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rtt qt s r - - PowerPoint PPT Presentation

Aug 24, 2022 138 likes •709 views

rtt qt s r tr t tt Pss t r

slide-1
SLIDE 1

❆r✐t❤♠❡t✐❝ q✉❛♥t✉♠ ❝❤❛♦s ❛♥❞ r❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✾t❤ ▼❛t❤❡♠❛t✐❝❛❧ P❤②s✐❝s ▼❡❡t✐♥❣

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❯♥✐✈❡rs✐t② ♦❢ ❇❡❧❣r❛❞❡ ❋❛❝✉❧t② ♦❢ ▼❛t❤❡♠❛t✐❝s

✶✽✳ ✾✳ ✷✵✶✼✳

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✶ ✴ ✷✽

slide-2
SLIDE 2

❆❜str❛❝t

❆ ❢✉♥❞❛♠❡♥t❛❧ ♣r♦❜❧❡♠ ✐♥ t❤❡ ❛r❡❛ ♦❢ q✉❛♥t✉♠ ❝❤❛♦s ✐s t♦ ✉♥❞❡rst❛♥❞ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ ❧❛r❣❡ ❢r❡q✉❡♥❝② ❡✐❣❡♥❢✉♥❝t✐♦♥s ♦❢ t❤❡ ▲❛♣❧❛❝✐❛♥ ♦♥ ❝❡rt❛✐♥ ♥❡❣❛t✐✈❡❧② ❝✉r✈❡❞ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞s✳ ❆r✐t❤♠❡t✐❝ q✉❛♥t✉♠ ❝❤❛♦s r❡❢❡rs t♦ q✉❛♥t✉♠ s②st❡♠s t❤❛t ❤❛✈❡ ❛r✐t❤♠❡t✐❝ str✉❝t✉r❡ ❛♥❞ s♦✱ ❛r❡ ♦❢ ✐♥t❡r❡st t♦ ❜♦t❤ ♥✉♠❜❡r t❤❡♦r✐sts ❛♥❞ ♠❛t❤❡♠❛t✐❝❛❧ ♣❤②s✐❝✐sts✳ ❙✉❝❤ ❡①❛♠♣❧❡s ❛r✐s❡ ❛s ❤②♣❡r❜♦❧✐❝ s✉r❢❛❝❡s ♦❜t❛✐♥❡❞ ❛s q✉♦t✐❡♥ts ♦❢ t❤❡ ✉♣♣❡r ❤❛❧❢✲♣❧❛♥❡ ❜② ❛ ❞✐s❝r❡t❡ ❛r✐t❤♠❡t✐❝ s✉❜❣r♦✉♣ ♦❢ SL✷(R)✳ ❚❤❡ r❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ s❛②s t❤❛t ❛♥ ❡✐❣❡♥❢♦r♠ ♦❢ ❧❛r❣❡ ▲❛♣❧❛❝✐❛♥ ❡✐❣❡♥✈❛❧✉❡ ✭✇❤✐❝❤ ✐s ❛❧s♦ ❛ ❥♦✐♥t ❡✐❣❡♥❢♦r♠ ♦❢ ❛❧❧ ❍❡❝❦❡ ♦♣❡r❛t♦rs✮ s❤♦✉❧❞ ❜❡❤❛✈❡ ❧✐❦❡ ❛ r❛♥❞♦♠ ✇❛✈❡✱ t❤❛t ✐s✱ ✐ts ❞✐str✐❜✉t✐♦♥ s❤♦✉❧❞ ❜❡ ●❛✉ss✐❛♥✳ ■♥ t❤✐s t❛❧❦ ✐♥ ♣❛rt✐❝✉❧❛r ✇❡ ❢♦❝✉s ♦♥ t❤✐s ❝♦♥❥❡❝t✉r❡ ✐♥ t❤❡ ❝❛s❡ ♦❢ ❊✐s❡♥st❡✐♥ s❡r✐❡s✳ ❚❤✐s ✐s ❜❛s❡❞ ♦♥ t❤❡ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❘✐③✇❛♥✉r ❑❤❛♥✳

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✷ ✴ ✷✽

slide-3
SLIDE 3

❊✐❣❡♥❢✉♥❝t✐♦♥s ✭st❛t❡s✱ ♠♦❞❡s✮ ♦❢ t❤❡ ▲❛♣❧❛❝✐❛♥

❈❧❛ss✐❝❛❧ ♠❡❝❤❛♥✐❝s✿ ❛ ♣♦✐♥t ♣❛rt✐❝❧❡ ♠♦✈✐♥❣ ✇✐t❤♦✉t ❢r✐❝t✐♦♥ ✐♥ ❛ ❜✐❧❧✐❛r❞ t❛❜❧❡ Ω ◗✉❛♥t✉♠ ♠❡❝❤❛♥✐❝❛❧ ❞❡s❝r✐♣t✐♦♥✿ ✵ ✐♥ ❡✳❣✳ ❛ ❜✐❧❧✐❛r❞ t❛❜❧❡ ✲ t❤❡ ❛♠♣❧✐t✉❞❡ ♦❢ ❛ st❛t✐♦♥❛r② s♦❧✉t✐♦♥ ♦❢ ❙❝❤rö❞✐♥❣❡r✬s ❡q✉❛t✐♦♥

✷

✷

✲ r❡s❝❛❧❡❞ q✉❛♥t❛❧ ❡♥❡r❣② ❧❡✈❡❧s ♦❢ t❤❡ s②st❡♠ ✵ ❉✐r✐❝❤❧❡t ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s

✷

✶ ♥♦r♠❛❧✐③❡❞✱ ♦❢ ✉♥✐t

✷✲♥♦r♠

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✸ ✴ ✷✽

slide-4
SLIDE 4

❊✐❣❡♥❢✉♥❝t✐♦♥s ✭st❛t❡s✱ ♠♦❞❡s✮ ♦❢ t❤❡ ▲❛♣❧❛❝✐❛♥

❈❧❛ss✐❝❛❧ ♠❡❝❤❛♥✐❝s✿ ❛ ♣♦✐♥t ♣❛rt✐❝❧❡ ♠♦✈✐♥❣ ✇✐t❤♦✉t ❢r✐❝t✐♦♥ ✐♥ ❛ ❜✐❧❧✐❛r❞ t❛❜❧❡ Ω ◗✉❛♥t✉♠ ♠❡❝❤❛♥✐❝❛❧ ❞❡s❝r✐♣t✐♦♥✿ ∆φj + λjφj = ✵, ✐♥ Ω (❡✳❣✳ ❛ ❜✐❧❧✐❛r❞ t❛❜❧❡) φj(x) ✲ t❤❡ ❛♠♣❧✐t✉❞❡ ♦❢ ❛ st❛t✐♦♥❛r② s♦❧✉t✐♦♥ ♦❢ ❙❝❤rö❞✐♥❣❡r✬s ❡q✉❛t✐♦♥ λj = ✷mEj

✷

✲ r❡s❝❛❧❡❞ q✉❛♥t❛❧ ❡♥❡r❣② ❧❡✈❡❧s ♦❢ t❤❡ s②st❡♠ ✵ ❉✐r✐❝❤❧❡t ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s

✷

✶ ♥♦r♠❛❧✐③❡❞✱ ♦❢ ✉♥✐t

✷✲♥♦r♠

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✸ ✴ ✷✽

slide-5
SLIDE 5

❊✐❣❡♥❢✉♥❝t✐♦♥s ✭st❛t❡s✱ ♠♦❞❡s✮ ♦❢ t❤❡ ▲❛♣❧❛❝✐❛♥

❈❧❛ss✐❝❛❧ ♠❡❝❤❛♥✐❝s✿ ❛ ♣♦✐♥t ♣❛rt✐❝❧❡ ♠♦✈✐♥❣ ✇✐t❤♦✉t ❢r✐❝t✐♦♥ ✐♥ ❛ ❜✐❧❧✐❛r❞ t❛❜❧❡ Ω ◗✉❛♥t✉♠ ♠❡❝❤❛♥✐❝❛❧ ❞❡s❝r✐♣t✐♦♥✿ ∆φj + λjφj = ✵, ✐♥ Ω (❡✳❣✳ ❛ ❜✐❧❧✐❛r❞ t❛❜❧❡) φj(x) ✲ t❤❡ ❛♠♣❧✐t✉❞❡ ♦❢ ❛ st❛t✐♦♥❛r② s♦❧✉t✐♦♥ ♦❢ ❙❝❤rö❞✐♥❣❡r✬s ❡q✉❛t✐♦♥ λj = ✷mEj

✷

✲ r❡s❝❛❧❡❞ q✉❛♥t❛❧ ❡♥❡r❣② ❧❡✈❡❧s ♦❢ t❤❡ s②st❡♠ φj|∂Ω = ✵, (❉✐r✐❝❤❧❡t ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s)

✷

✶ ♥♦r♠❛❧✐③❡❞✱ ♦❢ ✉♥✐t

✷✲♥♦r♠

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✸ ✴ ✷✽

slide-6
SLIDE 6

❊✐❣❡♥❢✉♥❝t✐♦♥s ✭st❛t❡s✱ ♠♦❞❡s✮ ♦❢ t❤❡ ▲❛♣❧❛❝✐❛♥

❈❧❛ss✐❝❛❧ ♠❡❝❤❛♥✐❝s✿ ❛ ♣♦✐♥t ♣❛rt✐❝❧❡ ♠♦✈✐♥❣ ✇✐t❤♦✉t ❢r✐❝t✐♦♥ ✐♥ ❛ ❜✐❧❧✐❛r❞ t❛❜❧❡ Ω ◗✉❛♥t✉♠ ♠❡❝❤❛♥✐❝❛❧ ❞❡s❝r✐♣t✐♦♥✿ ∆φj + λjφj = ✵, ✐♥ Ω (❡✳❣✳ ❛ ❜✐❧❧✐❛r❞ t❛❜❧❡) φj(x) ✲ t❤❡ ❛♠♣❧✐t✉❞❡ ♦❢ ❛ st❛t✐♦♥❛r② s♦❧✉t✐♦♥ ♦❢ ❙❝❤rö❞✐♥❣❡r✬s ❡q✉❛t✐♦♥ λj = ✷mEj

✷

✲ r❡s❝❛❧❡❞ q✉❛♥t❛❧ ❡♥❡r❣② ❧❡✈❡❧s ♦❢ t❤❡ s②st❡♠ φj|∂Ω = ✵, (❉✐r✐❝❤❧❡t ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s)

  • Ω

|φj|✷dµ = ✶, ♥♦r♠❛❧✐③❡❞✱ ♦❢ ✉♥✐t L✷✲♥♦r♠

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✸ ✴ ✷✽

slide-7
SLIDE 7

❇❛s✐❝ q✉❡st✐♦♥s✿

λ✶ ≤ λ✷ ≤ λ✸ ≤ . . . ≤ λj ≤ . . . ❍♦✇ ✐s t❤❡ ❝❧❛ss✐❝❛❧ ♠❡❝❤❛♥✐❝s ❞❡s❝r✐♣t✐♦♥ r❡✢❡❝t❡❞ ✐♥ t❤❡ q✉❛♥t✉♠ ❞❡s❝r✐♣t✐♦♥ ✇❤❡♥ P❧❛♥❝❦✬s ❝♦♥st❛♥t ✐s s♠❛❧❧ ✭♦r ❡q✉✐✈❛❧❡♥t❧② ✐♥ t❤❡ ❝❛s❡ ❛t ❤❛♥❞✱ ✇❤❡♥ ✮❄ ❆r❡ t❤❡r❡ ✉♥✐✈❡rs❛❧ ❧❛✇s ✐♥ t❤❡ ❡♥❡r❣② s♣❡❝tr✉♠❄ ❲❤❛t ❛r❡ t❤❡ st❛t✐st✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ ❤✐❣❤❧② ❡①❝✐t❡❞ ❡✐❣❡♥❢✉♥❝t✐♦♥s❄

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✹ ✴ ✷✽

slide-8
SLIDE 8

❇❛s✐❝ q✉❡st✐♦♥s✿

λ✶ ≤ λ✷ ≤ λ✸ ≤ . . . ≤ λj ≤ . . . ❍♦✇ ✐s t❤❡ ❝❧❛ss✐❝❛❧ ♠❡❝❤❛♥✐❝s ❞❡s❝r✐♣t✐♦♥ r❡✢❡❝t❡❞ ✐♥ t❤❡ q✉❛♥t✉♠ ❞❡s❝r✐♣t✐♦♥ ✇❤❡♥ P❧❛♥❝❦✬s ❝♦♥st❛♥t ✐s s♠❛❧❧ ✭♦r ❡q✉✐✈❛❧❡♥t❧② ✐♥ t❤❡ ❝❛s❡ ❛t ❤❛♥❞✱ ✇❤❡♥ λj → ∞✮❄ ❆r❡ t❤❡r❡ ✉♥✐✈❡rs❛❧ ❧❛✇s ✐♥ t❤❡ ❡♥❡r❣② s♣❡❝tr✉♠❄ ❲❤❛t ❛r❡ t❤❡ st❛t✐st✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ ❤✐❣❤❧② ❡①❝✐t❡❞ ❡✐❣❡♥❢✉♥❝t✐♦♥s❄

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✹ ✴ ✷✽

slide-9
SLIDE 9

Ω - an ellipse, 12 modes around 5600th eigenvalue, classical Hamiltonian dynamics – a billiard in Ω

  • - motion is integrable

P✐❝t✉r❡✿ ✶

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✺ ✴ ✷✽

slide-10
SLIDE 10

Ω - a stadium, 12 modes around 5600th eigenvalue, classical Hamiltonian dynamics – a billiard in Ω

  • - motion is ergodic (almost all of the trajectories are dense)

P✐❝t✉r❡✿ ✶

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✻ ✴ ✷✽

slide-11
SLIDE 11

P✐❝t✉r❡✿ ✶

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✼ ✴ ✷✽

slide-12
SLIDE 12

Ω - a dispersing Sinai billiard, 12 modes around 5600th eigenvalue, classical Hamiltonian dynamics – a billiard in Ω

  • - motion is ergodic and strongly chaotic (almost all of the trajectories are dense)

P✐❝t✉r❡✿ ✶

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✽ ✴ ✷✽

slide-13
SLIDE 13

❉❡♥s✐t② ♠❡❛s✉r❡s ♦♥ Ω

νφj := |φj|✷dµ ✕ t❤❡ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❜❡✐♥❣ ✐♥ t❤❡ st❛t❡ ❉♦ t❤❡s❡ ♠❡❛s✉r❡s ❡q✉✐❞✐str✐❜✉t❡ ❛s ✱ ♦r ❝❛♥ t❤❡② ❧♦❝❛❧✐③❡❄ ■♥ t❤❡ ❝❛s❡ ♦❢ ❛♥ ❡❧❧✐♣s❡ ✭♠♦r❡ ❣❡♥❡r❛❧❧② ✲ t❤❡ q✉❛♥t✐③❛t✐♦♥ ♦❢ ❛♥② ❝♦♠♣❧❡t❡❧② ✐♥t❡❣r❛❜❧❡ ❍❛♠✐❧t♦♥✐❛♥ s②st❡♠✮ t❤❡s❡ ♠❡❛s✉r❡s ✭♦r r❛t❤❡r t❤❡✐r ♠✐❝r♦❧♦❝❛❧ ❧✐❢ts t♦

✶

✳✳✳✮ ❧♦❝❛❧✐③❡ ♦♥ ✐♥✈❛r✐❛♥t t♦r✐ ✐♥ ❛ ✇❡❧❧ ✉♥❞❡rst♦♦❞ ♠❛♥♥❡r

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✾ ✴ ✷✽

slide-14
SLIDE 14

❉❡♥s✐t② ♠❡❛s✉r❡s ♦♥ Ω

νφj := |φj|✷dµ ✕ t❤❡ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❜❡✐♥❣ ✐♥ t❤❡ st❛t❡ φj ❉♦ t❤❡s❡ ♠❡❛s✉r❡s ❡q✉✐❞✐str✐❜✉t❡ ❛s ✱ ♦r ❝❛♥ t❤❡② ❧♦❝❛❧✐③❡❄ ■♥ t❤❡ ❝❛s❡ ♦❢ ❛♥ ❡❧❧✐♣s❡ ✭♠♦r❡ ❣❡♥❡r❛❧❧② ✲ t❤❡ q✉❛♥t✐③❛t✐♦♥ ♦❢ ❛♥② ❝♦♠♣❧❡t❡❧② ✐♥t❡❣r❛❜❧❡ ❍❛♠✐❧t♦♥✐❛♥ s②st❡♠✮ t❤❡s❡ ♠❡❛s✉r❡s ✭♦r r❛t❤❡r t❤❡✐r ♠✐❝r♦❧♦❝❛❧ ❧✐❢ts t♦

✶

✳✳✳✮ ❧♦❝❛❧✐③❡ ♦♥ ✐♥✈❛r✐❛♥t t♦r✐ ✐♥ ❛ ✇❡❧❧ ✉♥❞❡rst♦♦❞ ♠❛♥♥❡r

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✾ ✴ ✷✽

slide-15
SLIDE 15

❉❡♥s✐t② ♠❡❛s✉r❡s ♦♥ Ω

νφj := |φj|✷dµ ✕ t❤❡ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❜❡✐♥❣ ✐♥ t❤❡ st❛t❡ φj ⋆ ❉♦ t❤❡s❡ ♠❡❛s✉r❡s ❡q✉✐❞✐str✐❜✉t❡ ❛s λj → ∞✱ ♦r ❝❛♥ t❤❡② ❧♦❝❛❧✐③❡❄ ■♥ t❤❡ ❝❛s❡ ♦❢ ❛♥ ❡❧❧✐♣s❡ ✭♠♦r❡ ❣❡♥❡r❛❧❧② ✲ t❤❡ q✉❛♥t✐③❛t✐♦♥ ♦❢ ❛♥② ❝♦♠♣❧❡t❡❧② ✐♥t❡❣r❛❜❧❡ ❍❛♠✐❧t♦♥✐❛♥ s②st❡♠✮ t❤❡s❡ ♠❡❛s✉r❡s ✭♦r r❛t❤❡r t❤❡✐r ♠✐❝r♦❧♦❝❛❧ ❧✐❢ts t♦

✶

✳✳✳✮ ❧♦❝❛❧✐③❡ ♦♥ ✐♥✈❛r✐❛♥t t♦r✐ ✐♥ ❛ ✇❡❧❧ ✉♥❞❡rst♦♦❞ ♠❛♥♥❡r

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✾ ✴ ✷✽

slide-16
SLIDE 16

❉❡♥s✐t② ♠❡❛s✉r❡s ♦♥ Ω

νφj := |φj|✷dµ ✕ t❤❡ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❜❡✐♥❣ ✐♥ t❤❡ st❛t❡ φj ⋆ ❉♦ t❤❡s❡ ♠❡❛s✉r❡s ❡q✉✐❞✐str✐❜✉t❡ ❛s λj → ∞✱ ♦r ❝❛♥ t❤❡② ❧♦❝❛❧✐③❡❄ ■♥ t❤❡ ❝❛s❡ ♦❢ ❛♥ ❡❧❧✐♣s❡ ✭♠♦r❡ ❣❡♥❡r❛❧❧② ✲ t❤❡ q✉❛♥t✐③❛t✐♦♥ ♦❢ ❛♥② ❝♦♠♣❧❡t❡❧② ✐♥t❡❣r❛❜❧❡ ❍❛♠✐❧t♦♥✐❛♥ s②st❡♠✮ → t❤❡s❡ ♠❡❛s✉r❡s ✭♦r r❛t❤❡r t❤❡✐r ♠✐❝r♦❧♦❝❛❧ ❧✐❢ts µφj t♦ T✶(Ω)✳✳✳✮ ❧♦❝❛❧✐③❡ ♦♥ ✐♥✈❛r✐❛♥t t♦r✐ ✐♥ ❛ ✇❡❧❧ ✉♥❞❡rst♦♦❞ ♠❛♥♥❡r

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✾ ✴ ✷✽

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SLIDE 17

❈♦♠♣❛❝t ♠❛♥✐❢♦❧❞s

Ω = (M, g), ❛ ❝♦♠♣❛❝t ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞ t❤❡ ▲❛♣❧❛❝❡✲❇❡❧tr❛♠✐ ♦♣❡r❛t♦r ❢♦r t❤❡ ♠❡tr✐❝ t❤❡ ❝❧❛ss✐❝❛❧ ♠❡❝❤❛♥✐❝s ✐s t❤❛t ♦❢ ♠♦t✐♦♥ ❜② ❣❡♦❞❡s✐❝s ♦♥ t❤❡ ✉♥✐t t❛♥❣❡♥t ❜✉♥❞❧❡

✶

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✶✵ ✴ ✷✽

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SLIDE 18

❈♦♠♣❛❝t ♠❛♥✐❢♦❧❞s

Ω = (M, g), ❛ ❝♦♠♣❛❝t ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞ ∆g, t❤❡ ▲❛♣❧❛❝❡✲❇❡❧tr❛♠✐ ♦♣❡r❛t♦r ❢♦r t❤❡ ♠❡tr✐❝ g t❤❡ ❝❧❛ss✐❝❛❧ ♠❡❝❤❛♥✐❝s ✐s t❤❛t ♦❢ ♠♦t✐♦♥ ❜② ❣❡♦❞❡s✐❝s ♦♥ t❤❡ ✉♥✐t t❛♥❣❡♥t ❜✉♥❞❧❡

✶

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✶✵ ✴ ✷✽

slide-19
SLIDE 19

❈♦♠♣❛❝t ♠❛♥✐❢♦❧❞s

Ω = (M, g), ❛ ❝♦♠♣❛❝t ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞ ∆g, t❤❡ ▲❛♣❧❛❝❡✲❇❡❧tr❛♠✐ ♦♣❡r❛t♦r ❢♦r t❤❡ ♠❡tr✐❝ g t❤❡ ❝❧❛ss✐❝❛❧ ♠❡❝❤❛♥✐❝s ✐s t❤❛t ♦❢ ♠♦t✐♦♥ ❜② ❣❡♦❞❡s✐❝s ♦♥ t❤❡ ✉♥✐t t❛♥❣❡♥t ❜✉♥❞❧❡ T✶(M)

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✶✵ ✴ ✷✽

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SLIDE 20

❚❤❡ ❝❛s❡ ♦❢ ❡r❣♦❞✐❝ ❣❡♦❞❡s✐❝ ✢♦✇

❡r❣♦❞✐❝ ✕ t❤❡ ♦♥❧② ✢♦✇ ✐♥✈❛r✐❛♥t s✉❜s❡ts ♦❢ T✶(M) ❛r❡ ❡✐t❤❡r ♦❢ ③❡r♦ ♦r ❢✉❧❧ µ✲♠❡❛s✉r❡✱ ✇❤❡r❡ µ ✐s t❤❡ ▲✐♦✉✈✐❧❧❡ ♠❡❛s✉r❡ ♦♥ T✶(M) ❇✐r❦❤♦✛✬s ❡r❣♦❞✐❝ t❤❡♦r❡♠✿ ✲❛❧♠♦st ❛❧❧ ❣❡♦❞❡s✐❝s ❛r❡ ✲❡q✉✐❞✐str✐❜✉t❡❞ ✐♥

✶

❚❤❡♦r❡♠ ✭❙❤♥✐r❡❧♠❛♥ ✶✾✼✹✱ ❩❡❧❞✐t❝❤ ✶✾✽✼✱ ❈♦❧✐♥ ❞❡ ❱❡r❞✐❡r❡ ✶✾✽✺✮ ■❢ t❤❡ ❣❡♦❞❡s✐❝ ✢♦✇ ✐s ❡r❣♦❞✐❝✱ t❤❡♥ ❛❧♠♦st ❛❧❧ ✭✐♥ t❤❡ s❡♥s❡ ♦❢ ❞❡♥s✐t②✮ ♦❢ t❤❡ ❡✐❣❡♥❢✉♥❝t✐♦♥s ❜❡❝♦♠❡ ❡q✉✐❞✐str✐❜✉t❡❞ ✇✐t❤ r❡s♣❡❝t t♦ ✳ ❚❤❛t ✐s✱ ✐❢

✵ ✐s ❛♥ ♦rt❤♦♥♦r♠❛❧ ❜❛s✐s ♦❢ ❡✐❣❡♥❢✉♥❝t✐♦♥s ✇✐t❤

✵

✵ ✶ ✷ ✸

✱ t❤❡♥ t❤❡r❡ ✐s ❛ s✉❜s❡q✉❡♥❝❡ ♦❢ ✐♥t❡❣❡rs ♦❢ ❢✉❧❧ ❞❡♥s✐t②✱ s✉❝❤ t❤❛t ❛s

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✶✶ ✴ ✷✽

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SLIDE 21

❚❤❡ ❝❛s❡ ♦❢ ❡r❣♦❞✐❝ ❣❡♦❞❡s✐❝ ✢♦✇

❡r❣♦❞✐❝ ✕ t❤❡ ♦♥❧② ✢♦✇ ✐♥✈❛r✐❛♥t s✉❜s❡ts ♦❢ T✶(M) ❛r❡ ❡✐t❤❡r ♦❢ ③❡r♦ ♦r ❢✉❧❧ µ✲♠❡❛s✉r❡✱ ✇❤❡r❡ µ ✐s t❤❡ ▲✐♦✉✈✐❧❧❡ ♠❡❛s✉r❡ ♦♥ T✶(M) ❇✐r❦❤♦✛✬s ❡r❣♦❞✐❝ t❤❡♦r❡♠✿ µ✲❛❧♠♦st ❛❧❧ ❣❡♦❞❡s✐❝s ❛r❡ µ✲❡q✉✐❞✐str✐❜✉t❡❞ ✐♥ T✶(M) ❚❤❡♦r❡♠ ✭❙❤♥✐r❡❧♠❛♥ ✶✾✼✹✱ ❩❡❧❞✐t❝❤ ✶✾✽✼✱ ❈♦❧✐♥ ❞❡ ❱❡r❞✐❡r❡ ✶✾✽✺✮ ■❢ t❤❡ ❣❡♦❞❡s✐❝ ✢♦✇ ✐s ❡r❣♦❞✐❝✱ t❤❡♥ ❛❧♠♦st ❛❧❧ ✭✐♥ t❤❡ s❡♥s❡ ♦❢ ❞❡♥s✐t②✮ ♦❢ t❤❡ ❡✐❣❡♥❢✉♥❝t✐♦♥s ❜❡❝♦♠❡ ❡q✉✐❞✐str✐❜✉t❡❞ ✇✐t❤ r❡s♣❡❝t t♦ ✳ ❚❤❛t ✐s✱ ✐❢

✵ ✐s ❛♥ ♦rt❤♦♥♦r♠❛❧ ❜❛s✐s ♦❢ ❡✐❣❡♥❢✉♥❝t✐♦♥s ✇✐t❤

✵

✵ ✶ ✷ ✸

✱ t❤❡♥ t❤❡r❡ ✐s ❛ s✉❜s❡q✉❡♥❝❡ ♦❢ ✐♥t❡❣❡rs ♦❢ ❢✉❧❧ ❞❡♥s✐t②✱ s✉❝❤ t❤❛t ❛s

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✶✶ ✴ ✷✽

slide-22
SLIDE 22

❚❤❡ ❝❛s❡ ♦❢ ❡r❣♦❞✐❝ ❣❡♦❞❡s✐❝ ✢♦✇

❡r❣♦❞✐❝ ✕ t❤❡ ♦♥❧② ✢♦✇ ✐♥✈❛r✐❛♥t s✉❜s❡ts ♦❢ T✶(M) ❛r❡ ❡✐t❤❡r ♦❢ ③❡r♦ ♦r ❢✉❧❧ µ✲♠❡❛s✉r❡✱ ✇❤❡r❡ µ ✐s t❤❡ ▲✐♦✉✈✐❧❧❡ ♠❡❛s✉r❡ ♦♥ T✶(M) ❇✐r❦❤♦✛✬s ❡r❣♦❞✐❝ t❤❡♦r❡♠✿ µ✲❛❧♠♦st ❛❧❧ ❣❡♦❞❡s✐❝s ❛r❡ µ✲❡q✉✐❞✐str✐❜✉t❡❞ ✐♥ T✶(M) ❚❤❡♦r❡♠ ✭❙❤♥✐r❡❧♠❛♥ ✶✾✼✹✱ ❩❡❧❞✐t❝❤ ✶✾✽✼✱ ❈♦❧✐♥ ❞❡ ❱❡r❞✐❡r❡ ✶✾✽✺✮ ■❢ t❤❡ ❣❡♦❞❡s✐❝ ✢♦✇ ✐s ❡r❣♦❞✐❝✱ t❤❡♥ ❛❧♠♦st ❛❧❧ ✭✐♥ t❤❡ s❡♥s❡ ♦❢ ❞❡♥s✐t②✮ ♦❢ t❤❡ ❡✐❣❡♥❢✉♥❝t✐♦♥s ❜❡❝♦♠❡ ❡q✉✐❞✐str✐❜✉t❡❞ ✇✐t❤ r❡s♣❡❝t t♦ µ✳ ❚❤❛t ✐s✱ ✐❢ {φj}∞

j=✵ ✐s ❛♥ ♦rt❤♦♥♦r♠❛❧ ❜❛s✐s ♦❢ ❡✐❣❡♥❢✉♥❝t✐♦♥s ✇✐t❤

✵ = λ✵ < λ✶ ≤ λ✷ ≤ λ✸ ≤ . . .✱ t❤❡♥ t❤❡r❡ ✐s ❛ s✉❜s❡q✉❡♥❝❡ jk ♦❢ ✐♥t❡❣❡rs ♦❢ ❢✉❧❧ ❞❡♥s✐t②✱ s✉❝❤ t❤❛t µjk − → µ, ❛s k → ∞

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✶✶ ✴ ✷✽

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SLIDE 23

◗✉❛♥t✉♠ ✉♥✐q✉❡ ❡r❣♦❞✐❝✐t② ❝♦♥❥❡❝t✉r❡

⋆ ❚❤❡ ❜❛s✐❝ q✉❡st✐♦♥✿ ❝❛♥ t❤❡r❡ ❜❡ ♦t❤❡r q✉❛♥t✉♠ ❧✐♠✐ts✱ ✐✳❡✳ s✉❜s❡q✉❡♥❝❡s ♦♥ ✇❤✐❝❤ t❤❡ µφj✬s ❜❡❤❛✈❡ ❞✐✛❡r❡♥t❧② ❄ ■❢ ❤❛s ✭str✐❝t❧②✮ ♥❡❣❛t✐✈❡ ❝✉r✈❛t✉r❡✱ t❤❡♥ t❤❡ ❣❡♦❞❡s✐❝ ✢♦✇ ✐s ❡r❣♦❞✐❝ ❛♥❞ str♦♥❣❧② ❝❤❛♦t✐❝✱ t❤❡ ♣❡r✐♦❞✐❝ ❣❡♦❞❡s✐❝s ❛r❡ ✐s♦❧❛t❡❞ ❛♥❞ ✉♥st❛❜❧❡✱ ❡t❝✳✳✳ ❈❛♥ t❤❡ ❛r❝❧❡♥❣t❤ ♠❡❛s✉r❡ ♦♥ ❛ ❝❧♦s❡❞ ❣❡♦❞❡s✐❝ ✭✧str♦♥❣ s❝❛rs✧✱ t❤❡ ♠♦st s✐♥❣✉❧❛r ✢♦✇ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡✮ ❜❡ ❛ q✉❛♥t✉♠ ❧✐♠✐t❄ ❈♦♥❥❡❝t✉r❡ ✭◗❯❊✱ ❘✉❞♥✐❝❦✲❙❛r♥❛❦✱ ✶✾✾✹✮ ■❢ ✐s ❛ ❝♦♠♣❛❝t ♥❡❣❛t✐✈❡❧② ❝✉r✈❡❞ ♠❛♥✐❢♦❧❞✱ t❤❡♥ ❛s ✱ ✐✳❡✳ ✐s t❤❡ ♦♥❧② q✉❛♥t✉♠ ❧✐♠✐t✦

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✶✷ ✴ ✷✽

slide-24
SLIDE 24

◗✉❛♥t✉♠ ✉♥✐q✉❡ ❡r❣♦❞✐❝✐t② ❝♦♥❥❡❝t✉r❡

⋆ ❚❤❡ ❜❛s✐❝ q✉❡st✐♦♥✿ ❝❛♥ t❤❡r❡ ❜❡ ♦t❤❡r q✉❛♥t✉♠ ❧✐♠✐ts✱ ✐✳❡✳ s✉❜s❡q✉❡♥❝❡s ♦♥ ✇❤✐❝❤ t❤❡ µφj✬s ❜❡❤❛✈❡ ❞✐✛❡r❡♥t❧② ❄ ■❢ M ❤❛s ✭str✐❝t❧②✮ ♥❡❣❛t✐✈❡ ❝✉r✈❛t✉r❡✱ t❤❡♥ t❤❡ ❣❡♦❞❡s✐❝ ✢♦✇ ✐s ❡r❣♦❞✐❝ ❛♥❞ str♦♥❣❧② ❝❤❛♦t✐❝✱ t❤❡ ♣❡r✐♦❞✐❝ ❣❡♦❞❡s✐❝s ❛r❡ ✐s♦❧❛t❡❞ ❛♥❞ ✉♥st❛❜❧❡✱ ❡t❝✳✳✳ ❈❛♥ t❤❡ ❛r❝❧❡♥❣t❤ ♠❡❛s✉r❡ ♦♥ ❛ ❝❧♦s❡❞ ❣❡♦❞❡s✐❝ ✭✧str♦♥❣ s❝❛rs✧✱ t❤❡ ♠♦st s✐♥❣✉❧❛r ✢♦✇ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡✮ ❜❡ ❛ q✉❛♥t✉♠ ❧✐♠✐t❄ ❈♦♥❥❡❝t✉r❡ ✭◗❯❊✱ ❘✉❞♥✐❝❦✲❙❛r♥❛❦✱ ✶✾✾✹✮ ■❢ ✐s ❛ ❝♦♠♣❛❝t ♥❡❣❛t✐✈❡❧② ❝✉r✈❡❞ ♠❛♥✐❢♦❧❞✱ t❤❡♥ ❛s ✱ ✐✳❡✳ ✐s t❤❡ ♦♥❧② q✉❛♥t✉♠ ❧✐♠✐t✦

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✶✷ ✴ ✷✽

slide-25
SLIDE 25

◗✉❛♥t✉♠ ✉♥✐q✉❡ ❡r❣♦❞✐❝✐t② ❝♦♥❥❡❝t✉r❡

⋆ ❚❤❡ ❜❛s✐❝ q✉❡st✐♦♥✿ ❝❛♥ t❤❡r❡ ❜❡ ♦t❤❡r q✉❛♥t✉♠ ❧✐♠✐ts✱ ✐✳❡✳ s✉❜s❡q✉❡♥❝❡s ♦♥ ✇❤✐❝❤ t❤❡ µφj✬s ❜❡❤❛✈❡ ❞✐✛❡r❡♥t❧② ❄ ■❢ M ❤❛s ✭str✐❝t❧②✮ ♥❡❣❛t✐✈❡ ❝✉r✈❛t✉r❡✱ t❤❡♥ t❤❡ ❣❡♦❞❡s✐❝ ✢♦✇ ✐s ❡r❣♦❞✐❝ ❛♥❞ str♦♥❣❧② ❝❤❛♦t✐❝✱ t❤❡ ♣❡r✐♦❞✐❝ ❣❡♦❞❡s✐❝s ❛r❡ ✐s♦❧❛t❡❞ ❛♥❞ ✉♥st❛❜❧❡✱ ❡t❝✳✳✳ ⋆ ❈❛♥ t❤❡ ❛r❝❧❡♥❣t❤ ♠❡❛s✉r❡ ♦♥ ❛ ❝❧♦s❡❞ ❣❡♦❞❡s✐❝ ✭✧str♦♥❣ s❝❛rs✧✱ t❤❡ ♠♦st s✐♥❣✉❧❛r ✢♦✇ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡✮ ❜❡ ❛ q✉❛♥t✉♠ ❧✐♠✐t❄ ❈♦♥❥❡❝t✉r❡ ✭◗❯❊✱ ❘✉❞♥✐❝❦✲❙❛r♥❛❦✱ ✶✾✾✹✮ ■❢ ✐s ❛ ❝♦♠♣❛❝t ♥❡❣❛t✐✈❡❧② ❝✉r✈❡❞ ♠❛♥✐❢♦❧❞✱ t❤❡♥ ❛s ✱ ✐✳❡✳ ✐s t❤❡ ♦♥❧② q✉❛♥t✉♠ ❧✐♠✐t✦

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✶✷ ✴ ✷✽

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SLIDE 26

◗✉❛♥t✉♠ ✉♥✐q✉❡ ❡r❣♦❞✐❝✐t② ❝♦♥❥❡❝t✉r❡

⋆ ❚❤❡ ❜❛s✐❝ q✉❡st✐♦♥✿ ❝❛♥ t❤❡r❡ ❜❡ ♦t❤❡r q✉❛♥t✉♠ ❧✐♠✐ts✱ ✐✳❡✳ s✉❜s❡q✉❡♥❝❡s ♦♥ ✇❤✐❝❤ t❤❡ µφj✬s ❜❡❤❛✈❡ ❞✐✛❡r❡♥t❧② ❄ ■❢ M ❤❛s ✭str✐❝t❧②✮ ♥❡❣❛t✐✈❡ ❝✉r✈❛t✉r❡✱ t❤❡♥ t❤❡ ❣❡♦❞❡s✐❝ ✢♦✇ ✐s ❡r❣♦❞✐❝ ❛♥❞ str♦♥❣❧② ❝❤❛♦t✐❝✱ t❤❡ ♣❡r✐♦❞✐❝ ❣❡♦❞❡s✐❝s ❛r❡ ✐s♦❧❛t❡❞ ❛♥❞ ✉♥st❛❜❧❡✱ ❡t❝✳✳✳ ⋆ ❈❛♥ t❤❡ ❛r❝❧❡♥❣t❤ ♠❡❛s✉r❡ ♦♥ ❛ ❝❧♦s❡❞ ❣❡♦❞❡s✐❝ ✭✧str♦♥❣ s❝❛rs✧✱ t❤❡ ♠♦st s✐♥❣✉❧❛r ✢♦✇ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡✮ ❜❡ ❛ q✉❛♥t✉♠ ❧✐♠✐t❄ ❈♦♥❥❡❝t✉r❡ ✭◗❯❊✱ ❘✉❞♥✐❝❦✲❙❛r♥❛❦✱ ✶✾✾✹✮ ■❢ M ✐s ❛ ❝♦♠♣❛❝t ♥❡❣❛t✐✈❡❧② ❝✉r✈❡❞ ♠❛♥✐❢♦❧❞✱ t❤❡♥ µφ → µ ❛s λ → ∞✱ ✐✳❡✳ µ ✐s t❤❡ ♦♥❧② q✉❛♥t✉♠ ❧✐♠✐t✦

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✶✷ ✴ ✷✽

slide-27
SLIDE 27

❙t❛❞✐✉♠ ❞♦♠❛✐♥

❍❛ss❡❧❧ ✭❆♥♥❛❧s ♦❢ ▼❛t❤✳ ✷✵✶✵✳✮ ❋♦r ❛❧♠♦st ❛❧❧ st❛❞✐✉♠s✱ ❜✐❧❧✐❛r❞s ❛r❡ ♥♦t q✉❛♥t✉♠ ✉♥✐q✉❡ ❡r❣♦❞✐❝✦ ✭t❤❡r❡ ❡①✐st ❛ q✉❛♥t✉♠ ❧✐♠✐t ✇❤✐❝❤ ❣✐✈❡s ♣♦s✐t✐✈❡ ♠❛ss t♦ t❤❡ ❜♦✉♥❝✐♥❣ ❜❛❧❧ tr❛❥❡❝t♦r✐❡s✮

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✶✸ ✴ ✷✽

slide-28
SLIDE 28

❆r✐t❤♠❡t✐❝ ◗❯❊

Γ ≤ PSL✷(R), ❛ ❞✐s❝r❡t❡ s✉❜❣r♦✉♣, Γ H H − t❤❡ ❤②♣❡r❜♦❧✐❝ ♣❧❛♥❡ Γ = SL✷(Z) = a b c d

  • : a, b, c, d ∈ Z, ad − bc = ✶
  • ♠♦❞✉❧❛r s✉r❢❛❝❡✱ ♥♦♥❝♦♠♣❛❝t✱ ❜✉t ♦❢ ✜♥✐t❡ ❛r❡❛

♦❢ ❝♦♥st❛♥t ❝✉r✈❛t✉r❡ ✶ ✕ ❢♦r t❤❡s❡✱ t❤❡r❡ ✐s ❛❧s♦ ❝♦♥t✐♥✉♦✉s s♣❡❝tr✉♠ ❝♦♠✐♥❣ ❢r♦♠ t❤❡ t❤❡♦r② ♦❢ ❊✐s❡♥st❡✐♥ s❡r✐❡s ❞❡✈❡❧♦♣❡❞ ❜② ❙❡❧❜❡r❣

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✶✹ ✴ ✷✽

slide-29
SLIDE 29

❆r✐t❤♠❡t✐❝ ◗❯❊

Γ ≤ PSL✷(R), ❛ ❞✐s❝r❡t❡ s✉❜❣r♦✉♣, Γ H H − t❤❡ ❤②♣❡r❜♦❧✐❝ ♣❧❛♥❡ Γ = SL✷(Z) = a b c d

  • : a, b, c, d ∈ Z, ad − bc = ✶
  • M = Γ\H,

♠♦❞✉❧❛r s✉r❢❛❝❡✱ ♥♦♥❝♦♠♣❛❝t✱ ❜✉t ♦❢ ✜♥✐t❡ ❛r❡❛ ♦❢ ❝♦♥st❛♥t ❝✉r✈❛t✉r❡ K = −✶ ✕ ❢♦r t❤❡s❡✱ t❤❡r❡ ✐s ❛❧s♦ ❝♦♥t✐♥✉♦✉s s♣❡❝tr✉♠ ❝♦♠✐♥❣ ❢r♦♠ t❤❡ t❤❡♦r② ♦❢ ❊✐s❡♥st❡✐♥ s❡r✐❡s ❞❡✈❡❧♦♣❡❞ ❜② ❙❡❧❜❡r❣

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✶✹ ✴ ✷✽

slide-30
SLIDE 30

❆r✐t❤♠❡t✐❝ ◗❯❊

Γ ≤ PSL✷(R), ❛ ❞✐s❝r❡t❡ s✉❜❣r♦✉♣, Γ H H − t❤❡ ❤②♣❡r❜♦❧✐❝ ♣❧❛♥❡ Γ = SL✷(Z) = a b c d

  • : a, b, c, d ∈ Z, ad − bc = ✶
  • M = Γ\H,

♠♦❞✉❧❛r s✉r❢❛❝❡✱ ♥♦♥❝♦♠♣❛❝t✱ ❜✉t ♦❢ ✜♥✐t❡ ❛r❡❛ ♦❢ ❝♦♥st❛♥t ❝✉r✈❛t✉r❡ K = −✶ ✕ ❢♦r t❤❡s❡✱ t❤❡r❡ ✐s ❛❧s♦ ❝♦♥t✐♥✉♦✉s s♣❡❝tr✉♠ ❝♦♠✐♥❣ ❢r♦♠ t❤❡ t❤❡♦r② ♦❢ ❊✐s❡♥st❡✐♥ s❡r✐❡s ❞❡✈❡❧♦♣❡❞ ❜② ❙❡❧❜❡r❣

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✶✹ ✴ ✷✽

slide-31
SLIDE 31

▼♦❞✉❧❛r s✉r❢❛❝❡s

𝑇𝑀2(𝑎)\𝑰 Γ \𝑰

P✐❝t✉r❡✿ ✶

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✶✺ ✴ ✷✽

slide-32
SLIDE 32

❍❡❝❦❡ ♦♣❡r❛t♦rs

❛r✐t❤♠❡t✐❝ s✉r❢❛❝❡s M ❝❛rr② ❛ ❧❛r❣❡ ❢❛♠✐❧② ♦❢ ❛❧❣❡❜r❛✐❝ ❝♦rr❡s♣♦♥❞❡♥❝❡s ✭✧❛❞❞✐t✐♦♥❛❧ s②♠♠❡tr✐❡s✧✮ ✇❤✐❝❤ ❣✐✈❡ r✐s❡ t♦ t❤❡ ❢❛♠✐❧② ♦❢ ❍❡❝❦❡ ♦♣❡r❛t♦rs ♦♥ L✷(M) ♦♥

✷

❢♦r ❡✈❡r② ✶ ✇❡ ❤❛✈❡ t❤❡ ❍❡❝❦❡ ♦♣❡r❛t♦r

✵ ✷ ✷ ✶ ❝♦♠♠✉t✐♥❣ ❢❛♠✐❧② ♦❢ ♥♦r♠❛❧ ♦♣❡r❛t♦rs ✇❤✐❝❤ ❝♦♠♠✉t❡ ✇✐t❤

❤②♣❡r❜♦❧✐❝ ▲❛♣❧❛❝✐❛♥

✷

✷ ✷ ✷ ✷

✕ ❝❛♥ ❜❡ s✐♠✉❧t❛♥❡♦✉s❧② ❞✐❛❣♦♥❛❧✐③❡❞ ✭ ❍❡❝❦❡ ❡✐❣❡♥❢♦r♠s✮

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✶✻ ✴ ✷✽

slide-33
SLIDE 33

❍❡❝❦❡ ♦♣❡r❛t♦rs

❛r✐t❤♠❡t✐❝ s✉r❢❛❝❡s M ❝❛rr② ❛ ❧❛r❣❡ ❢❛♠✐❧② ♦❢ ❛❧❣❡❜r❛✐❝ ❝♦rr❡s♣♦♥❞❡♥❝❡s ✭✧❛❞❞✐t✐♦♥❛❧ s②♠♠❡tr✐❡s✧✮ ✇❤✐❝❤ ❣✐✈❡ r✐s❡ t♦ t❤❡ ❢❛♠✐❧② ♦❢ ❍❡❝❦❡ ♦♣❡r❛t♦rs ♦♥ L✷(M) ♦♥ SL✷(Z)\H ❢♦r ❡✈❡r② n ≥ ✶ ✇❡ ❤❛✈❡ t❤❡ ❍❡❝❦❡ ♦♣❡r❛t♦r Tnφ(z) =

  • ad=n

✵≤b<d

φ az + b d

  • ,

Tn : L✷(M) → L✷(M)

✶ ❝♦♠♠✉t✐♥❣ ❢❛♠✐❧② ♦❢ ♥♦r♠❛❧ ♦♣❡r❛t♦rs ✇❤✐❝❤ ❝♦♠♠✉t❡ ✇✐t❤

❤②♣❡r❜♦❧✐❝ ▲❛♣❧❛❝✐❛♥

✷

✷ ✷ ✷ ✷

✕ ❝❛♥ ❜❡ s✐♠✉❧t❛♥❡♦✉s❧② ❞✐❛❣♦♥❛❧✐③❡❞ ✭ ❍❡❝❦❡ ❡✐❣❡♥❢♦r♠s✮

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✶✻ ✴ ✷✽

slide-34
SLIDE 34

❍❡❝❦❡ ♦♣❡r❛t♦rs

❛r✐t❤♠❡t✐❝ s✉r❢❛❝❡s M ❝❛rr② ❛ ❧❛r❣❡ ❢❛♠✐❧② ♦❢ ❛❧❣❡❜r❛✐❝ ❝♦rr❡s♣♦♥❞❡♥❝❡s ✭✧❛❞❞✐t✐♦♥❛❧ s②♠♠❡tr✐❡s✧✮ ✇❤✐❝❤ ❣✐✈❡ r✐s❡ t♦ t❤❡ ❢❛♠✐❧② ♦❢ ❍❡❝❦❡ ♦♣❡r❛t♦rs ♦♥ L✷(M) ♦♥ SL✷(Z)\H ❢♦r ❡✈❡r② n ≥ ✶ ✇❡ ❤❛✈❡ t❤❡ ❍❡❝❦❡ ♦♣❡r❛t♦r Tnφ(z) =

  • ad=n

✵≤b<d

φ az + b d

  • ,

Tn : L✷(M) → L✷(M) {Tn}n≥✶ ❝♦♠♠✉t✐♥❣ ❢❛♠✐❧② ♦❢ ♥♦r♠❛❧ ♦♣❡r❛t♦rs ✇❤✐❝❤ ❝♦♠♠✉t❡ ✇✐t❤ ❤②♣❡r❜♦❧✐❝ ▲❛♣❧❛❝✐❛♥ ∆ = y✷

∂✷ ∂x✷ + ∂✷ ∂y✷

  • ✕ ❝❛♥ ❜❡ s✐♠✉❧t❛♥❡♦✉s❧② ❞✐❛❣♦♥❛❧✐③❡❞ ✭ ❍❡❝❦❡ ❡✐❣❡♥❢♦r♠s✮
  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✶✻ ✴ ✷✽

slide-35
SLIDE 35

❍❡❝❦❡✲▼❛❛ss ❢♦r♠s ♦♥ M = SL✷(Z)\H

φ(γz) = φ(z), ❢♦r ❛❧❧ γ ∈ SL✷(Z) ∆φ + λφ = ✵ φ ∈ L✷(M) Tnφ = λφ(n)φ, ❢♦r ❛❧❧ n ≥ ✶ ✕ s✉❝❤ ✬s ❛r❡ ❡①❛♠♣❧❡s ♦❢ ❛✉t♦♠♦r♣❤✐❝ ❢♦r♠s✱ ❛♥❞ ❛r❡ ❜❛s✐❝ ♦❜❥❡❝ts ✐♥ ♠♦❞❡r♥ ♥✉♠❜❡r t❤❡♦r② ❤♦♣❡✿ ◗❯❊ q✉❡st✐♦♥s ❢♦r ❛r✐t❤♠❡t✐❝ ♠❛♥✐❢♦❧❞s ❝❛♥ ❜❡ st✉❞✐❡❞ ❜② ♠❡t❤♦❞s ❢r♦♠ t❤❡ t❤❡♦r✐❡s ♦❢ ❛✉t♦♠♦r♣❤✐❝ ❢♦r♠s ❛♥❞ ❛ss♦❝✐❛t❡❞ ✲❢✉♥❝t✐♦♥s

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✶✼ ✴ ✷✽

slide-36
SLIDE 36

❍❡❝❦❡✲▼❛❛ss ❢♦r♠s ♦♥ M = SL✷(Z)\H

φ(γz) = φ(z), ❢♦r ❛❧❧ γ ∈ SL✷(Z) ∆φ + λφ = ✵ φ ∈ L✷(M) Tnφ = λφ(n)φ, ❢♦r ❛❧❧ n ≥ ✶ ✕ s✉❝❤ φ✬s ❛r❡ ❡①❛♠♣❧❡s ♦❢ ❛✉t♦♠♦r♣❤✐❝ ❢♦r♠s✱ ❛♥❞ ❛r❡ ❜❛s✐❝ ♦❜❥❡❝ts ✐♥ ♠♦❞❡r♥ ♥✉♠❜❡r t❤❡♦r② ❤♦♣❡✿ ◗❯❊ q✉❡st✐♦♥s ❢♦r ❛r✐t❤♠❡t✐❝ ♠❛♥✐❢♦❧❞s ❝❛♥ ❜❡ st✉❞✐❡❞ ❜② ♠❡t❤♦❞s ❢r♦♠ t❤❡ t❤❡♦r✐❡s ♦❢ ❛✉t♦♠♦r♣❤✐❝ ❢♦r♠s ❛♥❞ ❛ss♦❝✐❛t❡❞ ✲❢✉♥❝t✐♦♥s

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✶✼ ✴ ✷✽

slide-37
SLIDE 37

❍❡❝❦❡✲▼❛❛ss ❢♦r♠s ♦♥ M = SL✷(Z)\H

φ(γz) = φ(z), ❢♦r ❛❧❧ γ ∈ SL✷(Z) ∆φ + λφ = ✵ φ ∈ L✷(M) Tnφ = λφ(n)φ, ❢♦r ❛❧❧ n ≥ ✶ ✕ s✉❝❤ φ✬s ❛r❡ ❡①❛♠♣❧❡s ♦❢ ❛✉t♦♠♦r♣❤✐❝ ❢♦r♠s✱ ❛♥❞ ❛r❡ ❜❛s✐❝ ♦❜❥❡❝ts ✐♥ ♠♦❞❡r♥ ♥✉♠❜❡r t❤❡♦r② ❤♦♣❡✿ ◗❯❊ q✉❡st✐♦♥s ❢♦r ❛r✐t❤♠❡t✐❝ ♠❛♥✐❢♦❧❞s ❝❛♥ ❜❡ st✉❞✐❡❞ ❜② ♠❡t❤♦❞s ❢r♦♠ t❤❡ t❤❡♦r✐❡s ♦❢ ❛✉t♦♠♦r♣❤✐❝ ❢♦r♠s ❛♥❞ ❛ss♦❝✐❛t❡❞ L✲❢✉♥❝t✐♦♥s

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✶✼ ✴ ✷✽

slide-38
SLIDE 38

✶st, ✶✵th, ✶✼th ❛♥❞ ✸✸rd ❍❡❝❦❡✲▼❛❛ss ❡✐❣❡♥❢✉♥❝t✐♦♥

P✐❝t✉r❡✿ ✶

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✶✽ ✴ ✷✽

slide-39
SLIDE 39

❖♥❡ |φ|✷(z) ❢♦r ❛ ♥♦♥✲❛r✐t❤♠❡t✐❝ s✉r❢❛❝❡ Γ\H ♦❢ ❣❡♥✉s t✇♦

P✐❝t✉r❡✿ ✶

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✶✾ ✴ ✷✽

slide-40
SLIDE 40

❆r✐t❤♠❡t✐❝ ◗❯❊ ✐s tr✉❡✦

❚❤❡♦r❡♠ ✭❊✳ ▲✐♥❞❡♥str❛✉ss✱ ❆♥♥✳ ♦❢ ▼❛t❤✳ ✷✵✵✻✮ ▲❡t M ❜❡ ❛ ❝♦♠♣❛❝t ❛r✐t❤♠❡t✐❝ s✉r❢❛❝❡✳ ❚❤❡♥ t❤❡ ♦♥❧② q✉❛♥t✉♠ ❧✐♠✐t ✐s t❤❡ ▲✐♦✉✈✐❧❧❡ ♠❡❛s✉r❡ µ✳ ♣r♦♦❢✿ ♠❡❛s✉r❡ r✐❣✐❞✐t② ❢♦r ❤✐❣❤❡r r❛♥❦ ❞✐❛❣♦♥❛❧ ❛❝t✐♦♥s ♦♥ ❤♦♠♦❣❡♥❡♦✉s s♣❛❝❡s ❚❤❡♦r❡♠ ✭❊✳ ▲✐♥❞❡♥str❛✉ss✱ ✷✵✵✻ ✰ ❑✳ ❙♦✉♥❞❛r❛r❛❥❛♥✱ ❆♥♥✳ ♦❢ ▼❛t❤✳ ✷✵✶✵✮ ▲❡t ❜❡ ❛ ♥♦♥❝♦♠♣❛❝t ❛r✐t❤♠❡t✐❝ s✉r❢❛❝❡✳ ❚❤❡♥ ◗❯❊ ❤♦❧❞s ❢♦r ❜♦t❤ t❤❡ ❝♦♥t✐♥✉♦✉s ❛♥❞ ❞✐s❝r❡t❡ ❍❡❝❦❡ ❡✐❣❡♥❢✉♥❝t✐♦♥s ♦♥ ✳ ♣r♦♦❢✿ ✰ t❤❡♦r② ♦❢ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ❢✉♥❝t✐♦♥s t♦ ❡❧✐♠✐♥❛t❡ ❛♥ ✧❡s❝❛♣❡ ♦❢ ♠❛ss ✐♥t♦ ❝✉s♣✧

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✷✵ ✴ ✷✽

slide-41
SLIDE 41

❆r✐t❤♠❡t✐❝ ◗❯❊ ✐s tr✉❡✦

❚❤❡♦r❡♠ ✭❊✳ ▲✐♥❞❡♥str❛✉ss✱ ❆♥♥✳ ♦❢ ▼❛t❤✳ ✷✵✵✻✮ ▲❡t M ❜❡ ❛ ❝♦♠♣❛❝t ❛r✐t❤♠❡t✐❝ s✉r❢❛❝❡✳ ❚❤❡♥ t❤❡ ♦♥❧② q✉❛♥t✉♠ ❧✐♠✐t ✐s t❤❡ ▲✐♦✉✈✐❧❧❡ ♠❡❛s✉r❡ µ✳ ♣r♦♦❢✿ ♠❡❛s✉r❡ r✐❣✐❞✐t② ❢♦r ❤✐❣❤❡r r❛♥❦ ❞✐❛❣♦♥❛❧ ❛❝t✐♦♥s ♦♥ ❤♦♠♦❣❡♥❡♦✉s s♣❛❝❡s ❚❤❡♦r❡♠ ✭❊✳ ▲✐♥❞❡♥str❛✉ss✱ ✷✵✵✻ ✰ ❑✳ ❙♦✉♥❞❛r❛r❛❥❛♥✱ ❆♥♥✳ ♦❢ ▼❛t❤✳ ✷✵✶✵✮ ▲❡t M ❜❡ ❛ ♥♦♥❝♦♠♣❛❝t ❛r✐t❤♠❡t✐❝ s✉r❢❛❝❡✳ ❚❤❡♥ ◗❯❊ ❤♦❧❞s ❢♦r ❜♦t❤ t❤❡ ❝♦♥t✐♥✉♦✉s ❛♥❞ ❞✐s❝r❡t❡ ❍❡❝❦❡ ❡✐❣❡♥❢✉♥❝t✐♦♥s ♦♥ M✳ ♣r♦♦❢✿ ✰ t❤❡♦r② ♦❢ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ❢✉♥❝t✐♦♥s t♦ ❡❧✐♠✐♥❛t❡ ❛♥ ✧❡s❝❛♣❡ ♦❢ ♠❛ss ✐♥t♦ ❝✉s♣✧

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✷✵ ✴ ✷✽

slide-42
SLIDE 42

❘❛♥❞♦♠ ❲❛✈❡ ❈♦♥❥❡❝t✉r❡

✕ ❢♦r♠✉❧❛t❡❞ ❜② ❇❡rr② ✭✶✾✼✼✮ ❢♦r q✉❛♥t✐③❛t✐♦♥s ♦❢ ❝❤❛♦t✐❝ ❍❛♠✐❧t♦♥✐❛♥s ✕ ❡①t❡♥❞❡❞ ❜② ❍❡❥❤❛❧ ❛♥❞ ❘❛❝❦♥❡r ✭✶✾✾✷✮ t♦ ♥♦♥✲❝♦♠♣❛❝t s✉r❢❛❝❡s ♦❢ ✜♥✐t❡ ✈♦❧✉♠❡ ✰ ♥✉♠❡r✐❝❛❧ ❡✈✐❞❡♥❝❡ ❈♦♥❥❡❝t✉r❡ ✭❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡✮ ■♥ t❤❡ ❝❛s❡ ♦❢ ♥❡❣❛t✐✈❡ ❝✉r✈❛t✉r❡✱ t❤❡ ▲❛♣❧❛❝❡ ❡✐❣❡♥❢✉♥❝t✐♦♥s t❡♥❞ t♦ ❡①❤✐❜✐t ●❛✉ss✐❛♥ r❛♥❞♦♠ ❜❡❤❛✈✐♦r ✐♥ t❤❡ ❤✐❣❤ ❡♥❡r❣② ❧✐♠✐t✳

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✷✶ ✴ ✷✽

slide-43
SLIDE 43

❘❛♥❞♦♠ ❲❛✈❡ ❈♦♥❥❡❝t✉r❡

✕ ❢♦r♠✉❧❛t❡❞ ❜② ❇❡rr② ✭✶✾✼✼✮ ❢♦r q✉❛♥t✐③❛t✐♦♥s ♦❢ ❝❤❛♦t✐❝ ❍❛♠✐❧t♦♥✐❛♥s ✕ ❡①t❡♥❞❡❞ ❜② ❍❡❥❤❛❧ ❛♥❞ ❘❛❝❦♥❡r ✭✶✾✾✷✮ t♦ ♥♦♥✲❝♦♠♣❛❝t s✉r❢❛❝❡s ♦❢ ✜♥✐t❡ ✈♦❧✉♠❡ ✰ ♥✉♠❡r✐❝❛❧ ❡✈✐❞❡♥❝❡ ❈♦♥❥❡❝t✉r❡ ✭❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡✮ ■♥ t❤❡ ❝❛s❡ ♦❢ ♥❡❣❛t✐✈❡ ❝✉r✈❛t✉r❡✱ t❤❡ ▲❛♣❧❛❝❡ ❡✐❣❡♥❢✉♥❝t✐♦♥s t❡♥❞ t♦ ❡①❤✐❜✐t ●❛✉ss✐❛♥ r❛♥❞♦♠ ❜❡❤❛✈✐♦r ✐♥ t❤❡ ❤✐❣❤ ❡♥❡r❣② ❧✐♠✐t✳

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✷✶ ✴ ✷✽

slide-44
SLIDE 44

❘❛♥❞♦♠ ❲❛✈❡ ❈♦♥❥❡❝t✉r❡

✕ ❢♦r♠✉❧❛t❡❞ ❜② ❇❡rr② ✭✶✾✼✼✮ ❢♦r q✉❛♥t✐③❛t✐♦♥s ♦❢ ❝❤❛♦t✐❝ ❍❛♠✐❧t♦♥✐❛♥s ✕ ❡①t❡♥❞❡❞ ❜② ❍❡❥❤❛❧ ❛♥❞ ❘❛❝❦♥❡r ✭✶✾✾✷✮ t♦ ♥♦♥✲❝♦♠♣❛❝t s✉r❢❛❝❡s ♦❢ ✜♥✐t❡ ✈♦❧✉♠❡ ✰ ♥✉♠❡r✐❝❛❧ ❡✈✐❞❡♥❝❡ ❈♦♥❥❡❝t✉r❡ ✭❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡✮ ■♥ t❤❡ ❝❛s❡ ♦❢ ♥❡❣❛t✐✈❡ ❝✉r✈❛t✉r❡✱ t❤❡ ▲❛♣❧❛❝❡ ❡✐❣❡♥❢✉♥❝t✐♦♥s φλ t❡♥❞ t♦ ❡①❤✐❜✐t ●❛✉ss✐❛♥ r❛♥❞♦♠ ❜❡❤❛✈✐♦r ✐♥ t❤❡ ❤✐❣❤ ❡♥❡r❣② ❧✐♠✐t✳

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✷✶ ✴ ✷✽

slide-45
SLIDE 45

❚❤❡ ♠♦♠❡♥t ✈❡rs✐♦♥

❈♦♥❥❡❝t✉r❡ ✭❘❲❈ ✲ t❤❡ ♠♦♠❡♥t ✈❡rs✐♦♥✮ ❋♦r ❛♥② ✐♥t❡❣❡r ✷ ≤ p < ∞ ❛♥❞ ❛♥② ♥✐❝❡✱ ❝♦♠♣❛❝t Ω ⊂ SL✷(Z)\H ✇❡ ❤❛✈❡ ✶ ✈♦❧(Ω)

  • Ω

φp

λ(z)dxdy

y✷ − → σpcp, λ → ∞, ✇❤❡r❡ cp ✐s t❤❡ pt❤ ♠♦♠❡♥t ♦❢ t❤❡ ♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥ N(✵; ✶) ❛♥❞ σ✷ =

✶ ✈♦❧(SL✷(Z)\H) = ✸ π ✐s t❤❡ ❝♦♥❥❡❝t✉r❡❞ ✈❛r✐❛♥❝❡ ♦❢ t❤❡ r❛♥❞♦♠ ✇❛✈❡✳

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✷✷ ✴ ✷✽

slide-46
SLIDE 46

❙♣❡❝tr❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ∆ ♦♥ SL✷(Z)\H

f (z) =

  • j≥✵

f , φjφj(z) + ✶ ✹π +∞

−∞

f , E(·, ✶ ✷ + it)E(z, ✶ ✷ + it)dt ✇❤❡r❡

✵

✸

✶ ✷

✱ ✶ ✕ ❝✉s♣ ❢♦r♠s ✭❞✐s❝r❡t❡ s♣❡❝tr✉♠✮

✶ ✷

✕ ❊✐s❡♥st❡✐♥ s❡r✐❡s ✭❝♦♥t✐♥✉♦✉s s♣❡❝tr✉♠✮ ✕ ♦❜t❛✐♥❡❞ ❛s t❤❡ ❛♥❛❧②t✐❝ ❝♦♥t✐♥✉❛t✐♦♥ ✭❜② ❙❡❧❜❡r❣✮ ✐♥ ✲✈❛r✐❛❜❧❡ ♦❢ ✶

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✷✸ ✴ ✷✽

slide-47
SLIDE 47

❙♣❡❝tr❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ∆ ♦♥ SL✷(Z)\H

f (z) =

  • j≥✵

f , φjφj(z) + ✶ ✹π +∞

−∞

f , E(·, ✶ ✷ + it)E(z, ✶ ✷ + it)dt ✇❤❡r❡ φ✵(z) = (✸/π)✶/✷ φj(z)✱ j ≥ ✶ ✕ ❝✉s♣ ❢♦r♠s ✭❞✐s❝r❡t❡ s♣❡❝tr✉♠✮ E(z, ✶

✷ + it) ✕ ❊✐s❡♥st❡✐♥ s❡r✐❡s ✭❝♦♥t✐♥✉♦✉s s♣❡❝tr✉♠✮

✕ ♦❜t❛✐♥❡❞ ❛s t❤❡ ❛♥❛❧②t✐❝ ❝♦♥t✐♥✉❛t✐♦♥ ✭❜② ❙❡❧❜❡r❣✮ ✐♥ s✲✈❛r✐❛❜❧❡ ♦❢ E(z, s) :=

  • Γ∞\Γ

ℑ(γz)s, ℜ(s) > ✶

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✷✸ ✴ ✷✽

slide-48
SLIDE 48

❘❛♥❞♦♠ ❲❛✈❡ ❈♦♥❥❡❝t✉r❡ ❢♦r ❝✉s♣ ❢♦r♠s

❚❤❡♦r❡♠ ✭❏✳ ❇✉tt❝❛♥❡✱ ❘✳ ❑❤❛♥✱ ✷✵✶✻✮ ❆ss✉♠❡ t❤❡ ●❡♥❡r❛❧✐③❡❞ ▲✐♥❞❡❧ö❢ ❍②♣♦t❤❡s✐s✳ ▲❡t f ❜❡ ❛♥ ❡✈❡♥ ♦r ♦❞❞ ❍❡❝❦❡✲▼❛❛ss ❝✉s♣ ❢♦r♠ ❢♦r M = SL✷(Z)\H ✇✐t❤ ▲❛♣❧❛❝✐❛♥ ❡✐❣❡♥✈❛❧✉❡ λ = ✶

✹ + T ✷✱ ✇❤❡r❡ T > ✵✳ ▲❡t f ❜❡ ♥♦r♠❛❧✐③❡❞ t♦ ❤❛✈❡ ♣r♦❜❛❜✐❧✐t②

♠❡❛s✉r❡ ❡q✉❛❧ t♦ ✶✱ ❛s ❢♦❧❧♦✇s✿ ✶ ✈♦❧(M)

  • M

|f (z)|✷ dxdy y✷ = ✶. ❚❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t δ > ✵ s✉❝❤ t❤❛t ✶ ✈♦❧(M)

  • M

|f (z)|✹ dxdy y✷ = ✶ √ ✷π ∞

−∞

t✹ e−t✷/✷dt + O(T −δ), T → ∞. ✕ ❝♦♥✜r♠s t❤❡ ❘❲❈✱ ✇✐t❤ ❛ ♣♦✇❡r s❛✈✐♥❣✱ ❢♦r ❝✉s♣ ❢♦r♠s✱ ❝♦♥❞✐t✐♦♥❛❧❧② ♦♥

  • ▲❍
  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✷✹ ✴ ✷✽

slide-49
SLIDE 49

❘❛♥❞♦♠ ❲❛✈❡ ❈♦♥❥❡❝t✉r❡ ❢♦r ❝✉s♣ ❢♦r♠s

❚❤❡♦r❡♠ ✭❏✳ ❇✉tt❝❛♥❡✱ ❘✳ ❑❤❛♥✱ ✷✵✶✻✮ ❆ss✉♠❡ t❤❡ ●❡♥❡r❛❧✐③❡❞ ▲✐♥❞❡❧ö❢ ❍②♣♦t❤❡s✐s✳ ▲❡t f ❜❡ ❛♥ ❡✈❡♥ ♦r ♦❞❞ ❍❡❝❦❡✲▼❛❛ss ❝✉s♣ ❢♦r♠ ❢♦r M = SL✷(Z)\H ✇✐t❤ ▲❛♣❧❛❝✐❛♥ ❡✐❣❡♥✈❛❧✉❡ λ = ✶

✹ + T ✷✱ ✇❤❡r❡ T > ✵✳ ▲❡t f ❜❡ ♥♦r♠❛❧✐③❡❞ t♦ ❤❛✈❡ ♣r♦❜❛❜✐❧✐t②

♠❡❛s✉r❡ ❡q✉❛❧ t♦ ✶✱ ❛s ❢♦❧❧♦✇s✿ ✶ ✈♦❧(M)

  • M

|f (z)|✷ dxdy y✷ = ✶. ❚❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t δ > ✵ s✉❝❤ t❤❛t ✶ ✈♦❧(M)

  • M

|f (z)|✹ dxdy y✷ = ✶ √ ✷π ∞

−∞

t✹ e−t✷/✷dt + O(T −δ), T → ∞. ✕ ❝♦♥✜r♠s t❤❡ ❘❲❈✱ ✇✐t❤ ❛ ♣♦✇❡r s❛✈✐♥❣✱ ❢♦r ❝✉s♣ ❢♦r♠s✱ ❝♦♥❞✐t✐♦♥❛❧❧② ♦♥

  • ▲❍
  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✷✹ ✴ ✷✽

slide-50
SLIDE 50

❘❛♥❞♦♠ ❲❛✈❡ ❈♦♥❥❡❝t✉r❡ ❢♦r ❊✐s❡♥st❡✐♥ s❡r✐❡s

❚❤❡♦r❡♠ ✭●✳ ❉❥✳✱ ❘✳ ❑❤❛♥✱ ✷✵✶✼✮ ▲❡t {φj : j ≥ ✶} ❞❡♥♦t❡ ❛♥ ♦rt❤♦♥♦r♠❛❧ ❜❛s✐s ♦❢ ❡✈❡♥ ❛♥❞ ♦❞❞ ❍❡❝❦❡✲▼❛❛ss ❝✉s♣ ❢♦r♠s ❢♦r SL✷(Z)✱ ♦r❞❡r❡❞ ❜② ▲❛♣❧❛❝✐❛♥ ❡✐❣❡♥✈❛❧✉❡ ✶

✹ + it✷ j ✱ ❛♥❞ ❧❡t

Λ(s, φj) ❞❡♥♦t❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❝♦♠♣❧❡t❡❞ L✲❢✉♥❝t✐♦♥s✳ ▲❡t ξ(s) ❞❡♥♦t❡ t❤❡ ❝♦♠♣❧❡t❡❞ ❘✐❡♠❛♥♥ ζ ❢✉♥❝t✐♦♥✳ ❆s T → ∞✱ ✇❡ ❤❛✈❡ reg

M

|E(z, ✶/✷ + iT)|✹ dxdy y✷ = ✷✹ π ❧♦❣✷ T +

  • j≥✶

❝♦s❤(πtj) ✷ |Λ( ✶

✷ + ✷Ti, φj)|✷Λ✷( ✶ ✷, φj)

L(✶, s②♠✷φj) |ξ(✶ + ✷Ti)|✹ + O(❧♦❣✺/✸+ǫ T), ❢♦r ❛♥② ǫ > ✵✳

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✷✺ ✴ ✷✽

slide-51
SLIDE 51

❘❲❈ ❢♦r t❤❡ r❡❣✉❧❛r✐③❡❞ ✹th ♠♦♠❡♥t

❈♦♥❥❡❝t✉r❡ ✭●✳ ❉❥✳✱ ❘✳ ❑❤❛♥✱ ✷✵✶✼ ✕ ❘❲❈ ❢♦r t❤❡ r❡❣✉❧❛r✐③❡❞ ❢♦✉rt❤ ♠♦♠❡♥t ♦❢ ❊✐s❡♥st❡✐♥ s❡r✐❡s✮ reg

M

|E(z, ✶ ✷ + iT)|✹ dxdy y✷ ∼ ✼✷ π ❧♦❣✷ T. ❤♦♣❡✿ t♦ ✉♥❞❡rst❛♥❞ t❤❡ s✉♠ ♦❢ s♣❡❝✐❛❧ ✈❛❧✉❡s ♦❢ ✲❢✉♥❝t✐♦♥s ✐♥ ❢❛♠✐❧② ♦❢ ❍❡❝❦❡✲▼❛❛ss ❢♦r♠s✱ ❜② ♠❡t❤♦❞s ❢r♦♠ ❛♥❛❧②t✐❝ ♥✉♠❜❡r t❤❡♦r②

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✷✻ ✴ ✷✽

slide-52
SLIDE 52

❘❲❈ ❢♦r t❤❡ r❡❣✉❧❛r✐③❡❞ ✹th ♠♦♠❡♥t

❈♦♥❥❡❝t✉r❡ ✭●✳ ❉❥✳✱ ❘✳ ❑❤❛♥✱ ✷✵✶✼ ✕ ❘❲❈ ❢♦r t❤❡ r❡❣✉❧❛r✐③❡❞ ❢♦✉rt❤ ♠♦♠❡♥t ♦❢ ❊✐s❡♥st❡✐♥ s❡r✐❡s✮ reg

M

|E(z, ✶ ✷ + iT)|✹ dxdy y✷ ∼ ✼✷ π ❧♦❣✷ T. ❤♦♣❡✿ t♦ ✉♥❞❡rst❛♥❞ t❤❡ s✉♠ ♦❢ s♣❡❝✐❛❧ ✈❛❧✉❡s ♦❢ L✲❢✉♥❝t✐♦♥s ✐♥ ❢❛♠✐❧② ♦❢ ❍❡❝❦❡✲▼❛❛ss ❢♦r♠s✱ ❜② ♠❡t❤♦❞s ❢r♦♠ ❛♥❛❧②t✐❝ ♥✉♠❜❡r t❤❡♦r②

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✷✻ ✴ ✷✽

slide-53
SLIDE 53

❚r✐♣❧❡ ♣r♦❞✉❝t ❢♦r♠✉❧❛

❚✳ ❲❛ts♦♥✱ ✷✵✵✶ ✕ ❢♦r♠✉❧❛ r❡❧❛t✐♥❣ t❤❡ ✐♥t❡❣r❛❧ ♦❢ ❛ ♣r♦❞✉❝t ♦❢ t❤r❡❡ ❛✉t♦♠♦r♣❤✐❝ ❢♦r♠s ♦♥ ❛♥ ❛r✐t❤♠❡t✐❝ s✉r❢❛❝❡ t♦ t❤❡ s♣❡❝✐❛❧ ✈❛❧✉❡ ♦❢ ❛ ❞❡❣r❡❡ ✽ L✲❢✉♥❝t✐♦♥ ✕ ✐♥ t❤❡ ❝❛s❡ ♦❢

✷

❛♥❞ t❤r❡❡ ❍❡❝❦❡✲▼❛❛ss ❝✉s♣ ❢♦r♠s

✶ ✷ ✸ ✇❤♦s❡ ✷✲♥♦r♠s ❛r❡ ♥♦r♠❛❧✐③❡❞ t♦ ❜❡ ✶ ✿ ✶ ✷ ✸ ✷ ✹

✷✶✻ ✶ ✷

✶ ✷ ✸

✶ s②♠✷

✶

✶ s②♠✷

✷

✶ s②♠✷

✸ ✶ ✷ ✸ ✕

✲❢✉♥❝t✐♦♥ ♦❢ ❞❡❣r❡❡ ✽ ✭❘✐❡♠❛♥♥ ③❡t❛ ❢✉♥❝t✐♦♥ ✐s ♦❢ ❞❡❣r❡❡ ✶✮

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✷✼ ✴ ✷✽

slide-54
SLIDE 54

❚r✐♣❧❡ ♣r♦❞✉❝t ❢♦r♠✉❧❛

❚✳ ❲❛ts♦♥✱ ✷✵✵✶ ✕ ❢♦r♠✉❧❛ r❡❧❛t✐♥❣ t❤❡ ✐♥t❡❣r❛❧ ♦❢ ❛ ♣r♦❞✉❝t ♦❢ t❤r❡❡ ❛✉t♦♠♦r♣❤✐❝ ❢♦r♠s ♦♥ ❛♥ ❛r✐t❤♠❡t✐❝ s✉r❢❛❝❡ t♦ t❤❡ s♣❡❝✐❛❧ ✈❛❧✉❡ ♦❢ ❛ ❞❡❣r❡❡ ✽ L✲❢✉♥❝t✐♦♥ ✕ ✐♥ t❤❡ ❝❛s❡ ♦❢ M = SL✷(Z)\H ❛♥❞ t❤r❡❡ ❍❡❝❦❡✲▼❛❛ss ❝✉s♣ ❢♦r♠s φ✶, φ✷, φ✸ ✇❤♦s❡ L✷✲♥♦r♠s ❛r❡ ♥♦r♠❛❧✐③❡❞ t♦ ❜❡ ✶ ✿

  • M

φ✶(z)φ✷(z)φ✸(z)

  • ✷

= π✹ ✷✶✻ Λ(✶/✷, φ✶ ⊗ φ✷ ⊗ φ✸) Λ(✶, s②♠✷φ✶)Λ(✶, s②♠✷φ✷)Λ(✶, s②♠✷φ✸)

✶ ✷ ✸ ✕

✲❢✉♥❝t✐♦♥ ♦❢ ❞❡❣r❡❡ ✽ ✭❘✐❡♠❛♥♥ ③❡t❛ ❢✉♥❝t✐♦♥ ✐s ♦❢ ❞❡❣r❡❡ ✶✮

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✷✼ ✴ ✷✽

slide-55
SLIDE 55

❚r✐♣❧❡ ♣r♦❞✉❝t ❢♦r♠✉❧❛

❚✳ ❲❛ts♦♥✱ ✷✵✵✶ ✕ ❢♦r♠✉❧❛ r❡❧❛t✐♥❣ t❤❡ ✐♥t❡❣r❛❧ ♦❢ ❛ ♣r♦❞✉❝t ♦❢ t❤r❡❡ ❛✉t♦♠♦r♣❤✐❝ ❢♦r♠s ♦♥ ❛♥ ❛r✐t❤♠❡t✐❝ s✉r❢❛❝❡ t♦ t❤❡ s♣❡❝✐❛❧ ✈❛❧✉❡ ♦❢ ❛ ❞❡❣r❡❡ ✽ L✲❢✉♥❝t✐♦♥ ✕ ✐♥ t❤❡ ❝❛s❡ ♦❢ M = SL✷(Z)\H ❛♥❞ t❤r❡❡ ❍❡❝❦❡✲▼❛❛ss ❝✉s♣ ❢♦r♠s φ✶, φ✷, φ✸ ✇❤♦s❡ L✷✲♥♦r♠s ❛r❡ ♥♦r♠❛❧✐③❡❞ t♦ ❜❡ ✶ ✿

  • M

φ✶(z)φ✷(z)φ✸(z)

  • ✷

= π✹ ✷✶✻ Λ(✶/✷, φ✶ ⊗ φ✷ ⊗ φ✸) Λ(✶, s②♠✷φ✶)Λ(✶, s②♠✷φ✷)Λ(✶, s②♠✷φ✸) Λ(s, φ✶ ⊗ φ✷ ⊗ φ✸) ✕ L✲❢✉♥❝t✐♦♥ ♦❢ ❞❡❣r❡❡ ✽ ✭❘✐❡♠❛♥♥ ③❡t❛ ❢✉♥❝t✐♦♥ ζ(s) ✐s ♦❢ ❞❡❣r❡❡ ✶✮

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✷✼ ✴ ✷✽

slide-56
SLIDE 56

❚❤❛♥❦ ②♦✉✦

  • ♦r❛♥ ❉❥❛♥❦♦✈✐➣

❘❛♥❞♦♠ ✇❛✈❡ ❝♦♥❥❡❝t✉r❡ ✶✽✳ ✾✳ ✷✵✶✼✳ ✷✽ ✴ ✷✽