When Applied Maths Collided with Algebra

Nalini Joshi

Supported by the Australian Research Council

@monsoon0

A Reflection

↵1

↵2

A Reflection

↵1

↵2s1

A Reflection

↵1

↵2s1

A Reflection

↵1

↵2s1 w1(↵2)

//

//

A Reflection

↵1

↵2s1 w1(↵2)

//

//

w1(↵2) = ↵2 � 2(↵1,↵2)

(↵1,↵1)↵1

A Reflection

↵1

↵2s1 w1(↵2)

//

//

w1(↵2) = ↵2 � 2(↵1,↵2)

(↵1,↵1)↵1

= (�1,p3) + (2, 0)

A Reflection

↵1

↵2s1 w1(↵2)

//

//

w1(↵2) = ↵2 � 2(↵1,↵2)

(↵1,↵1)↵1

= (�1,p3) + (2, 0)

= (1,p3)

Root System

↵1

↵2

are “simple” roots↵1 and ↵2

Root System

↵1

↵2s1

are “simple” roots↵1 and ↵2

Root System

↵1

↵2s1

↵1 + ↵2

are “simple” roots↵1 and ↵2

Root System

↵1

↵2s1

↵1 + ↵2

�↵1

are “simple” roots↵1 and ↵2

Root System

↵1

↵2

s2

s1↵1 + ↵2

�↵1

are “simple” roots↵1 and ↵2

Root System

↵1

↵2

s2

s1↵1 + ↵2

�↵2

�↵1

are “simple” roots↵1 and ↵2

Root System

↵1

↵2

s2

s1↵1 + ↵2

�↵1 � ↵2 �↵2

�↵1

are “simple” roots↵1 and ↵2

Root System

↵1

↵2

s2

s1↵1 + ↵2

�↵1 � ↵2 �↵2

�↵1

are “simple” roots↵1 and ↵2

Reflection Groups• Roots:

• Reflections:

• Co-roots:

• Weights:

↵1,↵2, . . . ,↵n

wi(↵j) = ↵j � 2(↵i,↵j)

(↵i,↵i)↵i

↵̌i = 2↵i

(↵i,↵i)

h1, h2, . . . , hn

(hi, ↵̌j) = �ij

↵1

↵2 ↵1 + ↵2

�↵1 � ↵2 �↵2

�↵1

A2

↵1

↵2

s2

s1↵1 + ↵2

�↵1 � ↵2 �↵2

�↵1

A2

↵1

↵2

s2

s1↵1 + ↵2

�↵1 � ↵2 �↵2

�↵1

A2

h1

h2

↵1

↵2

s2

s1↵1 + ↵2

�↵1 � ↵2 �↵2

�↵1

A2

h1

h2

longest root

↵1

↵2

s2

s1↵1 + ↵2

�↵1 � ↵2 �↵2

�↵1

A2

h1

h2

longest root

Translations2

Figure 3. hexagons

A2(1)

Other Choices

2 cos

2(✓↵1↵2) = n

Crystallographic property: (↵i, ↵̌j) 2 Z

n = 0 ) ✓↵1↵2 = 3⇡/2

n = 1 ) ✓↵1↵2 = 2⇡/3

n = 2 ) ✓↵1↵2 = 3⇡/4

n = 3 ) ✓↵1↵2 = 5⇡/6

↵2

↵1� 2↵2 � ↵1

2↵2 + ↵1 = s1

↵1 + ↵2 = s2

B2

↵2

↵1� 2↵2 � ↵1

2↵2 + ↵1 = s1

↵1 + ↵2 = s2

B2

↵2

↵1� 2↵2 � ↵1

2↵2 + ↵1 = s1

↵1 + ↵2 = s2

B2

↵2�↵1

↵1� 2↵2 � ↵1

2↵2 + ↵1 = s1

↵1 + ↵2 = s2

B2

↵2

�↵2

�↵1

↵1� 2↵2 � ↵1

2↵2 + ↵1 = s1

↵1 + ↵2 = s2

B2

↵2

�↵1 � ↵2

�↵2

�↵1

↵1� 2↵2 � ↵1

2↵2 + ↵1 = s1

↵1 + ↵2 = s2

B2

↵2

�↵1 � ↵2

�↵2

�↵1

↵1� 2↵2 � ↵1

2↵2 + ↵1 = s1

↵1 + ↵2 = s2

B2

h2

h1

On the A2 Lattice

a0=0a 1

=0

a2 = 0

s0, s1, s2• Define to be reflections across each edge

On the A2 Lattice

a0=0a 1

=0

a2 = 0

s0, s1, s2• Define to be reflections across each edge

On the A2 Lattice

a0=0a 1

=0

a2 = 0

s0, s1, s2• Define to be reflections across each edge

On the A2 Lattice

a0=0a 1

=0

a2 = 0

s0, s1, s2• Define to be reflections across each edge

On the A2 Lattice

equilateral trianglea0=0a 1

=0

a2 = 0

s0, s1, s2• Define to be reflections across each edge

On the A2 Lattice

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

equilateral trianglea0=0a 1

=0

a2 = 0

s0, s1, s2• Define to be reflections across each edge

Coxeter Relations

fW(A(1)2 ) = hs0, s1, s2,⇡i

s2j = 1

(sj sj+1)3= 1

⇡ sj = sj+1 ⇡

9>=

>;j 2 N mod 3

⇡3 = 1diagram automorphism⇡ :

Discrete Dynamics I

• Translation

Figure 1. Triangles inside a cube

Translations

Figure 2. Coordinates

4 40

T1

1

0 2

a 1=

0

a0=

0

a2 = 0

Figure 3. 3 triangles

1

Discrete Dynamics II• Translation as reflections ?2

4 40

s1(4)

T1

1

0 2

0

12

1

0 2

a 1=

0

a0=

0

a2 = 0

Figure 4. Translation II

4 40

s1(4))

T1

1

0 2

0

12

1

0 2

a 1=

0

a0=

0

a2 = 0

Figure 5. Translation III

Discrete Dynamics II• Translation as reflections ?

2

4 40

s1(4)

T1

1

0 2

0

12

1

0 2

a 1=

0

a0=

0

a2 = 0

Figure 4. Translation II

4 40

s2(s1(4))

T1

1

0 2

0

12

0

12

1

0 2

a 1=

0

a0=

0

a2 = 0

Figure 5. Translation III

Discrete Dynamics II• Translation as reflections ?

3

4 40

⇡(s2(s1(4)))

⇡T1

1

0 2

0

12

1

20

1

0 2

a 1=

0

a0=

0

a2 = 0

Figure 6. Translation IV

Discrete Dynamics III• Translations as reflections

+ diagram automorphism

T1 = ⇡ s2 s1

T2 = s1 ⇡ s2

T0 = s2 s1 ⇡

Constancy of coordinatesFigure 1. Triangles inside a cube

Translations

Figure 2. Coordinates

1

a0 + a1 + a2 = k

Constancy of coordinatesFigure 1. Triangles inside a cube

Translations

Figure 2. Coordinates

1

a0 + a1 + a2 = k

Constancy of coordinatesFigure 1. Triangles inside a cube

Translations

Figure 2. Coordinates

1

a0 + a1 + a2 = k

Constancy of coordinatesFigure 1. Triangles inside a cube

Translations

Figure 2. Coordinates

1

a0 + a1 + a2 = k

TranslationsSo we have

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

f2

Cremona Isometries

Noumi 2004

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

f2

Cremona Isometries

Noumi 2004

Translations again

Define

UsingT1(a0) = a0 + 1, T1(a1) = a1 � 1, T1(a2) = a2

un = Tn1 (f1), vn = Tn

1 (f0)

Translations again

Define

UsingT1(a0) = a0 + 1, T1(a1) = a1 � 1, T1(a2) = a2

)

un = Tn1 (f1), vn = Tn

1 (f0)

(un + un+1 = t� vn � a0+n

vn

vn + vn�1 = t� un + a1�nun

Translations again

Define

UsingT1(a0) = a0 + 1, T1(a1) = a1 � 1, T1(a2) = a2

)

This is a discrete Painlevé equation, which arises in quantum gravity.

un = Tn1 (f1), vn = Tn

1 (f0)

(un + un+1 = t� vn � a0+n

vn

vn + vn�1 = t� un + a1�nun

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

!0.5

0

0.5

1

Re!x"!1

!0.50

0.51

Im!x"

!1!0.5

00.51

!1!0.5

00.5

1

Im!x"

!1!0.5

00.51

dP1.nb 1

Solution orbits of scalar dP1 on the Riemann sphere (where the north pole is infinity).

What does this have to do with applied mathematics?

Applications• Electrical structures of

interfaces in steady electrolysis L. Bass, Trans Faraday Soc 60 (1964)1656–1663

• Spin-spin correlation functions for the 2D Ising model TT Wu, BM McCoy, CA Tracy, E Barouch Phys Rev B13 (1976) 316–374

• Spherical electric probe in a continuum gas PCT de Boer, GSS Ludford, Plasma Phys 17 (1975) 29–41

• Cylindrical Waves in General Relativity S Chandrashekar, Proc. R. Soc. Lond. A 408 (1986) 209–232

• Non-perturbative 2D quantum gravity Gross & Migdal PRL 64(1990) 127-130

• Orthogonal polynomials with non-classical weight function AP Magnus J. Comput Appl. Anal. 57 (1995) 215–237

• Level spacing distributions and the Airy kernel CA Tracy, H Widom CMP 159 (1994) 151–174

• Spatially dependent ecological models: J & Morrison Anal Appl 6 (2008) 371-381

• Gradient catastrophe in fluids: Dubrovin, Grava & Klein J. Nonlin. Sci 19 (2009) 57-94

Summary

• New mathematical models of physics pose new questions for applied mathematics.

• Global dynamics of solutions can be found through geometry.

• It is currently the only analytic approach available in for discrete equations.

• Tantalising questions remain open.

C