When applied mathematics collided with algebra

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The University of Sydney

When applied mathematics collided with algebraNalini Joshi@monsoon0

earth.nullschool.net

How do we predict, simulate, or approximate reality?

98 (4) PNAS (2001) 1341–1346

Parabolic Cylinder Functions, Hermite Polynomials, and Gauss-Hermite Quadrature

x2 � x+ 2

x2 � 3

2F1(a, b; c;x)

}(x; g2, g3)

Polynomials

a continuous function on a finite subinterval of the real line can be uniformly approximated arbitrarily closely by a polynomial.

Karl Weierstrass

x2 � 1

x3 � 3x

x4 � 6x2 + 3

Example

�11 · x · e�x2/2dx = �

he�x2/2

i1�1

Orthogonality

�1Pm(x)Pn(x) e

�x2/2 dx = 0, if m 6= n

Orthogonality

�1Pm(x)Pn(x) e

�x2/2 dx = 0, if m 6= n

| {z }k

< Pm, Pn >

�11 · 1 · e�x2/2dx = 2

0e�x2/2dx

=p2⇡

�1Pn(x)Pn(x) e

�x2/2 dx =p2⇡n!

x2 � 1� x · x+ 1 · 1 = 0

3-term Recurrence Relation

P2(x)� xP1(x) + 1 · P0(x) = 0

Pn+1(x)� xPn(x) + nPn�1(x) = 0

xPn = Pn+1 +nX

a(n)j Pj

hxPn, Pmi = a(n)m hPm, Pmi, 0 m n

hPn, xPmi = 0, 0 m n� 2

hxPn, Pmi = hPn, xPmi) xPn = Pn+1 + anPn + bnPn�1

an =hxPn, PnihPn, Pni

bn =hxPn, Pni

hPn�1, Pn�1i

hxPn, Pmi = hPn, xPmi) xPn = Pn+1 + anPn + bnPn�1

Classical Polynomials

�1Pm(x)Pn(x) e

�x2/2 dx =p2⇡n!�nm

�1Pm(x)Pn(x) e

�x2/2 dx =p2⇡n!�nm

Classical weights

�1Pm(x)Pn(x) e

�x2/2 dx =p2⇡n!�nm

Classical weights

Hen(x)Hermite polynomials

What other weights are possible?

Shohat (1939)

“The method used is of a very elementary character.”

(AKA Jacques Chokhate)

w(x) =1

✓� x4

Shohat’s resultsZ 1

�1Pm(x)Pn(x) e

�x4/4 dx = 0, if m 6= n

P0(x) = 1

P1(x) = x� c1

Pn(x)� (x� cn)Pn�1(x) + �n Pn�2(x) = 0

�1Pm(x)Pn(x) e

�x4/4 dx = 0, if m 6= n

P0(x) = 1

P1(x) = x� c1

Pn(x)� (x� cn)Pn�1(x) + �n Pn�2(x) = 0

�1Pm(x)Pn(x) e

�x4/4 dx = 0, if m 6= n

P0(x) = 1

P1(x) = x� c1

Pn(x)� (x� cn)Pn�1(x) + �n Pn�2(x) = 0

�1Pm(x)Pn(x) e

�x4/4 dx = 0, if m 6= n

��n+1 + �n+2 + �n+3

�= n+ 1

Such weights also arise elsewhere…

Are there more such equations?

Algebra

A Reflection

↵2s1

A Reflection

↵2s1

A Reflection

↵2s1

A Reflection

↵2s1

A Reflection

↵2s1 w1(↵2)

A Reflection

↵2s1

w1(↵2) = ↵2 � 2(↵1,↵2)

(↵1,↵1)↵1

= (�1,p3) + (2, 0)

= (1,p3)

w1(↵2)

A Reflection

s1↵1 + ↵2

�↵1 � ↵2 �↵2

�↵1

Root system

s1↵1 + ↵2

�↵1 � ↵2 �↵2

�↵1are “simple” roots↵1 and ↵2

Root system

s1↵1 + ↵2

�↵1 � ↵2 �↵2

�↵1are “simple” roots↵1 and ↵2

Root system

This is a reflection group called A2

↵2 ↵1 + ↵2

�↵1 � ↵2 �↵2

�↵1

longest rootA2

Figure 3. hexagons

lattice

equilateral triangle

a0=0a 1

a2 = 0

s0, s1, s2✑Define to be

On the A2(1) lattice

reflections acrosseach edge

s0(a0, a1, a2) = (�a0, a1 + a0, a2 + a0)

equilateral triangle

a0=0a 1

a2 = 0

s0, s1, s2✑Define to be

On the A2(1) lattice

reflections acrosseach edge

fW(A(1)2 ) = hs0, s1, s2,⇡i

s2j = 1

(sj sj+1)3 = 1

⇡ sj = sj+1 ⇡

>;j 2 N mod 3

⇡3 = 1

diagram automorphism⇡ :

Coxeter Relations

Figure 1. Triangles inside a cube

Translations

Figure 2. Coordinates

a 1=0 a

a2 = 0

Figure 3. 3 triangles

Dynamics on the lattice

From reflections, we can show

T1(a0) = a0 + k, T1(a1) = a1 � k, T1(a2) = a2

T1(a0) = ⇡ s2 s1(a0)

= ⇡ s2 (a0 + a1)

= ⇡ (a0 + a1 + 2a2)

= a1 + a2 + 2 a0 = a0 + k

a0 a1 a2 f0 f1 f2

s0 �a0 a1 + a0 a2 + a0 f0 f1 +a0f0

f2 �a0f0

s1 a0 + a1 �a1 a2 + a1 f0 �a1f1

f1 f2 �a1f1

s2 a0 + a2 a1 + a2 �a2 f0 +a2f2

f1 �a2f1

Noumi 2004

Group Actions

a0 a1 a2 f0 f1 f2

s0 �a0 a1 + a0 a2 + a0 f0 f1 +a0f0

f2 �a0f0

s1 a0 + a1 �a1 a2 + a1 f0 �a1f1

f1 f2 �a1f1

s2 a0 + a2 a1 + a2 �a2 f0 +a2f2

f1 �a2f1

Noumi 2004

Group Actions

Define

T1(a0) = a0 + 1, T1(a1) = a1 � 1, T1(a2) = a2

un = Tn1 (f1), vn = Tn

1 (f0)

Define

T1(a0) = a0 + 1, T1(a1) = a1 � 1, T1(a2) = a2

1 (f0)

(un + un+1 = t� vn � a0+n

vn + vn�1 = t� un + a1�nun

Define

T1(a0) = a0 + 1, T1(a1) = a1 � 1, T1(a2) = a2

A scalar version of this is Shohat’s equation, again.

1 (f0)

(un + un+1 = t� vn � a0+n

vn + vn�1 = t� un + a1�nun

• “String equation” • “discrete first Painlevé equation”

wn (wn+1 + wn + wn�1) = ↵n+ � + � wn

Douglas & Shenker, Nuclear Phys B 1990 Fokas, Its & Kitaev CMP 1991

Eqn 1. Alpha ! 0.25‘, Beta ! 0.‘, Gamma ! "0.5 .!1

Re!x"!1

!1!0.5

dP1.nb 1

wn (wn+1 + wn + wn�1) = ↵n+ � + � wn

Eqn 1. Alpha ! 0.25‘, Beta ! 0.‘, Gamma ! "0.5 .!1

Re!x"!1

!1!0.5

dP1.nb 1

wn (wn+1 + wn + wn�1) = ↵n+ � + � wn

Autonomous Case

K(x, y) = xy2 + x2y � b(x+ y)� cxy

�xn+1 + xn + xn�1

�= b+ cxn

K(xn+1, xn)�K(xn, xn�1) = 0

-2 -1 0 1 2-2

A pencil of elliptic curves

Invariant9

The dynamics are best described by using tools from algebraic geometry.

The Setting

A complex projective surface X obtained by a finite number of blow ups

The Setting

Irreducible components Dj forming a root system R

The Setting

Intersection form (Dk I Dj)

The Setting

Intersection form (Dk I Dj)

Dynamical system whose solution trajectories form leaves in a foliated vector bundle

Symmetry Group

The automorphisms of Pic(X)

Symmetry Group

that preserve the blow-up structure

Symmetry Group

that preserve the blow-up structure that leave R invariant

Symmetry Group

⇔ orthogonal complement R⊥ of R

that preserve the blow-up structure that leave R invariant

Theorem (Sakai, 2001). Let X a compact smooth rational surface obtained by blowing up 9 points, equipped with a unique anti-canonical divisor D of canonical type. Then there exists two orthogonal root systems R and R⊥ in the root lattice of D. The Weyl group of R⊥ acts as automorphisms of Pic(X) giving bi-rational actions of X. The translation part of the Weyl group gives rise to discrete Painlevé equations.

Corollary. Degenerate cases of the above give the six Painlevé equations.

This geometric description also leads to dynamical information about their solutions.

Duistermaat & J 2011J & Howes 2014J & Radnovic 2016-2019

Sakai (2001) found 22 classes of such equations.

The translation part of the Weyl group of R⊥

gives iterations of a discrete Painlevé equation.

Sakai (2001) found 22 classes of such equations.

Did Sakai obtain all possible such equations?

The translation part of the Weyl group of R⊥

gives iterations of a discrete Painlevé equation.

New equations and solutions

The RCG equation

The RCG equation✑A reduction of the lattice equation Krichever-Novikov

equation (also known as Adler’s equation or Q4):

The RCG equation✑A reduction of the lattice equation Krichever-Novikov

equation (also known as Adler’s equation or Q4):

cn(�n)dn(�n)�1� k2sn4(zn)

�un+1 + un�1

� cn(zn)dn(zn)�1� k2sn2(zn)sn

2(�n)��un+1un�1 + un

+�cn2(zn)� cn2(�n)

�cn(zn)dn(zn)

�1 + k2un

2un+1un�1

�= 0

zn = (�e + �o)n+ z0, �n =

(�e, for n = 2 j

�o, for n = 2j + 1.

whereRamani, Carstea &Grammaticos, J Phys A 2009

What is the symmetry group of the equation?

• An 8-dimensional root system containing 240 root vectors.

• All root vectors have the same length √2.

• The roots span a polytope, known as the 421 polytope.

Wikipedia CC BY-SA 3.0

E8(1) Lattice

Z↵k, Q_ =8X

Z↵_k , P =

Augmented with

Finite root system

↵0 + e↵ = 0

e↵ = 2↵1 + 4↵2 + 6↵3 + 5↵4 + 4↵5 + 3↵6 + 2↵7 + 3↵8

E8(1) Reflections

s0 : (↵0,↵7) 7! (�↵0,↵7 + ↵0), s1 : (↵1,↵2) 7! (�↵1,↵2 + ↵1),

s2 : (↵1,↵2,↵3) 7! (↵1 + ↵2,�↵2,↵3 + ↵2),

s3 : (↵2,↵3,↵4,↵8) 7! (↵2 + ↵3,�↵3,↵4 + ↵3,↵8 + ↵3),

s4 : (↵3,↵4,↵5) 7! (↵3 + ↵4,�↵4,↵5 + ↵4),

s5 : (↵4,↵5,↵6) 7! (↵4 + ↵5,�↵5,↵6 + ↵5),

s6 : (↵5,↵6,↵7) 7! (↵5 + ↵6,�↵6,↵7 + ↵6),

s7 : (↵6,↵7) 7! (↵6 + ↵7,�↵7), s8 : (↵3,↵8) 7! (↵3 + ↵8,�↵8).

E8(1) TranslationsFor any roots ↵,� 2 Q the corresponding translations satisfy

T↵ � T� = T↵+�

For any w 2 W (E(1)8 ) we have

w � T↵ = Tw(↵) � w

T↵(�) = �

Moreover, translation acts as an identity on a root vector

|T↵|2 := h↵_,↵i

Define the length of a translation by

Lengths of translationsFor any ↵,� 2 Q if the translations T↵, T� are conjugate

then|T↵|2 = |T� |2

w � T↵ � w�1 = T�i.e. w 2 W (E(1)8 )for some

Proof.

Tw(↵) = T�

Conjugacy implies

|T↵|2 = |Tw(↵)|2 = |T� |2So we have

Two translations

T (M) = s1238432543865432765438076543212345670834567234568345234832

T (JN) = s25645348370675645234832156453483706756452348321706734830468Murata, 2003

J. & Nakazono, J. Phys. A 2017

T (M) = T↵1 ,

T (JN) = T↵0+2↵2+4↵3+4↵4+4↵5+3↵6+2↵7+2↵8

|T↵1 |2 = 2,

|T↵0+2↵2+4↵3+4↵4+4↵5+3↵6+2↵7+2↵8 |2 = 4.

Or, in terms of roots

of different lengths

Inside the lattice

Fix a point on the lattice.

It has 240 nearest neighbours lying at a distance whose squared length is equal to 2.

It has 2160 next-nearest neighbours lying at a distance whose squared length is equal to 4.

) T (M)

) T (JN)

E8 Weight Lattice

✑ For each vertex, 240 nearest-neighbours, reached by vectors of squared length 2.

E8 Weight Lattice

Sakai’s elliptic difference equation

E8 Weight Lattice

✑ For each vertex, 2160 next-nearest-neighbours, reached by vectors of squared length 4.

A new elliptic difference equation

E8 Weight Lattice

• Ramani, Carstea, Grammaticos (2009)• Atkinson, Howes, J. and Nakazono

(2016)

E8 Weight Lattice

• Ramani, Carstea, Grammaticos (2009)• Atkinson, Howes, J. and Nakazono

(2016)• J. and Nakazono (2017, 2019)• Carstea, Dzhamay, Takenawa (2017)

Symmetry of RCG equation

• The RCG equation has - initial value space A0(1)

- symmetry group F4(1) , a subgroup of E8(1)

- time iteration which is a “square root” of a translation on this lattice.

Ell: E(1)

Mul: E(1)

8 E(1)

7 E(1)

6 A(1)

3 A(1)

4 (A2 +A1)(1) (A1 +A1)(1)

Add: E(1)

8 E(1)

7 E(1)

6 D(1)

4 A(1)

3 2A(1)

1 A(1)

10 A(1)

2 A(1)

1 A(1)

Ell: E(1)

Mul: E(1)

8 E(1)

7 E(1)

6 A(1)

3 A(1)

4 (A2 +A1)(1) (A1 +A1)(1)

Add: E(1)

8 E(1)

7 E(1)

6 D(1)

4 A(1)

3 2A(1)

1 A(1)

10 A(1)

2 A(1)

1 A(1)

Ell: E(1)

Mul: E(1)

8 E(1)

7 E(1)

6 A(1)

3 A(1)

4 (A2 +A1)(1) (A1 +A1)(1)

Add: E(1)

8 E(1)

7 E(1)

6 D(1)

4 A(1)

3 2A(1)

1 A(1)

10 A(1)

2 A(1)

1 A(1)

The University of Sydney

At least four elliptic-difference equations

J. and Nakazono, 2017, 2019

RJ,2RJ,1

Sakai 2001

RCG 2009

Special solutions given by 10E9 elliptic hypergeometric functions.

Kajiwara, Masuda, Noumi, Ohta, Yamada, 2003Kajiwara, Noumi, Yamada, 2017

What are the dynamics of its general solutions?

When applied mathematics collided with algebra

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