S TAT E U N I V E R S I T Y O F N E W YO R K Recursive Moving Frames Francis Valiquette S TAT E U N I V E R S I T Y O F N E W YO R K Ongoing work with Peter J. Olver Fields Institute December 12, 2013 Francis Valiquette Recursive Moving Frames 12/12/2013 1 / 36
S TAT E U N I V E R S I T Y O F N E W YO R K Lie pseudo-groups Lie pseudo-groups infinite-dimensional generalization ú of local Lie group actions Given G acting on M , I’m interested in the induced action on S Ă M Example: U “ u ` e x X “ f p x q Y “ e p x , y q “ f x p x q y ` g p x q f x Francis Valiquette Recursive Moving Frames 12/12/2013 2 / 36
S TAT E U N I V E R S I T Y O F N E W YO R K Lie pseudo-groups Lie pseudo-groups infinite-dimensional generalization ú of local Lie group actions Given G acting on M , I’m interested in the induced action on S Ă M Example: U “ u ` e x X “ f p x q Y “ e p x , y q “ f x p x q y ` g p x q f x Francis Valiquette Recursive Moving Frames 12/12/2013 2 / 36
S TAT E U N I V E R S I T Y O F N E W YO R K Lie pseudo-groups Lie pseudo-groups infinite-dimensional generalization ú of local Lie group actions Given G acting on M , I’m interested in the induced action on S Ă M Example: U “ u ` e x X “ f p x q Y “ e p x , y q “ f x p x q y ` g p x q f x Francis Valiquette Recursive Moving Frames 12/12/2013 2 / 36
S TAT E U N I V E R S I T Y O F N E W YO R K Lie pseudo-groups in action symmetry of differential equations Navier–Stokes, Euler, K–P, Davey–Stewartson equivalence transformations fiber, point, contact equivalence of differential equations gauge transformations Maxwell, Yang–Mills, conformal, string, . . . invariant variational calculus – Noether’s second theorem (Stay tuned: Irina, Juha) . . . Francis Valiquette Recursive Moving Frames 12/12/2013 3 / 36
S TAT E U N I V E R S I T Y O F N E W YO R K Main theme & tools Compute differential invariants invariant differential forms invariant differential operators Tools available: Lie’s infinitesimal method Cartan’s method (EDS) Equivariant moving frames Lie algebroids Francis Valiquette Recursive Moving Frames 12/12/2013 4 / 36
S TAT E U N I V E R S I T Y O F N E W YO R K Main theme & tools Compute differential invariants invariant differential forms invariant differential operators Tools available: Lie’s infinitesimal method Cartan’s method (EDS) Equivariant moving frames Lie algebroids Francis Valiquette Recursive Moving Frames 12/12/2013 4 / 36
S TAT E U N I V E R S I T Y O F N E W YO R K Main theme & tools Compute differential invariants invariant differential forms invariant differential operators Tools available: Lie’s infinitesimal method Cartan’s method (EDS) Pick your favorite approach! Equivariant moving frames Lie algebroids Francis Valiquette Recursive Moving Frames 12/12/2013 4 / 36
S TAT E U N I V E R S I T Y O F N E W YO R K Main theme & tools Compute differential invariants invariant differential forms invariant differential operators Tools available: Lie’s infinitesimal method Cartan’s method (EDS) Pick your favorite approach! Equivariant moving frames Lie algebroids Francis Valiquette Recursive Moving Frames 12/12/2013 4 / 36
S TAT E U N I V E R S I T Y O F N E W YO R K Why use equivariant moving frames? Decouples the moving frame theory from reliance on any form of frame bundle or connection basic calculus even an undergraduate student can do this! Francis Valiquette Recursive Moving Frames 12/12/2013 5 / 36
S TAT E U N I V E R S I T Y O F N E W YO R K Why use equivariant moving frames? Decouples the moving frame theory from reliance on any form of frame bundle or connection basic calculus even an undergraduate student can do this! Francis Valiquette Recursive Moving Frames 12/12/2013 5 / 36
S TAT E U N I V E R S I T Y O F N E W YO R K Why use equivariant moving frames? Decouples the moving frame theory from reliance on any form of frame bundle or connection basic calculus even an undergraduate student can do this! Francis Valiquette Recursive Moving Frames 12/12/2013 5 / 36
S TAT E U N I V E R S I T Y O F N E W YO R K Why use equivariant moving frames? Decouples the moving frame theory from reliance on any form of frame bundle or connection basic calculus even an undergraduate student can do this! Recurrence relations symbolic basic linear algebra reveal the structure of the algebra of differential invariants Francis Valiquette Recursive Moving Frames 12/12/2013 5 / 36
S TAT E U N I V E R S I T Y O F N E W YO R K Why use equivariant moving frames? Decouples the moving frame theory from reliance on any form of frame bundle or connection basic calculus even an undergraduate student can do this! Recurrence relations symbolic basic linear algebra reveal the structure of the algebra of differential invariants Francis Valiquette Recursive Moving Frames 12/12/2013 5 / 36
S TAT E U N I V E R S I T Y O F N E W YO R K Why use equivariant moving frames? Decouples the moving frame theory from reliance on any form of frame bundle or connection basic calculus even an undergraduate student can do this! Recurrence relations symbolic basic linear algebra reveal the structure of the algebra of differential invariants Francis Valiquette Recursive Moving Frames 12/12/2013 5 / 36
S TAT E U N I V E R S I T Y O F N E W YO R K Why use equivariant moving frames? Decouples the moving frame theory from reliance on any form of frame bundle or connection basic calculus even an undergraduate student can do this! Recurrence relations symbolic basic linear algebra reveal the structure of the algebra of differential invariants Francis Valiquette Recursive Moving Frames 12/12/2013 5 / 36
S TAT E U N I V E R S I T Y O F N E W YO R K Growing number of applications computer vision group foliation invariant calculus of variation invariant geometric flows invariant numerical schemes computation of Laplace invariants of differential operators invariants and covariants of Killing tensors invariants of Lie algebras Francis Valiquette Recursive Moving Frames 12/12/2013 6 / 36
S TAT E U N I V E R S I T Y O F N E W YO R K Moving frame algorithm 1 Lie (pseudo-)group action 2 prolonged action (freeness) 3 cross-section 4 normalization 5 invariantization Francis Valiquette Recursive Moving Frames 12/12/2013 7 / 36
S TAT E U N I V E R S I T Y O F N E W YO R K Moving frame algorithm 1 Lie (pseudo-)group action 2 prolonged action (freeness) 3 cross-section ✲ ✲ 4 normalization ✲ 5 invariantization Francis Valiquette Recursive Moving Frames 12/12/2013 7 / 36
S TAT E U N I V E R S I T Y O F N E W YO R K Moving frame algorithm 1 Lie (pseudo-)group action 2 prolonged action K n (freeness) 3 cross-section ✲ ✲ 4 normalization ✲ 5 invariantization Francis Valiquette Recursive Moving Frames 12/12/2013 7 / 36
S TAT E U N I V E R S I T Y O F N E W YO R K Moving frame algorithm 1 Lie (pseudo-)group action 2 prolonged action K n ρ p n q p z p n q q (freeness) ✉ z p n q ✉ ✠ 3 cross-section ✲ ✲ 4 normalization ✲ 5 invariantization Francis Valiquette Recursive Moving Frames 12/12/2013 7 / 36
S TAT E U N I V E R S I T Y O F N E W YO R K Moving frame algorithm 1 Lie (pseudo-)group action 2 prolonged action K n ρ p n q p z p n q q (freeness) ✉ z p n q ✉ ✠ 3 cross-section ✲ ✲ 4 normalization ✲ 5 invariantization Francis Valiquette Recursive Moving Frames 12/12/2013 7 / 36
S TAT E U N I V E R S I T Y O F N E W YO R K Example U “ u ` e x X “ f p x q Y “ e p x , y q “ f x p x q y ` g p x q f x Prolonged action: (Lifted invariants) U X “ u x ` e xx ´ e x u y ´ 2 f xx e x U Y “ u y ` f xx f 2 f 3 f 2 f x f x x x x ´ 2 f 2 ` f xxx ´ f xx u y ´ e x u yy U XY “ u xy U YY “ u yy xx f 2 f 3 f 4 f 2 x x x x ` e xxx ´ e xx u y ´ 2 e x u xy ´ f xx u x U XX “ u xx f 2 f 3 x x ` e 2 ` 8 e x f 2 x u yy ` 3 e x f xxx u y ´ 4 e xx f xx ´ 3 e x f xxx xx f 4 f 5 x x Francis Valiquette Recursive Moving Frames 12/12/2013 8 / 36
S TAT E U N I V E R S I T Y O F N E W YO R K Example U “ u ` e x X “ f p x q Y “ e p x , y q “ f x p x q y ` g p x q f x Prolonged action: (Lifted invariants) U X “ u x ` e xx ´ e x u y ´ 2 f xx e x U Y “ u y ` f xx f 2 f 3 f 2 f x f x x x x ´ 2 f 2 ` f xxx ´ f xx u y ´ e x u yy U XY “ u xy U YY “ u yy xx f 2 f 3 f 4 f 2 x x x x ` e xxx ´ e xx u y ´ 2 e x u xy ´ f xx u x U XX “ u xx f 2 f 3 x x ` e 2 ` 8 e x f 2 x u yy ` 3 e x f xxx u y ´ 4 e xx f xx ´ 3 e x f xxx xx f 4 f 5 x x Francis Valiquette Recursive Moving Frames 12/12/2013 8 / 36
S TAT E U N I V E R S I T Y O F N E W YO R K Example U “ u ` e x X “ f p x q Y “ e p x , y q “ f x p x q y ` g p x q f x Prolonged action: (Lifted invariants) U X “ u x ` e xx ´ e x u y ´ 2 f xx e x U Y “ u y ` f xx f 2 f 3 f 2 f x f x x x x ´ 2 f 2 U XY “ u xy ` f xxx ´ f xx u y ´ e x u yy U YY “ u yy xx f 2 f 3 f 4 f 2 x x x x ` e xxx ´ e xx u y ´ 2 e x u xy ´ f xx u x U XX “ u xx f 2 f 3 x x ` e 2 ` 8 e x f 2 x u yy ` 3 e x f xxx u y ´ 4 e xx f xx ´ 3 e x f xxx xx f 4 f 5 x x Francis Valiquette Recursive Moving Frames 12/12/2013 8 / 36
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