Yang-Baxter and Braid Dynamics

Yang-Baxter and braid dynamics

Alexander Veselov, Loughborough University

GADUDIS, Glasgow, April 1, 2009 Thanks to Leonid Chekhov, Boris Dubrovin, Andy Hone, Oleg Lisovyy for illuminating discussions and to Vsevolod Adler for very useful comments and help with computer simulations.

I Links with Painleve-VI and Frobenius manifolds

I Markov dynamics

I Diophantine analysis and Markov equation

I Yang-Baxter versus braid relations

I Braid group and modular group

Plan

I Dynamics and groups Thanks to Leonid Chekhov, Boris Dubrovin, Andy Hone, Oleg Lisovyy for illuminating discussions and to Vsevolod Adler for very useful comments and help with computer simulations.

I Links with Painleve-VI and Frobenius manifolds

I Markov dynamics

I Diophantine analysis and Markov equation

I Yang-Baxter versus braid relations

Plan

I Dynamics and groups

I Braid group and modular group Thanks to Leonid Chekhov, Boris Dubrovin, Andy Hone, Oleg Lisovyy for illuminating discussions and to Vsevolod Adler for very useful comments and help with computer simulations.

I Links with Painleve-VI and Frobenius manifolds

I Markov dynamics

I Diophantine analysis and Markov equation

Plan

I Dynamics and groups

I Braid group and modular group

I Yang-Baxter versus braid relations Thanks to Leonid Chekhov, Boris Dubrovin, Andy Hone, Oleg Lisovyy for illuminating discussions and to Vsevolod Adler for very useful comments and help with computer simulations.

I Links with Painleve-VI and Frobenius manifolds

I Markov dynamics

Plan

I Dynamics and groups

I Braid group and modular group

I Yang-Baxter versus braid relations

I Diophantine analysis and Markov equation Thanks to Leonid Chekhov, Boris Dubrovin, Andy Hone, Oleg Lisovyy for illuminating discussions and to Vsevolod Adler for very useful comments and help with computer simulations.

I Links with Painleve-VI and Frobenius manifolds

Plan

I Dynamics and groups

I Braid group and modular group

I Yang-Baxter versus braid relations

I Diophantine analysis and Markov equation

I Markov dynamics Thanks to Leonid Chekhov, Boris Dubrovin, Andy Hone, Oleg Lisovyy for illuminating discussions and to Vsevolod Adler for very useful comments and help with computer simulations.

Plan

I Dynamics and groups

I Braid group and modular group

I Yang-Baxter versus braid relations

I Diophantine analysis and Markov equation

I Markov dynamics

I Links with Painleve-VI and Frobenius manifolds Plan

I Dynamics and groups

I Braid group and modular group

I Yang-Baxter versus braid relations

I Diophantine analysis and Markov equation

I Markov dynamics

I Links with Painleve-VI and Frobenius manifolds

Thanks to Leonid Chekhov, Boris Dubrovin, Andy Hone, Oleg Lisovyy for illuminating discussions and to Vsevolod Adler for very useful comments and help with computer simulations. I would like to discuss today the actions of the braid group B3, which has exponential growth.

Examples: Julia, Fatou, Ritt ; AV: commuting maps Okamoto; Noumi, Yamada: aﬃne Weyl group actions

INTEGRABILITY ∼ POLYNOMIAL GROWTH = ALMOST NILPOTENT GROUPS (Gromov’s theorem)

Groups and dynamics

Dynamical Erlangen programme: GROUPS ——– DYNAMICS Usual discrete dynamical system can be considered as nonlinear representation of Z. I would like to discuss today the actions of the braid group B3, which has exponential growth.

Examples: Julia, Fatou, Ritt ; AV: commuting maps Okamoto; Noumi, Yamada: aﬃne Weyl group actions

Groups and dynamics

Dynamical Erlangen programme: GROUPS ——– DYNAMICS Usual discrete dynamical system can be considered as nonlinear representation of Z.

INTEGRABILITY ∼ POLYNOMIAL GROWTH = ALMOST NILPOTENT GROUPS (Gromov’s theorem) I would like to discuss today the actions of the braid group B3, which has exponential growth.

Groups and dynamics

Dynamical Erlangen programme: GROUPS ——– DYNAMICS Usual discrete dynamical system can be considered as nonlinear representation of Z.

INTEGRABILITY ∼ POLYNOMIAL GROWTH = ALMOST NILPOTENT GROUPS (Gromov’s theorem)

Examples: Julia, Fatou, Ritt ; AV: commuting maps Okamoto; Noumi, Yamada: aﬃne Weyl group actions Groups and dynamics

Dynamical Erlangen programme: GROUPS ——– DYNAMICS Usual discrete dynamical system can be considered as nonlinear representation of Z.

INTEGRABILITY ∼ POLYNOMIAL GROWTH = ALMOST NILPOTENT GROUPS (Gromov’s theorem)

Examples: Julia, Fatou, Ritt ; AV: commuting maps Okamoto; Noumi, Yamada: aﬃne Weyl group actions

I would like to discuss today the actions of the braid group B3, which has exponential growth. If we add the relations 2 σi = 1

the braid group Bn is reduced to the symmetric group Sn.

It is generated by n − 1 elements σ1, . . . , σn−1 with the following relations

σi σi+1σi = σi+1σi σi+1

σi σj = σj σi , |i − j| > 1.

Braid group

Braid group Bn can be defined as the fundamental group π1(Mn) of the configuration space Mn of n different (unordered) points in C. If we add the relations 2 σi = 1

the braid group Bn is reduced to the symmetric group Sn.

Braid group

Braid group Bn can be defined as the fundamental group π1(Mn) of the configuration space Mn of n different (unordered) points in C.

It is generated by n − 1 elements σ1, . . . , σn−1 with the following relations

σi σi+1σi = σi+1σi σi+1

σi σj = σj σi , |i − j| > 1. Braid group

Braid group Bn can be defined as the fundamental group π1(Mn) of the configuration space Mn of n different (unordered) points in C.

It is generated by n − 1 elements σ1, . . . , σn−1 with the following relations

σi σi+1σi = σi+1σi σi+1

σi σj = σj σi , |i − j| > 1.

If we add the relations 2 σi = 1

the braid group Bn is reduced to the symmetric group Sn. Braid group B3

bx bz

bz bx

b b x = z

bz bx

3 b bx b = b b b z z x z x (bx bz) braidgroupdefiningrelations A natural homomorphism ∼ PSL(2, Z) → SL(2, Z2) = S3 maps it to the symmetric group.

The quotient B3/Z is isomorphic to the modular group

PSL(2, Z) = SL(2, Z)/{±I },

which is itself the free product Z2 ∗ Z3.

Braid group B3 and the modular group

Braid group B3 is generated by σ1 and σ2 with the only relation

σ1σ2σ1 = σ2σ1σ2.

It has the centre Z generated by

3 (σ1σ2) = (σ1σ2σ1)(σ2σ1σ2). A natural homomorphism ∼ PSL(2, Z) → SL(2, Z2) = S3 maps it to the symmetric group.

Braid group B3 and the modular group

Braid group B3 is generated by σ1 and σ2 with the only relation

σ1σ2σ1 = σ2σ1σ2.

It has the centre Z generated by

3 (σ1σ2) = (σ1σ2σ1)(σ2σ1σ2).

The quotient B3/Z is isomorphic to the modular group

PSL(2, Z) = SL(2, Z)/{±I },

which is itself the free product Z2 ∗ Z3. Braid group B3 and the modular group

Braid group B3 is generated by σ1 and σ2 with the only relation

σ1σ2σ1 = σ2σ1σ2.

It has the centre Z generated by

3 (σ1σ2) = (σ1σ2σ1)(σ2σ1σ2).

The quotient B3/Z is isomorphic to the modular group

PSL(2, Z) = SL(2, Z)/{±I },

which is itself the free product Z2 ∗ Z3. A natural homomorphism ∼ PSL(2, Z) → SL(2, Z2) = S3 maps it to the symmetric group. The reversible Yang-Baxter maps satisfy the relation

R21R = Id, corresponding to 2 Si = Id.

The twisted maps Si = Pii+1Rii+1 satisfy the braid relations

Si Si+1Si = Si+1Si Si+1, S12S23S12 = S23S12S23.

The map R is called Yang-Baxter map if it satisﬁes the Yang-Baxter relation

R12R13R23 = R23R13R12.

Yang-Baxter versus braid relations

Let X be any set and R be a map: R : X × X → X × X .

n n n Let Rij : X → X , X = X × X × ..... × X be the maps which acts as R on i-th and j-th factors and identically on the others. If P : X 2 → X 2 is the permutation: P(x, y) = (y, x), then R21 = PRP. The reversible Yang-Baxter maps satisfy the relation

R21R = Id, corresponding to 2 Si = Id.

The twisted maps Si = Pii+1Rii+1 satisfy the braid relations

Si Si+1Si = Si+1Si Si+1, S12S23S12 = S23S12S23.

Yang-Baxter versus braid relations

Let X be any set and R be a map: R : X × X → X × X .

n n n Let Rij : X → X , X = X × X × ..... × X be the maps which acts as R on i-th and j-th factors and identically on the others. If P : X 2 → X 2 is the permutation: P(x, y) = (y, x), then R21 = PRP. The map R is called Yang-Baxter map if it satisﬁes the Yang-Baxter relation

R12R13R23 = R23R13R12. The reversible Yang-Baxter maps satisfy the relation

R21R = Id, corresponding to 2 Si = Id.

Yang-Baxter versus braid relations

Let X be any set and R be a map: R : X × X → X × X .

R12R13R23 = R23R13R12. The twisted maps Si = Pii+1Rii+1 satisfy the braid relations

Si Si+1Si = Si+1Si Si+1, S12S23S12 = S23S12S23. Yang-Baxter versus braid relations

Let X be any set and R be a map: R : X × X → X × X .

R12R13R23 = R23R13R12. The twisted maps Si = Pii+1Rii+1 satisfy the braid relations

Si Si+1Si = Si+1Si Si+1, S12S23S12 = S23S12S23. The reversible Yang-Baxter maps satisfy the relation

R21R = Id, corresponding to 2 Si = Id. Yang-Baxter relations

1 1

2 = 2

3 3

Figure: Yang-Baxter relation

1 1 =

2 2

Figure: Reversibility Transfer dynamics corresponds to the translations for a natural action of the ˜(1) extended aﬃne Weyl group An−1. Note that the action of the corresponding braid group is reduced simply to the symmetric group Sn.

(n) For any reversible Yang-Baxter map R the transfer maps Ti commute with each other: (n) (n) (n) (n) Ti Tj = Tj Ti and satisfy the property (n) (n) (n) T1 T2 ... Tn = Id. (n) Conversely, if Ti satisfy these properties then R is a reversible YB map.

Transfer dynamics

Deﬁne the transfer maps

(n) n n Ti : X → X , i = 1,..., n by (n) Ti = Rii+n−1Rii+n−2 ... Rii+1, where the indices are considered modulo n. In particular (n) T1 = R1nR1n−1 ... R12. Transfer dynamics corresponds to the translations for a natural action of the ˜(1) extended aﬃne Weyl group An−1. Note that the action of the corresponding braid group is reduced simply to the symmetric group Sn.

Transfer dynamics

Deﬁne the transfer maps

(n) n n Ti : X → X , i = 1,..., n by (n) Ti = Rii+n−1Rii+n−2 ... Rii+1, where the indices are considered modulo n. In particular (n) T1 = R1nR1n−1 ... R12. (n) For any reversible Yang-Baxter map R the transfer maps Ti commute with each other: (n) (n) (n) (n) Ti Tj = Tj Ti and satisfy the property (n) (n) (n) T1 T2 ... Tn = Id. (n) Conversely, if Ti satisfy these properties then R is a reversible YB map. Transfer dynamics corresponds to the translations for a natural action of the ˜(1) extended aﬃne Weyl group An−1. Note that the action of the corresponding braid group is reduced simply to the symmetric group Sn. Commutativity of the transfer maps

i + 1 i i - 1 i + 1 i i - 1

j - 1 j - 1 j j j + 1 j + 1

Figure: Commutativity of the transfer maps Classiﬁcation

1 Quadrirational case, X = CP : Adler, Bobenko, Suris

Y X V U

Figure: A quadrirational map on a pair of conics In other words, φ is the most irrational number. What is the next one ?

History: golden ratio and Hurwitz theorem

It is well-known that any irrational real number α has a rational approximation p q (given by continued fraction expansion) such that p 1 |α − | < . q q2 In other words, φ is the most irrational number. What is the next one ?

History: golden ratio and Hurwitz theorem

It is well-known that any irrational real number α has a rational approximation p q (given by continued fraction expansion) such that p 1 |α − | < . q q2

p Hurwitz: For any α one can ﬁnd inﬁnitely many q such that p 1 |α − | < √ . q 5q2 For the golden ratio √ 1 + 5 α = φ = = [1, 1, 1, ...] 2 (and their equivalents) this is the best possible approximation. In other words, φ is the most irrational number. What is the next one ? For example, the next two most irrational numbers after φ are √ 2 = [1, 2, 2, 2....]

and √ 9 + 221 α = = [2, 2, 1, 1]. 10

Markov theorem. The Markov spectrum above 1/3 is discrete and consists of the numbers m µ = √ , 9m2 − 4 where m is one of the Markov numbers:

m = 1, 2, 5, 13, 29, 34, 89, 169, 194, 233, 433, 610, 985, ...

Markov spectrum

p c Deﬁne Markov constant as the minimal possible c in |α − q | < q2 : µ(α) = lim inf q||αq||, q→∞

where ||x|| = min |x − n|, n ∈ Z. The set of all possible values of µ(α) is called Markov spectrum. For example, the next two most irrational numbers after φ are √ 2 = [1, 2, 2, 2....]

and √ 9 + 221 α = = [2, 2, 1, 1]. 10

Markov spectrum

p c Deﬁne Markov constant as the minimal possible c in |α − q | < q2 : µ(α) = lim inf q||αq||, q→∞

where ||x|| = min |x − n|, n ∈ Z. The set of all possible values of µ(α) is called Markov spectrum.

Markov theorem. The Markov spectrum above 1/3 is discrete and consists of the numbers m µ = √ , 9m2 − 4 where m is one of the Markov numbers:

m = 1, 2, 5, 13, 29, 34, 89, 169, 194, 233, 433, 610, 985, ... Markov spectrum

p c Deﬁne Markov constant as the minimal possible c in |α − q | < q2 : µ(α) = lim inf q||αq||, q→∞

where ||x|| = min |x − n|, n ∈ Z. The set of all possible values of µ(α) is called Markov spectrum.

Markov theorem. The Markov spectrum above 1/3 is discrete and consists of the numbers m µ = √ , 9m2 − 4 where m is one of the Markov numbers:

m = 1, 2, 5, 13, 29, 34, 89, 169, 194, 233, 433, 610, 985, ...

For example, the next two most irrational numbers after φ are √ 2 = [1, 2, 2, 2....]

and √ 9 + 221 α = = [2, 2, 1, 1]. 10 Markov dynamics is the action of PSL2(Z) (and hence of the braid group B3) 3 on C determined by σ and τ(x, y, z) = (z, x, y). It has an obvious integral F = x 2 + y 2 + z 2 − 3xyz.

(1, 1, 1) → (1, 1, 2) → (1, 2, 1) → (1, 2, 5) → (1, 5, 2) → (1, 5, 13) → ...

Theorem (Markov) All Markov triples can be obtained from (1, 1, 1) by the involution σ :(x, y, z) → (x, y, 3xy − z) and permutations.

Markov triples

Markov numbers are parts of the Markov triples, which are the positive integer solutions of the Markov equation

x 2 + y 2 + z 2 − 3xyz = 0. Markov dynamics is the action of PSL2(Z) (and hence of the braid group B3) 3 on C determined by σ and τ(x, y, z) = (z, x, y). It has an obvious integral F = x 2 + y 2 + z 2 − 3xyz.

(1, 1, 1) → (1, 1, 2) → (1, 2, 1) → (1, 2, 5) → (1, 5, 2) → (1, 5, 13) → ...

Markov triples

Markov numbers are parts of the Markov triples, which are the positive integer solutions of the Markov equation

x 2 + y 2 + z 2 − 3xyz = 0.

Theorem (Markov) All Markov triples can be obtained from (1, 1, 1) by the involution σ :(x, y, z) → (x, y, 3xy − z) and permutations. Markov dynamics is the action of PSL2(Z) (and hence of the braid group B3) 3 on C determined by σ and τ(x, y, z) = (z, x, y). It has an obvious integral F = x 2 + y 2 + z 2 − 3xyz.

Markov triples

Markov numbers are parts of the Markov triples, which are the positive integer solutions of the Markov equation

x 2 + y 2 + z 2 − 3xyz = 0.

Theorem (Markov) All Markov triples can be obtained from (1, 1, 1) by the involution σ :(x, y, z) → (x, y, 3xy − z) and permutations.

(1, 1, 1) → (1, 1, 2) → (1, 2, 1) → (1, 2, 5) → (1, 5, 2) → (1, 5, 13) → ... Markov triples

Markov numbers are parts of the Markov triples, which are the positive integer solutions of the Markov equation

x 2 + y 2 + z 2 − 3xyz = 0.

Theorem (Markov) All Markov triples can be obtained from (1, 1, 1) by the involution σ :(x, y, z) → (x, y, 3xy − z) and permutations.

(1, 1, 1) → (1, 1, 2) → (1, 2, 1) → (1, 2, 5) → (1, 5, 2) → (1, 5, 13) → ...

Markov dynamics is the action of PSL2(Z) (and hence of the braid group B3) 3 on C determined by σ and τ(x, y, z) = (z, x, y). It has an obvious integral F = x 2 + y 2 + z 2 − 3xyz. Markov dynamics: F = 0.

Markov triples in logarithmic scale: Markov dynamics: F = 0.

Markov triples in logarithmic scale: Markov tree Markov tree Markov tree Markov tree Markov tree Markov tree Markov tree Markov tree 4 F = 9 : ”integrable chaos”

4 On the special level F = 9 the dynamics can be linearised by substitution 2 2 2 2 x = cos φ, y = cos ψ, z = cos(φ + ψ), z 0 = cos(φ − ψ). 3 3 3 3 The hyperbolic elements of SL(2, Z) act as Anosov maps, so the dynamics is nevertheless chaotic. 4 F = 9 : ”integrable chaos”

What happens at generic level is an interesting question (cf. Hone) Generic level of F : chaos ?

What happens at generic level is an interesting question (cf. Hone) Tropical Markov dynamics

It is not clear how to tropicalize the formula z 0 = 3xy − z, so we can use another relation zz 0 = x 2 + y 2 + c to deﬁne the following tropical Markov dynamics:

σ :(X , Y , Z) → (X , Y , 2 max(X , Y , C) − Z)

and τ :(X , Y , Z) → (Y , Z, X ). It has the following tropical integral

F = 2 max(X , Y , Z, C) − (X + Y + Z),

corresponding to x 2 + y 2 + z 2 + c F = . xyz Tropical Markov orbit Kontsevich, Manin: WDVV equation leads to recurrent formula for Nd : ! ! X 2 3d − 1 3d − 1 N = N N k l[l − k ]. d k l 3k − 2 3k − 1 k+l=d

The corresponding potential is 1 F = t (t t + t2) + G(t , t ), 2 1 1 3 2 2 3 where ∞ 3d−1 X t dt G(t , t ) = N 3 e 2 , 2 3 d (3d − 1)! d=1 2 Nd is the number of rational curves in CP passing by 3d − 1 points in general position.

Markov orbit and quantum cohomology

A remarkable observation due to Dubrovin is that in the theory of Frobenius manifolds

2 Markov orbit corresponds to quantum cohomology of projective plane CP Kontsevich, Manin: WDVV equation leads to recurrent formula for Nd : ! ! X 2 3d − 1 3d − 1 N = N N k l[l − k ]. d k l 3k − 2 3k − 1 k+l=d

Markov orbit and quantum cohomology

A remarkable observation due to Dubrovin is that in the theory of Frobenius manifolds

2 Markov orbit corresponds to quantum cohomology of projective plane CP The corresponding potential is 1 F = t (t t + t2) + G(t , t ), 2 1 1 3 2 2 3 where ∞ 3d−1 X t dt G(t , t ) = N 3 e 2 , 2 3 d (3d − 1)! d=1 2 Nd is the number of rational curves in CP passing by 3d − 1 points in general position. Markov orbit and quantum cohomology

A remarkable observation due to Dubrovin is that in the theory of Frobenius manifolds

Kontsevich, Manin: WDVV equation leads to recurrent formula for Nd : ! ! X 2 3d − 1 3d − 1 N = N N k l[l − k ]. d k l 3k − 2 3k − 1 k+l=d Analytic continuation induces an action of the braid group B3 on the triples of the monodromy matrices M1, M2, M3 :

−1 −1 σ1(M1, M2, M3) = (M2, M2M1M2 , M3), σ2(M1, M2, M3) = (M1, M3, M3M2M3 ). In 2 × 2 case this leads to the generalised Markov dynamics with

σ :(x, y, z) → (x, y, xy − z − ω3)

and τ :(x, y, z; ω1, ω2, ω3) → (y, z, x; ω2, ω3, ω1), preserving 2 2 2 F = x + y + z − xyz − ω1x − ω2y − ω3z.

Painlev´e-VIand braid group orbits

Painlev´eVI equation describes monodromy preserving deformations of Fuchsian systems „ « dΦ A1 A2 A3 = + + Φ, Φ ∈ SL(2, C). dλ λ − u1 λ − u2 λ − u3 Painlev´e-VIand braid group orbits

Painlev´eVI equation describes monodromy preserving deformations of Fuchsian systems „ « dΦ A1 A2 A3 = + + Φ, Φ ∈ SL(2, C). dλ λ − u1 λ − u2 λ − u3

Analytic continuation induces an action of the braid group B3 on the triples of the monodromy matrices M1, M2, M3 :

−1 −1 σ1(M1, M2, M3) = (M2, M2M1M2 , M3), σ2(M1, M2, M3) = (M1, M3, M3M2M3 ). In 2 × 2 case this leads to the generalised Markov dynamics with

σ :(x, y, z) → (x, y, xy − z − ω3)

and τ :(x, y, z; ω1, ω2, ω3) → (y, z, x; ω2, ω3, ω1), preserving 2 2 2 F = x + y + z − xyz − ω1x − ω2y − ω3z. Relation with Okamoto: aﬃne Weyl groups are acting on the set of all solutions while braid group describes the monodromy of a given solution.

Comparison with Watanabe result suggests that in general Markov dynamics can not be ”integrated in classical functions” (see, however, Picard solutions).

Dubrovin: Finite orbits correspond to the algebraic solutions of Painleve-VI (Dubrovin, Mazzocco)

Lisovyy, Tikhiy: classiﬁcation of the ﬁnite braid orbits

4 Integrable case F = 9 corresponds to the Picard solutions (Fuchs, Hitchin)

Markov dynamics and special solutions of PVI

Markov orbit corresponds to quantum cohomology solution (Dubrovin) Relation with Okamoto: aﬃne Weyl groups are acting on the set of all solutions while braid group describes the monodromy of a given solution.

Comparison with Watanabe result suggests that in general Markov dynamics can not be ”integrated in classical functions” (see, however, Picard solutions).

Dubrovin: Finite orbits correspond to the algebraic solutions of Painleve-VI (Dubrovin, Mazzocco)

Lisovyy, Tikhiy: classiﬁcation of the ﬁnite braid orbits

Markov dynamics and special solutions of PVI

Markov orbit corresponds to quantum cohomology solution (Dubrovin)

4 Integrable case F = 9 corresponds to the Picard solutions (Fuchs, Hitchin) Relation with Okamoto: aﬃne Weyl groups are acting on the set of all solutions while braid group describes the monodromy of a given solution.

Comparison with Watanabe result suggests that in general Markov dynamics can not be ”integrated in classical functions” (see, however, Picard solutions).

Markov dynamics and special solutions of PVI

Markov orbit corresponds to quantum cohomology solution (Dubrovin)

4 Integrable case F = 9 corresponds to the Picard solutions (Fuchs, Hitchin)

Dubrovin: Finite orbits correspond to the algebraic solutions of Painleve-VI (Dubrovin, Mazzocco)

Lisovyy, Tikhiy: classification of the finite braid orbits Relation with Okamoto: affine Weyl groups are acting on the set of all solutions while braid group describes the monodromy of a given solution.

Markov dynamics and special solutions of PVI

Markov orbit corresponds to quantum cohomology solution (Dubrovin)

4 Integrable case F = 9 corresponds to the Picard solutions (Fuchs, Hitchin)

Dubrovin: Finite orbits correspond to the algebraic solutions of Painleve-VI (Dubrovin, Mazzocco)

Lisovyy, Tikhiy: classiﬁcation of the ﬁnite braid orbits

Comparison with Watanabe result suggests that in general Markov dynamics can not be ”integrated in classical functions” (see, however, Picard solutions). Markov dynamics and special solutions of PVI

Markov orbit corresponds to quantum cohomology solution (Dubrovin)

4 Integrable case F = 9 corresponds to the Picard solutions (Fuchs, Hitchin)

Dubrovin: Finite orbits correspond to the algebraic solutions of Painleve-VI (Dubrovin, Mazzocco)

Lisovyy, Tikhiy: classiﬁcation of the ﬁnite braid orbits

Comparison with Watanabe result suggests that in general Markov dynamics can not be ”integrated in classical functions” (see, however, Picard solutions). Relation with Okamoto: aﬃne Weyl groups are acting on the set of all solutions while braid group describes the monodromy of a given solution. Relations with arithmetic (Markov, Mordell, Silverman)

Tropical and geometrical versions

1 3 Generalisations to K3 surfaces in (CP ) (Silverman)

Poisson structures (Goldman, Chekhov-Fock)

Cluster algebras and Laurent phenomenon (Berenstein, Fomin, Zelevinsky; Hone)

2 2 xn+1xn = xn+1 + xn+2 + c.

Other relations and questions

Teichm¨ullerspaces (Chekhov-Penner, Fock-Goncharov) Relations with arithmetic (Markov, Mordell, Silverman)

Tropical and geometrical versions

1 3 Generalisations to K3 surfaces in (CP ) (Silverman)

Poisson structures (Goldman, Chekhov-Fock)

Other relations and questions

Teichm¨ullerspaces (Chekhov-Penner, Fock-Goncharov) Cluster algebras and Laurent phenomenon (Berenstein, Fomin, Zelevinsky; Hone)

2 2 xn+1xn = xn+1 + xn+2 + c. Relations with arithmetic (Markov, Mordell, Silverman)

Tropical and geometrical versions

1 3 Generalisations to K3 surfaces in (CP ) (Silverman)

Other relations and questions

Teichm¨ullerspaces (Chekhov-Penner, Fock-Goncharov) Cluster algebras and Laurent phenomenon (Berenstein, Fomin, Zelevinsky; Hone)

2 2 xn+1xn = xn+1 + xn+2 + c.

Poisson structures (Goldman, Chekhov-Fock) Relations with arithmetic (Markov, Mordell, Silverman)

Tropical and geometrical versions

Other relations and questions

Teichm¨ullerspaces (Chekhov-Penner, Fock-Goncharov) Cluster algebras and Laurent phenomenon (Berenstein, Fomin, Zelevinsky; Hone)

2 2 xn+1xn = xn+1 + xn+2 + c.

Poisson structures (Goldman, Chekhov-Fock)

1 3 Generalisations to K3 surfaces in (CP ) (Silverman) Relations with arithmetic (Markov, Mordell, Silverman)

Other relations and questions

Teichm¨ullerspaces (Chekhov-Penner, Fock-Goncharov) Cluster algebras and Laurent phenomenon (Berenstein, Fomin, Zelevinsky; Hone)

2 2 xn+1xn = xn+1 + xn+2 + c.

Poisson structures (Goldman, Chekhov-Fock)

1 3 Generalisations to K3 surfaces in (CP ) (Silverman) Tropical and geometrical versions Other relations and questions

Teichm¨ullerspaces (Chekhov-Penner, Fock-Goncharov) Cluster algebras and Laurent phenomenon (Berenstein, Fomin, Zelevinsky; Hone)

2 2 xn+1xn = xn+1 + xn+2 + c.

Poisson structures (Goldman, Chekhov-Fock)

1 3 Generalisations to K3 surfaces in (CP ) (Silverman) Tropical and geometrical versions Relations with arithmetic (Markov, Mordell, Silverman)