Basic Todaism Carlos Tomei, PUC-Rio Geometry, Dynamics and - PowerPoint PPT Presentation

The physical track – a completely integrable system (Toda 67) For n particles in R , take n n − 1 H ( x , y ) = 1 � y 2 � k + exp( x k − x k + 1 ) , 2 k = 1 k = 1 k = ∂ H k = − ∂ H x ′ = y k , y ′ = exp( x k − 1 − x k ) − exp( x k − x k + 1 ) . ∂ y k ∂ x k x k − x k + 1 (Flaschka 74) Change variables, a k = − y k / 2 , b k = 1 2 exp( ) 2 and cleverly arrange them in matrices,   a 1 b 1 0 0 b 1 a 2 b 2 0   J =  ,   0 b 2 a 3 b 3  0 0 b 3 a 4 x + M u ( x , t ) , KdV is S ′ = [Π f ( S ) , S ] . (Lax 68) The celebrated KdV equation is a Lax pair. For S ( t ) = − D 2 3/10

The physical track – a completely integrable system (Toda 67) For n particles in R , take n n − 1 H ( x , y ) = 1 � y 2 � k + exp( x k − x k + 1 ) , 2 k = 1 k = 1 k = ∂ H k = − ∂ H x ′ = y k , y ′ = exp( x k − 1 − x k ) − exp( x k − x k + 1 ) . ∂ y k ∂ x k x k − x k + 1 (Flaschka 74) Change variables, a k = − y k / 2 , b k = 1 2 exp( ) 2 and cleverly arrange them in matrices,     a 1 b 1 0 0 0 − b 1 0 0 b 1 a 2 b 2 0 b 1 0 − b 2 0     J =  , Π skew J =     0 b 2 a 3 b 3 0 b 2 0 − b 3    0 0 b 3 a 4 0 0 b 3 0 x + M u ( x , t ) , KdV is S ′ = [Π f ( S ) , S ] . (Lax 68) The celebrated KdV equation is a Lax pair. For S ( t ) = − D 2 3/10

The physical track – a completely integrable system (Toda 67) For n particles in R , take n n − 1 H ( x , y ) = 1 � y 2 � k + exp( x k − x k + 1 ) , 2 k = 1 k = 1 k = ∂ H k = − ∂ H x ′ = y k , y ′ = exp( x k − 1 − x k ) − exp( x k − x k + 1 ) . ∂ y k ∂ x k x k − x k + 1 (Flaschka 74) Change variables, a k = − y k / 2 , b k = 1 2 exp( ) 2 and cleverly arrange them in matrices,     a 1 b 1 0 0 0 − b 1 0 0 b 1 a 2 b 2 0 b 1 0 − b 2 0     J =  , Π skew J =     0 b 2 a 3 b 3 0 b 2 0 − b 3    0 0 b 3 a 4 0 0 b 3 0 J ′ = [ J , Π skew J ] , a Lax pair . x + M u ( x , t ) , KdV is S ′ = [Π f ( S ) , S ] . (Lax 68) The celebrated KdV equation is a Lax pair. For S ( t ) = − D 2 3/10

The physical track – a completely integrable system (Toda 67) For n particles in R , take n n − 1 H ( x , y ) = 1 � y 2 � k + exp( x k − x k + 1 ) , 2 k = 1 k = 1 k = ∂ H k = − ∂ H x ′ = y k , y ′ = exp( x k − 1 − x k ) − exp( x k − x k + 1 ) . ∂ y k ∂ x k x k − x k + 1 (Flaschka 74) Change variables, a k = − y k / 2 , b k = 1 2 exp( ) 2 and cleverly arrange them in matrices,     a 1 b 1 0 0 0 − b 1 0 0 b 1 a 2 b 2 0 b 1 0 − b 2 0     J =  , Π skew J =     0 b 2 a 3 b 3 0 b 2 0 − b 3    0 0 b 3 a 4 0 0 b 3 0 J ′ = [ J , Π skew J ] , a Lax pair . The two conjugations preserve spectrum, symmetry, profile. x + M u ( x , t ) , KdV is S ′ = [Π f ( S ) , S ] . (Lax 68) The celebrated KdV equation is a Lax pair. For S ( t ) = − D 2 3/10

The physical track – a completely integrable system (Toda 67) For n particles in R , take n n − 1 H ( x , y ) = 1 � y 2 � k + exp( x k − x k + 1 ) , 2 k = 1 k = 1 k = ∂ H k = − ∂ H x ′ = y k , y ′ = exp( x k − 1 − x k ) − exp( x k − x k + 1 ) . ∂ y k ∂ x k x k − x k + 1 (Flaschka 74) Change variables, a k = − y k / 2 , b k = 1 2 exp( ) 2 and cleverly arrange them in matrices,     a 1 b 1 0 0 0 − b 1 0 0 b 1 a 2 b 2 0 b 1 0 − b 2 0     J =  , Π skew J =     0 b 2 a 3 b 3 0 b 2 0 − b 3    0 0 b 3 a 4 0 0 b 3 0 J ′ = [ J , Π skew J ] , a Lax pair . The two conjugations preserve spectrum, symmetry, profile. (Adler 79) Jacobis of zero trace form a coadjoint orbit. x + M u ( x , t ) , KdV is S ′ = [Π f ( S ) , S ] . (Lax 68) The celebrated KdV equation is a Lax pair. For S ( t ) = − D 2 3/10

The physical track – a completely integrable system (Toda 67) For n particles in R , take n n − 1 H ( x , y ) = 1 � y 2 � k + exp( x k − x k + 1 ) , 2 k = 1 k = 1 k = ∂ H k = − ∂ H x ′ = y k , y ′ = exp( x k − 1 − x k ) − exp( x k − x k + 1 ) . ∂ y k ∂ x k x k − x k + 1 (Flaschka 74) Change variables, a k = − y k / 2 , b k = 1 2 exp( ) 2 and cleverly arrange them in matrices,     a 1 b 1 0 0 0 − b 1 0 0 b 1 a 2 b 2 0 b 1 0 − b 2 0     J =  , Π skew J =     0 b 2 a 3 b 3 0 b 2 0 − b 3    0 0 b 3 a 4 0 0 b 3 0 J ′ = [ J , Π skew J ] , a Lax pair . The two conjugations preserve spectrum, symmetry, profile. (Adler 79) Jacobis of zero trace form a coadjoint orbit. The vector fields induced by H i ( x , y ) = λ i ( a , b ) commute. x + M u ( x , t ) , KdV is S ′ = [Π f ( S ) , S ] . (Lax 68) The celebrated KdV equation is a Lax pair. For S ( t ) = − D 2 3/10

Scattering and inverse variables (Moser 79) GGKM, AKNS, Lax, Fadeev... Reyman, Semenov-Tian-Shansky, Adler, Kostant... 4/10

Scattering and inverse variables (Moser 79) −∞ ∞ diag( λ 1 < . . . < λ n ) ← − J ( t ) − → diag( λ n > . . . > λ 1 ) GGKM, AKNS, Lax, Fadeev... Reyman, Semenov-Tian-Shansky, Adler, Kostant... 4/10

Scattering and inverse variables (Moser 79) −∞ ∞ diag( λ 1 < . . . < λ n ) ← − J ( t ) − → diag( λ n > . . . > λ 1 ) For fixed spectra, the resulting sets are Liouville tori J Λ ≃ R n − 1 , GGKM, AKNS, Lax, Fadeev... Reyman, Semenov-Tian-Shansky, Adler, Kostant... 4/10

Scattering and inverse variables (Moser 79) −∞ ∞ diag( λ 1 < . . . < λ n ) ← − J ( t ) − → diag( λ n > . . . > λ 1 ) For fixed spectra, the resulting sets are Liouville tori J Λ ≃ R n − 1 , parameterized by inverse variables (norming constants) J Λ ∼ = { c = ( c 1 > 0 , . . . , c n > 0 ) } . The c ′ k s are the first coordinates of the eigenvectors. GGKM, AKNS, Lax, Fadeev... Reyman, Semenov-Tian-Shansky, Adler, Kostant... 4/10

Scattering and inverse variables (Moser 79) −∞ ∞ diag( λ 1 < . . . < λ n ) ← − J ( t ) − → diag( λ n > . . . > λ 1 ) For fixed spectra, the resulting sets are Liouville tori J Λ ≃ R n − 1 , parameterized by inverse variables (norming constants) J Λ ∼ = { c = ( c 1 > 0 , . . . , c n > 0 ) } . The c ′ k s are the first coordinates of the eigenvectors. Under Toda, λ ′ i = 0 and c ( t ) = exp( t Λ) c ( 0 ) / � exp( t Λ) c ( 0 ) � . GGKM, AKNS, Lax, Fadeev... Reyman, Semenov-Tian-Shansky, Adler, Kostant... 4/10

Scattering and inverse variables (Moser 79) −∞ ∞ diag( λ 1 < . . . < λ n ) ← − J ( t ) − → diag( λ n > . . . > λ 1 ) For fixed spectra, the resulting sets are Liouville tori J Λ ≃ R n − 1 , parameterized by inverse variables (norming constants) J Λ ∼ = { c = ( c 1 > 0 , . . . , c n > 0 ) } . The c ′ k s are the first coordinates of the eigenvectors. Under Toda, λ ′ i = 0 and c ( t ) = exp( t Λ) c ( 0 ) / � exp( t Λ) c ( 0 ) � . (Symes 82, Deift, Nanda, T. 83) For standard QR, J n = J ( n ) , where J ′ = [ J , Π skew log J ] , J ( 0 ) = J 0 . GGKM, AKNS, Lax, Fadeev... Reyman, Semenov-Tian-Shansky, Adler, Kostant... 4/10

Scattering and inverse variables (Moser 79) −∞ ∞ diag( λ 1 < . . . < λ n ) ← − J ( t ) − → diag( λ n > . . . > λ 1 ) For fixed spectra, the resulting sets are Liouville tori J Λ ≃ R n − 1 , parameterized by inverse variables (norming constants) J Λ ∼ = { c = ( c 1 > 0 , . . . , c n > 0 ) } . The c ′ k s are the first coordinates of the eigenvectors. Under Toda, λ ′ i = 0 and c ( t ) = exp( t Λ) c ( 0 ) / � exp( t Λ) c ( 0 ) � . (Symes 82, Deift, Nanda, T. 83) For standard QR, J n = J ( n ) , where J ′ = [ J , Π skew log J ] , J ( 0 ) = J 0 . More generally, the solution of J ′ = [ J , Π skew f ( J )] , J ( 0 ) = J 0 is J ( t ) = [ exp ( f ( J 0 ))] ∗ Q J ( 0 ) [ exp ( f ( J 0 ))] Q . GGKM, AKNS, Lax, Fadeev... Reyman, Semenov-Tian-Shansky, Adler, Kostant... 4/10

Scattering and inverse variables (Moser 79) −∞ ∞ diag( λ 1 < . . . < λ n ) ← − J ( t ) − → diag( λ n > . . . > λ 1 ) For fixed spectra, the resulting sets are Liouville tori J Λ ≃ R n − 1 , parameterized by inverse variables (norming constants) J Λ ∼ = { c = ( c 1 > 0 , . . . , c n > 0 ) } . The c ′ k s are the first coordinates of the eigenvectors. Under Toda, λ ′ i = 0 and c ( t ) = exp( t Λ) c ( 0 ) / � exp( t Λ) c ( 0 ) � . (Symes 82, Deift, Nanda, T. 83) For standard QR, J n = J ( n ) , where J ′ = [ J , Π skew log J ] , J ( 0 ) = J 0 . More generally, the solution of J ′ = [ J , Π skew f ( J )] , J ( 0 ) = J 0 is J ( t ) = [ exp ( f ( J 0 ))] ∗ Q J ( 0 ) [ exp ( f ( J 0 ))] Q . A Hamiltonian flow with limit points... GGKM, AKNS, Lax, Fadeev... Reyman, Semenov-Tian-Shansky, Adler, Kostant... 4/10

Shifts According to Parlett, there are shifts for all seasons. 5/10

Shifts Algorithms are usually performed on Jacobi matrices According to Parlett, there are shifts for all seasons. 5/10

Shifts Algorithms are usually performed on Jacobi matrices (drop the signs of the ( k , k + 1 ) -entries of J and the spectrum does not change !). According to Parlett, there are shifts for all seasons. 5/10

Shifts Algorithms are usually performed on Jacobi matrices (drop the signs of the ( k , k + 1 ) -entries of J and the spectrum does not change !). Shifts are remarkable accelerators: According to Parlett, there are shifts for all seasons. 5/10

Shifts Algorithms are usually performed on Jacobi matrices (drop the signs of the ( k , k + 1 ) -entries of J and the spectrum does not change !). Shifts are remarkable accelerators: J ( 0 ) − sI = QR �→ J ( 1 ) = Q ∗ J ( 0 ) Q = R J ( 0 ) R − 1 . According to Parlett, there are shifts for all seasons. 5/10

Shifts Algorithms are usually performed on Jacobi matrices (drop the signs of the ( k , k + 1 ) -entries of J and the spectrum does not change !). Shifts are remarkable accelerators: J ( 0 ) − sI = QR �→ J ( 1 ) = Q ∗ J ( 0 ) Q = R J ( 0 ) R − 1 . Rayleigh would take s to be J n , n . According to Parlett, there are shifts for all seasons. 5/10

Shifts Algorithms are usually performed on Jacobi matrices (drop the signs of the ( k , k + 1 ) -entries of J and the spectrum does not change !). Shifts are remarkable accelerators: J ( 0 ) − sI = QR �→ J ( 1 ) = Q ∗ J ( 0 ) Q = R J ( 0 ) R − 1 . Rayleigh would take s to be J n , n . For Wilkinson, s is the eigenvalue of the bottom 2 × 2 block closer to J n , n . According to Parlett, there are shifts for all seasons. 5/10

Shifts Algorithms are usually performed on Jacobi matrices (drop the signs of the ( k , k + 1 ) -entries of J and the spectrum does not change !). Shifts are remarkable accelerators: J ( 0 ) − sI = QR �→ J ( 1 ) = Q ∗ J ( 0 ) Q = R J ( 0 ) R − 1 . Rayleigh would take s to be J n , n . For Wilkinson, s is the eigenvalue of the bottom 2 × 2 block closer to J n , n . Rayleigh’s shift strategy generates periodic orbits. According to Parlett, there are shifts for all seasons. 5/10

Shifts Algorithms are usually performed on Jacobi matrices (drop the signs of the ( k , k + 1 ) -entries of J and the spectrum does not change !). Shifts are remarkable accelerators: J ( 0 ) − sI = QR �→ J ( 1 ) = Q ∗ J ( 0 ) Q = R J ( 0 ) R − 1 . Rayleigh would take s to be J n , n . For Wilkinson, s is the eigenvalue of the bottom 2 × 2 block closer to J n , n . Rayleigh’s shift strategy generates periodic orbits. Wilkinson’s does not. According to Parlett, there are shifts for all seasons. 5/10

Shifts Algorithms are usually performed on Jacobi matrices (drop the signs of the ( k , k + 1 ) -entries of J and the spectrum does not change !). Shifts are remarkable accelerators: J ( 0 ) − sI = QR �→ J ( 1 ) = Q ∗ J ( 0 ) Q = R J ( 0 ) R − 1 . Rayleigh would take s to be J n , n . For Wilkinson, s is the eigenvalue of the bottom 2 × 2 block closer to J n , n . Rayleigh’s shift strategy generates periodic orbits. Wilkinson’s does not. (Leite, Saldanha, T., 12) No continuous shift strategy yields an always convergent iteration. According to Parlett, there are shifts for all seasons. 5/10

Shifts Algorithms are usually performed on Jacobi matrices (drop the signs of the ( k , k + 1 ) -entries of J and the spectrum does not change !). Shifts are remarkable accelerators: J ( 0 ) − sI = QR �→ J ( 1 ) = Q ∗ J ( 0 ) Q = R J ( 0 ) R − 1 . Rayleigh would take s to be J n , n . For Wilkinson, s is the eigenvalue of the bottom 2 × 2 block closer to J n , n . Rayleigh’s shift strategy generates periodic orbits. Wilkinson’s does not. (Leite, Saldanha, T., 12) No continuous shift strategy yields an always convergent iteration. (Leite, Saldanha, T., 10) If Λ has no three eigenvalues in arithmetic progression, Wilkinson iteration leads to cubic convergence of J n , n − 1 . Otherwise, there may be a Cantor-like set in which iteration is quadratic. According to Parlett, there are shifts for all seasons. 5/10

Shifts Algorithms are usually performed on Jacobi matrices (drop the signs of the ( k , k + 1 ) -entries of J and the spectrum does not change !). Shifts are remarkable accelerators: J ( 0 ) − sI = QR �→ J ( 1 ) = Q ∗ J ( 0 ) Q = R J ( 0 ) R − 1 . Rayleigh would take s to be J n , n . For Wilkinson, s is the eigenvalue of the bottom 2 × 2 block closer to J n , n . Rayleigh’s shift strategy generates periodic orbits. Wilkinson’s does not. (Leite, Saldanha, T., 12) No continuous shift strategy yields an always convergent iteration. (Leite, Saldanha, T., 10) If Λ has no three eigenvalues in arithmetic progression, Wilkinson iteration leads to cubic convergence of J n , n − 1 . Otherwise, there may be a Cantor-like set in which iteration is quadratic. And remember: | λ − J n , n | = O ( J 2 n , n − 1 ) ! According to Parlett, there are shifts for all seasons. 5/10

Shifts Algorithms are usually performed on Jacobi matrices (drop the signs of the ( k , k + 1 ) -entries of J and the spectrum does not change !). Shifts are remarkable accelerators: J ( 0 ) − sI = QR �→ J ( 1 ) = Q ∗ J ( 0 ) Q = R J ( 0 ) R − 1 . Rayleigh would take s to be J n , n . For Wilkinson, s is the eigenvalue of the bottom 2 × 2 block closer to J n , n . Rayleigh’s shift strategy generates periodic orbits. Wilkinson’s does not. (Leite, Saldanha, T., 12) No continuous shift strategy yields an always convergent iteration. (Leite, Saldanha, T., 10) If Λ has no three eigenvalues in arithmetic progression, Wilkinson iteration leads to cubic convergence of J n , n − 1 . Otherwise, there may be a Cantor-like set in which iteration is quadratic. And remember: | λ − J n , n | = O ( J 2 n , n − 1 ) ! Deflate ! According to Parlett, there are shifts for all seasons. 5/10

The isospectral manifold T Λ The eye wants its part (Italian proverb). 6/10

The isospectral manifold T Λ The eye wants its part (Italian proverb). 6/10

The isospectral manifold T Λ The eye wants its part (Italian proverb). 6/10

The isospectral manifold T Λ (5,7,4) (7,5,4) (5,7,4) g a (5,4,7) b b (5,4,7) c h (4,5,7) (4,5,7) ++ −+ d d (7,4,5) (4,7,5) (4,7,5) e e (4,5,7) −− +− (4,5,7) h c f f (5,4,7) (5,4,7) g a (5,7,4) (5,7,4) (7,5,4) The eye wants its part (Italian proverb). 6/10

The isospectral manifold T Λ (5,7,4) (7,5,4) (5,7,4) g a (5,4,7) b b (5,4,7) c h (4,5,7) (4,5,7) ++ −+ d d (7,4,5) (4,7,5) (4,7,5) e e (4,5,7) −− +− (4,5,7) h c f f (5,4,7) (5,4,7) g a (5,7,4) (5,7,4) (7,5,4) (7,5,4) (7,4,5) (4,7,5) (4,5,7) (5,4,7) (5,7,4) The eye wants its part (Italian proverb). 6/10

The isospectral manifold T Λ (5,7,4) (7,5,4) (5,7,4) g a (5,4,7) b b (5,4,7) c h (4,5,7) (4,5,7) ++ −+ d d (7,4,5) (4,7,5) (4,7,5) e e (4,5,7) −− +− (4,5,7) h c f f (5,4,7) (5,4,7) g a (5,7,4) (5,7,4) (7,5,4) (7,5,4) (7,4,5) (4,7,5) (4,5,7) (5,4,7) (5,7,4) Four hexagons, three (black) deflation components. The eye wants its part (Italian proverb). 6/10

The isospectral manifold T Λ (5,7,4) (7,5,4) (5,7,4) g a (5,4,7) b b (5,4,7) c h (4,5,7) (4,5,7) ++ −+ d d (7,4,5) (4,7,5) (4,7,5) e e (4,5,7) −− +− (4,5,7) h c f f (5,4,7) (5,4,7) g a (5,7,4) (5,7,4) (7,5,4) (7,5,4) (7,4,5) (4,7,5) (4,5,7) (5,4,7) (5,7,4) Four hexagons, three (black) deflation components. This is why continuous shift strategies are problematic. The eye wants its part (Italian proverb). 6/10

T Λ — an isospectral manifold A conformal diffeomorphic projection of T Λ ⊂ R 5 . 7/10

T Λ — an isospectral manifold A conformal diffeomorphic projection of T Λ ⊂ R 5 . 7/10

T Λ — an isospectral manifold Real, tridiagonal symmetric matrices with eigenvalues 4, 5 e 7. A conformal diffeomorphic projection of T Λ ⊂ R 5 . 7/10

T Λ — an isospectral manifold Real, tridiagonal symmetric matrices with eigenvalues 4, 5 e 7. For points in red, T 2 , 1 = 0; black stands for T 3 , 2 = 0. A conformal diffeomorphic projection of T Λ ⊂ R 5 . 7/10

Some symplectic and topological properties of T Λ An interesting object indeed. 8/10

Some symplectic and topological properties of T Λ (T. 84) The closure of Jacobi matrices with fixed spectrum J Λ is a permutohedron P = conv { ( λ π ( 1 ) , . . . , λ π ( n ) ) , π ∈ S n } . An interesting object indeed. 8/10

Some symplectic and topological properties of T Λ (T. 84) The closure of Jacobi matrices with fixed spectrum J Λ is a permutohedron P = conv { ( λ π ( 1 ) , . . . , λ π ( n ) ) , π ∈ S n } . (Bloch, Flaschka, Ratiu 90) It is the image of a moment map. An interesting object indeed. 8/10

Some symplectic and topological properties of T Λ (T. 84) The closure of Jacobi matrices with fixed spectrum J Λ is a permutohedron P = conv { ( λ π ( 1 ) , . . . , λ π ( n ) ) , π ∈ S n } . (Bloch, Flaschka, Ratiu 90) It is the image of a moment map. (Gibson, Saldanha, T) It is an ‘extreme projectivization’ of ( 0 , ∞ ) n . An interesting object indeed. 8/10

Some symplectic and topological properties of T Λ (T. 84) The closure of Jacobi matrices with fixed spectrum J Λ is a permutohedron P = conv { ( λ π ( 1 ) , . . . , λ π ( n ) ) , π ∈ S n } . (Bloch, Flaschka, Ratiu 90) It is the image of a moment map. (Gibson, Saldanha, T) It is an ‘extreme projectivization’ of ( 0 , ∞ ) n . In T Λ , h ( T ) = � n i = 1 i S ii is a height function for Toda and QR. An interesting object indeed. 8/10

Some symplectic and topological properties of T Λ (T. 84) The closure of Jacobi matrices with fixed spectrum J Λ is a permutohedron P = conv { ( λ π ( 1 ) , . . . , λ π ( n ) ) , π ∈ S n } . (Bloch, Flaschka, Ratiu 90) It is the image of a moment map. (Gibson, Saldanha, T) It is an ‘extreme projectivization’ of ( 0 , ∞ ) n . In T Λ , h ( T ) = � n i = 1 i S ii is a height function for Toda and QR. The Wielandt-Hoffman theorem. An interesting object indeed. 8/10

Some symplectic and topological properties of T Λ (T. 84) The closure of Jacobi matrices with fixed spectrum J Λ is a permutohedron P = conv { ( λ π ( 1 ) , . . . , λ π ( n ) ) , π ∈ S n } . (Bloch, Flaschka, Ratiu 90) It is the image of a moment map. (Gibson, Saldanha, T) It is an ‘extreme projectivization’ of ( 0 , ∞ ) n . In T Λ , h ( T ) = � n i = 1 i S ii is a height function for Toda and QR. The Wielandt-Hoffman theorem. (T. 84) The cohomology ring of T Λ has simple, explicit generators. An interesting object indeed. 8/10

Some symplectic and topological properties of T Λ (T. 84) The closure of Jacobi matrices with fixed spectrum J Λ is a permutohedron P = conv { ( λ π ( 1 ) , . . . , λ π ( n ) ) , π ∈ S n } . (Bloch, Flaschka, Ratiu 90) It is the image of a moment map. (Gibson, Saldanha, T) It is an ‘extreme projectivization’ of ( 0 , ∞ ) n . In T Λ , h ( T ) = � n i = 1 i S ii is a height function for Toda and QR. The Wielandt-Hoffman theorem. (T. 84) The cohomology ring of T Λ has simple, explicit generators. (Fried 86) It is free. An interesting object indeed. 8/10

Some symplectic and topological properties of T Λ (T. 84) The closure of Jacobi matrices with fixed spectrum J Λ is a permutohedron P = conv { ( λ π ( 1 ) , . . . , λ π ( n ) ) , π ∈ S n } . (Bloch, Flaschka, Ratiu 90) It is the image of a moment map. (Gibson, Saldanha, T) It is an ‘extreme projectivization’ of ( 0 , ∞ ) n . In T Λ , h ( T ) = � n i = 1 i S ii is a height function for Toda and QR. The Wielandt-Hoffman theorem. (T. 84) The cohomology ring of T Λ has simple, explicit generators. (Fried 86) It is free. (T. 84) Its covering is R n − 1 An interesting object indeed. 8/10

Some symplectic and topological properties of T Λ (T. 84) The closure of Jacobi matrices with fixed spectrum J Λ is a permutohedron P = conv { ( λ π ( 1 ) , . . . , λ π ( n ) ) , π ∈ S n } . (Bloch, Flaschka, Ratiu 90) It is the image of a moment map. (Gibson, Saldanha, T) It is an ‘extreme projectivization’ of ( 0 , ∞ ) n . In T Λ , h ( T ) = � n i = 1 i S ii is a height function for Toda and QR. The Wielandt-Hoffman theorem. (T. 84) The cohomology ring of T Λ has simple, explicit generators. (Fried 86) It is free. (T. 84) Its covering is R n − 1 (from results on Coxeter groups, Davis 83). An interesting object indeed. 8/10

Some symplectic and topological properties of T Λ (T. 84) The closure of Jacobi matrices with fixed spectrum J Λ is a permutohedron P = conv { ( λ π ( 1 ) , . . . , λ π ( n ) ) , π ∈ S n } . (Bloch, Flaschka, Ratiu 90) It is the image of a moment map. (Gibson, Saldanha, T) It is an ‘extreme projectivization’ of ( 0 , ∞ ) n . In T Λ , h ( T ) = � n i = 1 i S ii is a height function for Toda and QR. The Wielandt-Hoffman theorem. (T. 84) The cohomology ring of T Λ has simple, explicit generators. (Fried 86) It is free. (T. 84) Its covering is R n − 1 (from results on Coxeter groups, Davis 83). (Gaifullin, 14) Any homology class of a manifold has a multiple which is the image of a finite covering of T Λ . An interesting object indeed. 8/10

Some symplectic and topological properties of T Λ (T. 84) The closure of Jacobi matrices with fixed spectrum J Λ is a permutohedron P = conv { ( λ π ( 1 ) , . . . , λ π ( n ) ) , π ∈ S n } . (Bloch, Flaschka, Ratiu 90) It is the image of a moment map. (Gibson, Saldanha, T) It is an ‘extreme projectivization’ of ( 0 , ∞ ) n . In T Λ , h ( T ) = � n i = 1 i S ii is a height function for Toda and QR. The Wielandt-Hoffman theorem. (T. 84) The cohomology ring of T Λ has simple, explicit generators. (Fried 86) It is free. (T. 84) Its covering is R n − 1 (from results on Coxeter groups, Davis 83). (Gaifullin, 14) Any homology class of a manifold has a multiple which is the image of a finite covering of T Λ . (Casian, Kodama, 01) Twisted isospectral manifolds. An interesting object indeed. 8/10

Some symplectic and topological properties of T Λ (T. 84) The closure of Jacobi matrices with fixed spectrum J Λ is a permutohedron P = conv { ( λ π ( 1 ) , . . . , λ π ( n ) ) , π ∈ S n } . (Bloch, Flaschka, Ratiu 90) It is the image of a moment map. (Gibson, Saldanha, T) It is an ‘extreme projectivization’ of ( 0 , ∞ ) n . In T Λ , h ( T ) = � n i = 1 i S ii is a height function for Toda and QR. The Wielandt-Hoffman theorem. (T. 84) The cohomology ring of T Λ has simple, explicit generators. (Fried 86) It is free. (T. 84) Its covering is R n − 1 (from results on Coxeter groups, Davis 83). (Gaifullin, 14) Any homology class of a manifold has a multiple which is the image of a finite covering of T Λ . (Casian, Kodama, 01) Twisted isospectral manifolds. What about charts? An interesting object indeed. 8/10

Some symplectic and topological properties of T Λ (T. 84) The closure of Jacobi matrices with fixed spectrum J Λ is a permutohedron P = conv { ( λ π ( 1 ) , . . . , λ π ( n ) ) , π ∈ S n } . (Bloch, Flaschka, Ratiu 90) It is the image of a moment map. (Gibson, Saldanha, T) It is an ‘extreme projectivization’ of ( 0 , ∞ ) n . In T Λ , h ( T ) = � n i = 1 i S ii is a height function for Toda and QR. The Wielandt-Hoffman theorem. (T. 84) The cohomology ring of T Λ has simple, explicit generators. (Fried 86) It is free. (T. 84) Its covering is R n − 1 (from results on Coxeter groups, Davis 83). (Gaifullin, 14) Any homology class of a manifold has a multiple which is the image of a finite covering of T Λ . (Casian, Kodama, 01) Twisted isospectral manifolds. What about charts? Larger isospectral manifolds? An interesting object indeed. 8/10

Charts – bidiagonal variables Way beyond integrability. 9/10

Charts – bidiagonal variables π Λ π Q π . (Leite, Saldanha, T. 08) For π ∈ S n and T ∈ T Λ , write T = Q ∗ Way beyond integrability. 9/10

Charts – bidiagonal variables π Λ π Q π . (Leite, Saldanha, T. 08) For π ∈ S n and T ∈ T Λ , write T = Q ∗ A matrix T ∈ U π Λ ⊂ T Λ admits an LU factorization of Q π , Q π = L π U π : L π uni-lower and U π pos-upper, Λ π L π so that T = U − 1 L − 1 � � U π . π π Way beyond integrability. 9/10

Charts – bidiagonal variables π Λ π Q π . (Leite, Saldanha, T. 08) For π ∈ S n and T ∈ T Λ , write T = Q ∗ A matrix T ∈ U π Λ ⊂ T Λ admits an LU factorization of Q π , Q π = L π U π : L π uni-lower and U π pos-upper, Λ π L π so that T = U − 1 L − 1 � � U π . π π Set B π = L − 1 π Λ π L π (lower) = U π TU − 1 (upper Hessenberg): π Way beyond integrability. 9/10

Charts – bidiagonal variables π Λ π Q π . (Leite, Saldanha, T. 08) For π ∈ S n and T ∈ T Λ , write T = Q ∗ A matrix T ∈ U π Λ ⊂ T Λ admits an LU factorization of Q π , Q π = L π U π : L π uni-lower and U π pos-upper, Λ π L π so that T = U − 1 L − 1 � � U π . π π Set B π = L − 1 π Λ π L π (lower) = U π TU − 1 (upper Hessenberg): π B π is lower bidiagonal with diagonal Λ π ! Way beyond integrability. 9/10

Charts – bidiagonal variables π Λ π Q π . (Leite, Saldanha, T. 08) For π ∈ S n and T ∈ T Λ , write T = Q ∗ A matrix T ∈ U π Λ ⊂ T Λ admits an LU factorization of Q π , Q π = L π U π : L π uni-lower and U π pos-upper, Λ π L π so that T = U − 1 L − 1 � � U π . π π Set B π = L − 1 π Λ π L π (lower) = U π TU − 1 (upper Hessenberg): π B π is lower bidiagonal with diagonal Λ π ! Λ → R n − 1 . The entries β π j = ( B π ) j + 1 , j define a chart ψ π : U π Way beyond integrability. 9/10

Charts – bidiagonal variables π Λ π Q π . (Leite, Saldanha, T. 08) For π ∈ S n and T ∈ T Λ , write T = Q ∗ A matrix T ∈ U π Λ ⊂ T Λ admits an LU factorization of Q π , Q π = L π U π : L π uni-lower and U π pos-upper, Λ π L π so that T = U − 1 L − 1 � � U π . π π Set B π = L − 1 π Λ π L π (lower) = U π TU − 1 (upper Hessenberg): π B π is lower bidiagonal with diagonal Λ π ! Λ → R n − 1 . The entries β π j = ( B π ) j + 1 , j define a chart ψ π : U π B ′ = [ B , f (Λ)] Simple evolutions: Way beyond integrability. 9/10

Charts – bidiagonal variables π Λ π Q π . (Leite, Saldanha, T. 08) For π ∈ S n and T ∈ T Λ , write T = Q ∗ A matrix T ∈ U π Λ ⊂ T Λ admits an LU factorization of Q π , Q π = L π U π : L π uni-lower and U π pos-upper, Λ π L π so that T = U − 1 L − 1 � � U π . π π Set B π = L − 1 π Λ π L π (lower) = U π TU − 1 (upper Hessenberg): π B π is lower bidiagonal with diagonal Λ π ! Λ → R n − 1 . The entries β π j = ( B π ) j + 1 , j define a chart ψ π : U π B ′ = [ B , f (Λ)] , i.e., � � � � Simple evolutions: new β π = exp ( f ( λ π i + 1 ) / f ( λ π i )) old β π . i i Way beyond integrability. 9/10

Charts – bidiagonal variables π Λ π Q π . (Leite, Saldanha, T. 08) For π ∈ S n and T ∈ T Λ , write T = Q ∗ A matrix T ∈ U π Λ ⊂ T Λ admits an LU factorization of Q π , Q π = L π U π : L π uni-lower and U π pos-upper, Λ π L π so that T = U − 1 L − 1 � � U π . π π Set B π = L − 1 π Λ π L π (lower) = U π TU − 1 (upper Hessenberg): π B π is lower bidiagonal with diagonal Λ π ! Λ → R n − 1 . The entries β π j = ( B π ) j + 1 , j define a chart ψ π : U π B ′ = [ B , f (Λ)] , i.e., � � � � Simple evolutions: new β π = exp ( f ( λ π i + 1 ) / f ( λ π i )) old β π . i i Dropping signs (going Jacobi) is as bad as inserting absolute values. Way beyond integrability. 9/10

Charts – bidiagonal variables π Λ π Q π . (Leite, Saldanha, T. 08) For π ∈ S n and T ∈ T Λ , write T = Q ∗ A matrix T ∈ U π Λ ⊂ T Λ admits an LU factorization of Q π , Q π = L π U π : L π uni-lower and U π pos-upper, Λ π L π so that T = U − 1 L − 1 � � U π . π π Set B π = L − 1 π Λ π L π (lower) = U π TU − 1 (upper Hessenberg): π B π is lower bidiagonal with diagonal Λ π ! Λ → R n − 1 . The entries β π j = ( B π ) j + 1 , j define a chart ψ π : U π B ′ = [ B , f (Λ)] , i.e., � � � � Simple evolutions: new β π = exp ( f ( λ π i + 1 ) / f ( λ π i )) old β π . i i Dropping signs (going Jacobi) is as bad as inserting absolute values. Limits now belong to the charts: asymptotics is local theory. Way beyond integrability. 9/10

Charts – bidiagonal variables π Λ π Q π . (Leite, Saldanha, T. 08) For π ∈ S n and T ∈ T Λ , write T = Q ∗ A matrix T ∈ U π Λ ⊂ T Λ admits an LU factorization of Q π , Q π = L π U π : L π uni-lower and U π pos-upper, Λ π L π so that T = U − 1 L − 1 � � U π . π π Set B π = L − 1 π Λ π L π (lower) = U π TU − 1 (upper Hessenberg): π B π is lower bidiagonal with diagonal Λ π ! Λ → R n − 1 . The entries β π j = ( B π ) j + 1 , j define a chart ψ π : U π B ′ = [ B , f (Λ)] , i.e., � � � � Simple evolutions: new β π = exp ( f ( λ π i + 1 ) / f ( λ π i )) old β π . i i Dropping signs (going Jacobi) is as bad as inserting absolute values. Limits now belong to the charts: asymptotics is local theory. Bidiagonal variables are stabler than norming constants. Way beyond integrability. 9/10

Charts – bidiagonal variables π Λ π Q π . (Leite, Saldanha, T. 08) For π ∈ S n and T ∈ T Λ , write T = Q ∗ A matrix T ∈ U π Λ ⊂ T Λ admits an LU factorization of Q π , Q π = L π U π : L π uni-lower and U π pos-upper, Λ π L π so that T = U − 1 L − 1 � � U π . π π Set B π = L − 1 π Λ π L π (lower) = U π TU − 1 (upper Hessenberg): π B π is lower bidiagonal with diagonal Λ π ! Λ → R n − 1 . The entries β π j = ( B π ) j + 1 , j define a chart ψ π : U π B ′ = [ B , f (Λ)] , i.e., � � � � Simple evolutions: new β π = exp ( f ( λ π i + 1 ) / f ( λ π i )) old β π . i i Dropping signs (going Jacobi) is as bad as inserting absolute values. Limits now belong to the charts: asymptotics is local theory. Bidiagonal variables are stabler than norming constants. This begs for a new inverse algorithm: Gragg and Harrod, ’84. Way beyond integrability. 9/10

Larger matrices Duistermaat, Kolk, Varadarajan, Shub, Vasquez... 10/10

Larger matrices (Deift, Li, Nanda, T., 86, 89) Toda flows are completely integrable on symmetric and non-symmetric generic orbits. Duistermaat, Kolk, Varadarajan, Shub, Vasquez... 10/10

Larger matrices (Deift, Li, Nanda, T., 86, 89) Toda flows are completely integrable on symmetric and non-symmetric generic orbits. The Toda flow is Morse-Smale on isospectral matrices of full matrices (for sl(n), Shub-Vasquez 87; for other special cases, Chernyakov,Sharigin, Sorin 14, 17, 19; for non-compact semisimple lie algebras, Torres-T. 20) Duistermaat, Kolk, Varadarajan, Shub, Vasquez... 10/10

Larger matrices (Deift, Li, Nanda, T., 86, 89) Toda flows are completely integrable on symmetric and non-symmetric generic orbits. The Toda flow is Morse-Smale on isospectral matrices of full matrices (for sl(n), Shub-Vasquez 87; for other special cases, Chernyakov,Sharigin, Sorin 14, 17, 19; for non-compact semisimple lie algebras, Torres-T. 20) A staircase induces a matrix profile p . * * * * * * * * * * * * * * * * * * * * * * * * 0 * * * * * * * * * * * 0 0 0 * * * * * * * * * 0 0 0 0 0 0 * * * * * * 0 0 0 0 0 0 0 0 * * * * Duistermaat, Kolk, Varadarajan, Shub, Vasquez... 10/10

Larger matrices (Deift, Li, Nanda, T., 86, 89) Toda flows are completely integrable on symmetric and non-symmetric generic orbits. The Toda flow is Morse-Smale on isospectral matrices of full matrices (for sl(n), Shub-Vasquez 87; for other special cases, Chernyakov,Sharigin, Sorin 14, 17, 19; for non-compact semisimple lie algebras, Torres-T. 20) A staircase induces a matrix profile p . * * * * * * * * * * * * * * * * * * * * * * * * 0 * * * * * * * * * * * 0 0 0 * * * * * * * * * 0 0 0 0 0 0 * * * * * * 0 0 0 0 0 0 0 0 * * * * (Torres-T. 20) Charts extend to isospectral manifolds of given profile. Duistermaat, Kolk, Varadarajan, Shub, Vasquez... 10/10

Larger matrices (Deift, Li, Nanda, T., 86, 89) Toda flows are completely integrable on symmetric and non-symmetric generic orbits. The Toda flow is Morse-Smale on isospectral matrices of full matrices (for sl(n), Shub-Vasquez 87; for other special cases, Chernyakov,Sharigin, Sorin 14, 17, 19; for non-compact semisimple lie algebras, Torres-T. 20) A staircase induces a matrix profile p . * * * * * * * * * * * * * * * * * * * * * * * * 0 * * * * * * * * * * * 0 0 0 * * * * * * * * * 0 0 0 0 0 0 * * * * * * 0 0 0 0 0 0 0 0 * * * * (Torres-T. 20) Charts extend to isospectral manifolds of given profile. Thank you ! Duistermaat, Kolk, Varadarajan, Shub, Vasquez... 10/10

QR steps commute Giant steps. 11/10

QR steps commute A general QR step: for f ( S ) = QR = [ f ( S )] Q [ f ( S )] R , Giant steps. 11/10

QR steps commute A general QR step: for f ( S ) = QR = [ f ( S )] Q [ f ( S )] R , ˜ S = [ f ( S )] ∗ Q S [ f ( S )] Q . Giant steps. 11/10

QR steps commute A general QR step: for f ( S ) = QR = [ f ( S )] Q [ f ( S )] R , ˜ S = [ f ( S )] ∗ Q S [ f ( S )] Q . Take two steps, S �→ [ f ( S )] ∗ Q S [ f ( S )] Q Giant steps. 11/10

QR steps commute A general QR step: for f ( S ) = QR = [ f ( S )] Q [ f ( S )] R , ˜ S = [ f ( S )] ∗ Q S [ f ( S )] Q . Take two steps, S �→ [ f ( S )] ∗ Q S [ f ( S )] Q [ f ( S )] ∗ ] ∗ [ f ( S )] ∗ [ f ( S )] ∗ � � � � � � �→ [ g Q S [ f ( S )] Q Q S [ f ( S )] Q [ g Q S [ f ( S )] Q ] Q Q Giant steps. 11/10