Verified computations for hyperbolic 3-manifolds

Transcription

1 Verified computations for hyperbolic 3-manifolds Neil Hoffman Kazuhiro Ichihara Masahide Kashiwagi Hidetoshi Masai Shin ichi Oishi Akitoshi Takayasu (U. Melbourne) (Nihon U.) (Waseda U.) (U. Tokyo) (Waseda U.) (Waseda U.) A Satellite Conference of Seoul ICM 2014 Knots and Low Dimensional Manifolds August 22, BEXCO, Busan, Korea 1/14

2 Classification of 3-manifolds Every closed orientable 3-manifold is; Reducible Toroidal Seifert fibered Hyperbolic ( Riem.metric of curv. 1). as a consequence of the Geometrization Conjecture including famous Poincaré Conjecture (1904) Question: conjectured by Thurston (late 70s) established by Perelman ( ) How to prove a given 3-manifold is really hyperbolic? 2/14

3 Classification of 3-manifolds Every closed orientable 3-manifold is; Reducible Toroidal Seifert fibered Hyperbolic ( Riem.metric of curv. 1). as a consequence of the Geometrization Conjecture including famous Poincaré Conjecture (1904) Question: conjectured by Thurston (late 70s) established by Perelman ( ) How to prove a given 3-manifold is really hyperbolic? 2/14

4 Thurston s Idea [W. Thurston] M: triangulated 3-manifold, possibly with torus boundary. equation s.t. whose solution (IF ANY) gives rise to a hyperbolic structure on M. (Gluing equation) n (z j ) (a j,m c j,m) (1 z j ) ( b j,m+c j,m ) = j=1 for m = 1,, n + 2k + h and n ( 1) c j,m j=1 n arg((z j ) (a j,m c j,m ) ) + arg((1 z j ) ( b j,m+c j,m ) ) = ϵ m j=1 How to solve? Use Computer! n c j,m πi. j=1 3/14

5 Thurston s Idea [W. Thurston] M: triangulated 3-manifold, possibly with torus boundary. equation s.t. whose solution (IF ANY) gives rise to a hyperbolic structure on M. (Gluing equation) n (z j ) (a j,m c j,m) (1 z j ) ( b j,m+c j,m ) = j=1 for m = 1,, n + 2k + h and n ( 1) c j,m j=1 n arg((z j ) (a j,m c j,m ) ) + arg((1 z j ) ( b j,m+c j,m ) ) = ϵ m j=1 How to solve? Use Computer! n c j,m πi. j=1 3/14

6 Weeks Software [J. Weeks] SnapPea: a computer program to make/solve glueing equations. [M. Culler - N. Dunfield] SnapPy: SnapPea on python SnapPea uses Newton s method. floating point error? finitely many calculations can ensure convergece? 4/14

7 Newton s iteration converges but... Exact solution? In numerical computations, there are truncation errors when stopping an infinite series, e.g. sin x x x3 3! + x5 5! x7 7! + x9 9! x11 11! Influence of rounding error? It is well-known that a rounding error occurs in every floating-point operation /14

8 Newton s iteration converges but... Exact solution? In numerical computations, there are truncation errors when stopping an infinite series, e.g. sin x x x3 3! + x5 5! x7 7! + x9 9! x11 11! Influence of rounding error? It is well-known that a rounding error occurs in every floating-point operation /14

9 Moser s Proposal H. Moser, Proving a manifold to be hyperbolic once it has been approximated to be so, Algebraic & Geometric Topology, 9 (2009), The literature says that Kantorovich s convergence theorem guarantees the convergence of Newton s method. There is an exact solution in the neighborhood of an approximate solution. Can you believe this computer-aided procedure? (In particular, rounding error in floating-point operations.) 6/14

10 Moser s Proposal H. Moser, Proving a manifold to be hyperbolic once it has been approximated to be so, Algebraic & Geometric Topology, 9 (2009), The literature says that Kantorovich s convergence theorem guarantees the convergence of Newton s method. There is an exact solution in the neighborhood of an approximate solution. Can you believe this computer-aided procedure? (In particular, rounding error in floating-point operations.) 6/14

11 To prove the hyperbolicity (our approach) Truncation error Krawczyk s test Rounding error Interval analysis We develop a python module: HIKMOT. Our package performs a rigorous numerical existence test for solutions of gluing equations. 7/14

12 To prove the hyperbolicity (our approach) Truncation error Krawczyk s test Rounding error Interval analysis We develop a python module: HIKMOT. Our package performs a rigorous numerical existence test for solutions of gluing equations. 7/14

13 To prove the hyperbolicity (our approach) Truncation error Krawczyk s test Rounding error Interval analysis We develop a python module: HIKMOT. Our package performs a rigorous numerical existence test for solutions of gluing equations. 7/14

14 Interval analysis 8/14

15 Basic idea of interval analysis A computer cannot deal with 2 as an exact number, it can compute [ , ] We write an interval X := {x R : x x x, x, x R} = [x, x]. IR: the set of such intervals. 9/14

16 Interval arithmetic To calculate four basic operations {+,,, /}, the interval arithmetic can be executed by X + Y = [ x + y, x + y ] X Y = [ x y, x y ] X Y = [ min{x y, x y, x y, x y}, max{x y, x y, x y, x y} ] [ 1 X/Y = X y, 1 ], (0 Y ) y for X = [x, x] and Y = [y, y]. 10/14

17 Krawczyk s mapping The gluing equation can be rewritten as Find x R 2n s.t. f(x) = 0, where f : R 2n R 2n is a differentiable nonlinear real mapping. Set m = 2n. Krawczyk s mapping K : IR m IR m is defined by K(X) := c Rf(c) + (I Rf (X))(X c), where I R m m is a unit matrix, R R m m a certain matrix to be an approximate inverse of f (c) and c X is an approximate solution of f(x) = 0. 11/14

18 Krawczyk s mapping The gluing equation can be rewritten as Find x R 2n s.t. f(x) = 0, where f : R 2n R 2n is a differentiable nonlinear real mapping. Set m = 2n. Krawczyk s mapping K : IR m IR m is defined by K(X) := c Rf(c) + (I Rf (X))(X c), where I R m m is a unit matrix, R R m m a certain matrix to be an approximate inverse of f (c) and c X is an approximate solution of f(x) = 0. 11/14

19 Theorem (Krawczyk s test) For a given interval X IR m, let int(x) be the interior of X. If the condition K(X) int(x) holds, then there uniquely exists an exact solution x in X. Furthermore, it is also shown that R and all matrices C f (X) including f (x ) are nonsingular. R. Krawczyk, Newton-Algorithm zur Bestimmung von Nullstellen mit Fehlerschranken, Computing 4(1969), /14

20 Advantages of interval arithmetic Fast (especially compared to exact arithmetic) Uses less memory Overwrites the basic four operations (+,,, ) Extends to functions naturally Accumulates numerical error by itself Further techniques to implement Automatic differentiation Complex arithmetic Verified computations for arg(z) (atan2 function) 13/14

21 Advantages of interval arithmetic Fast (especially compared to exact arithmetic) Uses less memory Overwrites the basic four operations (+,,, ) Extends to functions naturally Accumulates numerical error by itself Further techniques to implement Automatic differentiation Complex arithmetic Verified computations for arg(z) (atan2 function) 13/14

22 HIKMOT Hoffman, Ichihara, Kashiwagi, Masai, Oishi, & Takayasu, Verified computations for hyperbolic 3-manifolds, submitted (arxiv: ), The python module is available on Thank you for kind attention! 14/14

23 HIKMOT Hoffman, Ichihara, Kashiwagi, Masai, Oishi, & Takayasu, Verified computations for hyperbolic 3-manifolds, submitted (arxiv: ), The python module is available on Thank you for kind attention! 14/14