Towards Generalized Hydrodynamic Integrability via the Characteristic Variety Abraham D. Smith Fordham University, Bronx, NY, USA December 9, 2013 Fields Institute, Toronto A.Smith (Fordham) EDS 2013-12-09 1 / 13
Short talk – Get to the point! Lie Pseudogroup = “sub-pseudogroup of the diffeomorphisms of a manifold that can be written as solutions of an involutive PDE/EDS, by Cartan–K¨ ahler.” A.Smith (Fordham) EDS 2013-12-09 2 / 13
Short talk – Get to the point! Lie Pseudogroup = “sub-pseudogroup of the diffeomorphisms of a manifold that can be written as solutions of an involutive PDE/EDS, by Cartan–K¨ ahler.” The traditional representation theory of Lie groups begins with the question “how hard is it to integrate the infinitesimal structure of the Lie algebra into the local structure of the Lie group?” A.Smith (Fordham) EDS 2013-12-09 2 / 13
Short talk – Get to the point! Lie Pseudogroup = “sub-pseudogroup of the diffeomorphisms of a manifold that can be written as solutions of an involutive PDE/EDS, by Cartan–K¨ ahler.” The traditional representation theory of Lie groups begins with the question “how hard is it to integrate the infinitesimal structure of the Lie algebra into the local structure of the Lie group?” Answering gives useful classes like abelian , solvable , and semi-simple and Levi’s thm. A.Smith (Fordham) EDS 2013-12-09 2 / 13
Short talk – Get to the point! Lie Pseudogroup = “sub-pseudogroup of the diffeomorphisms of a manifold that can be written as solutions of an involutive PDE/EDS, by Cartan–K¨ ahler.” The traditional representation theory of Lie groups begins with the question “how hard is it to integrate the infinitesimal structure of the Lie algebra into the local structure of the Lie group?” Answering gives useful classes like abelian , solvable , and semi-simple and Levi’s thm. Cartan–K¨ ahler (Cauchy–Kowalevski) is the weakest (requiring strongest regularity) in a family of Cauchy integration theorems. A.Smith (Fordham) EDS 2013-12-09 2 / 13
Short talk – Get to the point! Lie Pseudogroup = “sub-pseudogroup of the diffeomorphisms of a manifold that can be written as solutions of an involutive PDE/EDS, by Cartan–K¨ ahler.” The traditional representation theory of Lie groups begins with the question “how hard is it to integrate the infinitesimal structure of the Lie algebra into the local structure of the Lie group?” Answering gives useful classes like abelian , solvable , and semi-simple and Levi’s thm. Cartan–K¨ ahler (Cauchy–Kowalevski) is the weakest (requiring strongest regularity) in a family of Cauchy integration theorems. Therefore, examination of regularity and micro-local analysis (“how difficult is integration?”) of involutive PDE/EDS should help build our knowledge Lie pseudogroups. A.Smith (Fordham) EDS 2013-12-09 2 / 13
Short talk – Get to the point! Lie Pseudogroup = “sub-pseudogroup of the diffeomorphisms of a manifold that can be written as solutions of an involutive PDE/EDS, by Cartan–K¨ ahler.” The traditional representation theory of Lie groups begins with the question “how hard is it to integrate the infinitesimal structure of the Lie algebra into the local structure of the Lie group?” Answering gives useful classes like abelian , solvable , and semi-simple and Levi’s thm. Cartan–K¨ ahler (Cauchy–Kowalevski) is the weakest (requiring strongest regularity) in a family of Cauchy integration theorems. Therefore, examination of regularity and micro-local analysis (“how difficult is integration?”) of involutive PDE/EDS should help build our knowledge Lie pseudogroups. Where to find invariant notions of “difficult to integrate?” A.Smith (Fordham) EDS 2013-12-09 2 / 13
Dans certains cas, le nombre de ces familles de car- act´ eristiques peut ˆ etre r´ eduit, certaines familles de- venant double, triples, etc. Ces cas de r´ eduction sont analogues ` a ceux qui se pr´ esentent dans la r´ eduction d’une substitution lin´ eaire ` a sa forme normale et la rerecherche des caract´ eristiques d´ epend d’ailleurs —Cartan, 102 years ago. d’une telle r´ eduction. A.Smith (Fordham) EDS 2013-12-09 3 / 13
Dans certains cas, le nombre de ces familles de car- act´ eristiques peut ˆ etre r´ eduit, certaines familles de- venant double, triples, etc. Ces cas de r´ eduction sont analogues ` a ceux qui se pr´ esentent dans la r´ eduction d’une substitution lin´ eaire ` a sa forme normale et la rerecherche des caract´ eristiques d´ epend d’ailleurs —Cartan, 102 years ago. d’une telle r´ eduction. Moreover, those multiplicities in the charac- teristic variety can be accessed via the incidence correspondence given by the rank-one variety of the tableau. For involutive systems with higher Cartan int, submanifolds secant to the rank-one cone give hydrodynamic reductions, and the secant system indicates hydrodynamic integrability in local coordinates. A.Smith (Fordham) EDS 2013-12-09 3 / 13
Characteristic and Rank-One Variety Two interwoven stories: rank-one variety & hydrodynamic integrability. Tableau and Symbol: κ M (2) , I (2) T E M (2) Z (1) η M (1) , I (1) T e M (1) Z θ M , I T p M W ω V A.Smith (Fordham) EDS 2013-12-09 4 / 13
Characteristic and Rank-One Variety Two interwoven stories: rank-one variety & hydrodynamic integrability. Tableau and Symbol: κ M (2) , I (2) T E M (2) Z (1) � a d θ a ≡ i ∧ ω i + 1 ij ω i ∧ ω j � 2 T a τ ( η ) mod θ η M (1) , I (1) T e M (1) Z θ M , I T p M W ω V A.Smith (Fordham) EDS 2013-12-09 4 / 13
Characteristic and Rank-One Variety Two interwoven stories: rank-one variety & hydrodynamic integrability. Tableau and Symbol: κ M (2) , I (2) T E M (2) Z (1) � a d θ a ≡ i ∧ ω i + 1 ij ω i ∧ ω j � 2 T a τ ( η ) mod θ η → U ∗ → 0 τ σ → W ⊗ V ∗ 0 → Z M (1) , I (1) T e M (1) Z θ M , I T p M W ω V A.Smith (Fordham) EDS 2013-12-09 4 / 13
Characteristic and Rank-One Variety Two interwoven stories: rank-one variety & hydrodynamic integrability. Tableau and Symbol: κ M (2) , I (2) T E M (2) Z (1) � a d θ a ≡ i ∧ ω i + 1 ij ω i ∧ ω j � 2 T a τ ( η ) mod θ η → U ∗ → 0 τ σ → W ⊗ V ∗ 0 → Z M (1) , I (1) T e M (1) Z 0 → Z (1) → Z ⊗ V ∗ → W ⊗ ∧ 2 V ∗ → H 0 , 2 ( Z ) → 0 δ θ M , I T p M W ω V A.Smith (Fordham) EDS 2013-12-09 4 / 13
Characteristic and Rank-One Variety Two interwoven stories: rank-one variety & hydrodynamic integrability. Tableau and Symbol: κ M (2) , I (2) T E M (2) Z (1) � a d θ a ≡ i ∧ ω i + 1 ij ω i ∧ ω j � 2 T a τ ( η ) mod θ η → U ∗ → 0 τ σ → W ⊗ V ∗ 0 → Z M (1) , I (1) T e M (1) Z 0 → Z (1) → Z ⊗ V ∗ → W ⊗ ∧ 2 V ∗ → H 0 , 2 ( Z ) → 0 δ θ Characteristic Variety and Rank-One Variety: M , I T p M W ω V A.Smith (Fordham) EDS 2013-12-09 4 / 13
Characteristic and Rank-One Variety Two interwoven stories: rank-one variety & hydrodynamic integrability. Tableau and Symbol: κ M (2) , I (2) T E M (2) Z (1) � a d θ a ≡ i ∧ ω i + 1 ij ω i ∧ ω j � 2 T a τ ( η ) mod θ η → U ∗ → 0 τ σ → W ⊗ V ∗ 0 → Z M (1) , I (1) T e M (1) Z 0 → Z (1) → Z ⊗ V ∗ → W ⊗ ∧ 2 V ∗ → H 0 , 2 ( Z ) → 0 δ θ Characteristic Variety and Rank-One Variety: M , I T p M W Ξ = { ξ ∈ V ∗ : ∃ w , σ ξ ( w ) = σ ( w ⊗ ξ ) = 0 } ω C = { z ∈ Z : τ ( z ) = w ⊗ ξ, has rank 1 } V (slides sloppy about P ’s) A.Smith (Fordham) EDS 2013-12-09 4 / 13
Characteristic and Rank-One Variety W ξ ⊗ ξ C Gr • ( W ) W ξ = ker σ ξ Ξ ξ Some properties if I is involutive: A.Smith (Fordham) EDS 2013-12-09 5 / 13
Characteristic and Rank-One Variety W ξ ⊗ ξ C Gr • ( W ) W ξ = ker σ ξ Ξ ξ Some properties if I is involutive: 1 The eikonal system E (Ξ C ) is involutive on any ordinary integral N . (Typically, difficult.) A.Smith (Fordham) EDS 2013-12-09 5 / 13
Characteristic and Rank-One Variety W ξ ⊗ ξ C Gr • ( W ) W ξ = ker σ ξ Ξ ξ Some properties if I is involutive: 1 The eikonal system E (Ξ C ) is involutive on any ordinary integral N . (Typically, difficult.) 2 Ξ is essentially preserved under prolongation. A.Smith (Fordham) EDS 2013-12-09 5 / 13
Characteristic and Rank-One Variety W ξ ⊗ ξ C Gr • ( W ) W ξ = ker σ ξ Ξ ξ Some properties if I is involutive: 1 The eikonal system E (Ξ C ) is involutive on any ordinary integral N . (Typically, difficult.) 2 Ξ is essentially preserved under prolongation. Ξ C = deg ˆ 3 The last Cartan character is s dim ˆ Ξ C . A.Smith (Fordham) EDS 2013-12-09 5 / 13
Characteristic and Rank-One Variety W ξ ⊗ ξ C Gr • ( W ) W ξ = ker σ ξ Ξ ξ Some properties if I is involutive: 1 The eikonal system E (Ξ C ) is involutive on any ordinary integral N . (Typically, difficult.) 2 Ξ is essentially preserved under prolongation. Ξ C = deg ˆ 3 The last Cartan character is s dim ˆ Ξ C . 4 If I has no Cauchy characteristics, then ˆ Ξ C spans e ∗ C . A.Smith (Fordham) EDS 2013-12-09 5 / 13
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