' \ \ ~r w,,,(r,-i)

~ c ~~ ~ Y'J. t '11/Q.

~ \ ~~l - <tr'2)' ~ c -t'll/:i. a~,

• t

't

.. Pr~

~ \ !L~~k)_ J~1 .:$ c r fl/~ '2>e\

~ I LL~.,, ... >\ ~ C r ~"­'C!l B1

\ \ \.\..'(' \ ~ c y-l'lf-~ )/2 ~ (} Cl~\

) r< r..,

~\u.~~~0 - l(o~,~~\ ~ ( \)(~-'to\ 1 'd~\

)

i. ~~ fr-u.- ~·1 . ~~t-s ) D.u. "'"o C:vt. B~j:.

1 IA, ">,..- 0 I - l.l,.., °'? 0 J I

(.(...£..LI<.,: 0 t»\, 13 \

.. W1 s°'1 tkt XoE r(v..) ~s ~~ l~

l:,..,,_~~{. \/\(1<.)('\~~(Y-..)\ = Q

r- o \ B~ I

)(,,

i.i . )( 0 E: 2_ ~ K (~o) = N(o+11.<.) =- .il.M J hi f N . :

~ = u \,_WI WIEN

Let C ,, -. ( .. ) ~I'\ <..e IR . ~

L ~ MWi:-tft. t-h..-

\ ~ u.. M~~

IC..= ~WI

~~ ~s ~ ~.eJL ':tV'v' , #' E N L'i. L~ (~) }. ~ Ll.o 'S Gt<; a1uwe, ~ AU...o =- 0 '-~ IRVI

~ ~·s ~QM~ry h~

~. i. L~ L~ -l: o c r :l.w, c 2- T~

Mi""(r) ~ (~_.t,,..-1 \(~-p;i. .... )~ / :{o:-r rE(o,11) 0~("

... ¥~ ~at P~W\ • f:i.w, ~5 o.4-.. ~~ 2-JWI

Let -w- = u..- p,,._..,

::: ~~... \'t-V ( ><. <\/V('4) - 2.~-w) '4B,..

~ r~+ ..... -, \ ,.,. ( ><. ~w--~~w-) 'O ~r

t~ ... ) - u. ('<')'.) ~ lt. (2.oo) ll '(' (;ic.)- f:i..... 0

i~-~s ~~

We. kl-VOW +"'-tt.-t LL~:i. ... ) =- PJ..wi

Tb.- M ( '(") u.) P:1-w.) = ~ ( IJ..~"'> - p:L ... )~ ~ o """'~ -r;::. •; __.., 0-1-

. ;~' ~ p f'a-p=<? 0.-ttcwi

.. Pf''°4- £~~

0~1

(tJ~~)

T~

(.\-l (r) \ Vl.. =- h.. = o ( r1'") \ r-1 J ,..

Mi..,.\ o + ) 1-<.) pi..,) =- ~ p~ 'd P.> I

yJ ~ ~ -t ka,v.e_

M ~WI ( f') l.L) p :2...,) >,.... M 2V'I' ( 0 + ) 4-) p ,._.., )

\ (u_-p1MJ2 ij ~N-1 ~ r!.

~~~ ~~.

~ ~ L\.,2 - ~u..p1"" :;:,. 0 ~ ~ h~ u...; - :>..h.rf1""p'l. ... ~ Q

~~r ~;I

~ - 2 p;,,, ~ 0 -? pi,.. ;;. 0 ~--w ~Dt011 <!¥> \

~l.. 4 V Ouf~ \.. ~ ~ -l---e,rs

f.o,y- 'f..,,, E:- f'.t."' C. ~ C-O'Y\~W r.e.S~"l-tS ~o\ ~"fS (.:lM-~J

M l(.,. ( l<. \ ~ ( ) T' 0 ) \

\ " 1. r 1 4.. J p i... 1 =.. Lt. ,c 0 t x - 2.,..

r"'~Ji ... -\

.. 2. 5 Pr~~~ As'fwt-p-l-o-h:'c.. ~~~ON\ ~ )<o E- t;_wi we- h.ew-e_

L.L(;x) =- r:: (')(-')(•J + 0 (J"t-)(o\2.wi) ~~¥~ ~~se+.s ~ G...,

... '3. S ·ty·LA..<-~-e. ___ R:4:. ___ ~-~ S:.~t

.,. S~s.e Xo E- r~ c.. 2.. ~

. '

· l

... ? . i fr:-~~-\:\ o.+1

J-cL:.~~~

~- t

'

r.: \ 1.

r. 0 Lj

\"; \

• 1·

t !

r~: i:S CDYL--kU~ <-l'l a. .. ~~ ~~ o{-

• c ~ws

2.. :l '2. !!. 2. J... 4 LA.. = ')< 1 )C. 2 - ( ')(.. \ + 'I- i ) '/.. ~ + '$ '/..~

.. ~~C- -V)(-t.e.-vt ~Cl.f1 -bo IR.~ . o-4-

/\ =- \_ )(" =-o J X .. -.1.. ~ ~ (J'") 1 r = { )(M:. OJ :x.,-\ =. i<)(")}

x..,

StA..f?O~ 1..(... a.M.-d U ~c. (.."'

~:f? \\ S~d I/ ~OA\j ~GV(~

B1 '-.A ) v~s4~ ~~~s~ 91;! A

U > o l..11. B 1'-A

1. ~ _ [ ~ s L ~ p ~ ~ E: C "'- ( B 112. }

'-t.i.. ~~ ( \-\,..:~ cw-J.ru-~ HOv\"~ ~ De.Sctva..-S~0~)

r f: ck., Ill ~ .!!::- E c k, ~ ()I. I) - } )(" _, I r) r =- ) c:L. 'i t ( )(. ~ r) l. + x.,:i l)

k. '>,:-- .1..

I

.. R ~ 1 Z ~ 0 E: G.1i 'de l..L.- '"ij o fo-v"' a.. ~ of cl1,Y'-e..c(;-..; 01-f-~ -e. E L~

. , Itt.. pM"il;~ /\(11v) l.; ~Y-tM w\i:k L~p ~

.. ~ ,, s-t~J ,, ~Y Ua.r~ (,0'..e.,.s.

;e1. u, E: C <i1. ( B 'h) ~ ~ E ( Vi ( r E- C 1' ot ) o~.,,u...

f E C k+i 1 -.