Quadratic Forms and Closed Geodesics

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Pell’s Equation Quadratic Forms and Closed Geodesics Y. Petridis ◮ Algorithm for solving x 2 − 2y 2 = ±1 (Pythagoreans) ◮ Start with ◮ We get x 1 = 1, y 1 = 1 (x 2 , y 2 ) = (3, 2) (x 3 , y 3 ) = (7, 5) (x 4 , y 4 ) = (17, 12) ◮ Recurrences x n+1 = x n + 2y n y n+1 = x n + y n ◮ Fundamental solution: 1 + √ 2 x n + √ 2y n = (1 + √ 2) n Quadratic Forms Hyperbolic surfaces Closed Geodesics Spectral Theory
Pell’s Equation Quadratic Forms and Closed Geodesics Y. Petridis ◮ Algorithm for solving x 2 − 2y 2 = ±1 (Pythagoreans) ◮ Start with ◮ We get x 1 = 1, y 1 = 1 (x 2 , y 2 ) = (3, 2) (x 3 , y 3 ) = (7, 5) (x 4 , y 4 ) = (17, 12) ◮ Recurrences x n+1 = x n + 2y n y n+1 = x n + y n ◮ Fundamental solution: 1 + √ 2 x n + √ 2y n = (1 + √ 2) n Quadratic Forms Hyperbolic surfaces Closed Geodesics Spectral Theory
Page 1 and 2: Quadratic Forms and Closed Geodesic
Page 3: Pell’s Equation Quadratic Forms a
Page 7 and 8: Pell’s Equation Quadratic Forms a
Page 9 and 10: Indefinite Quadratic Forms ◮ Q(x,
Page 15 and 16: Class Numbers h(d) = number of ineq
Page 17 and 18: From Quadratic Forms to Hyperbolic
Page 19 and 20: Linear Fractional Transformations a
Page 21 and 22: Geodesics in Hyperbolic Disc Quadra
Page 23 and 24: Fundamental Domains z ∼ w ⇔ w =
Page 25 and 26: ¡ Tesselations Quadratic Forms and
Page 27 and 28: Pell’s Equation and Lengths of Cl
Page 29 and 30: Distribution Closed geodesics of H/
Page 31: The Laplace Operator Quadratic Form
Page 34 and 35: Contour plots of eigenfunctions of
Page 36 and 37: The Riemann Hypothesis ◮ Think of
Page 38 and 39: Relations of RH with spectral theor
Page 40 and 41: Duality between periods and eigenva
Page 42 and 43: Simple trace formula Quadratic Form

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