On Kleins icosahedral solution of the quintic (Irish Mathematics - - PowerPoint PPT Presentation

The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions

On Klein’s icosahedral solution of the quintic (Irish Mathematics Students Assoc. Conf.)

Oliver Nash March 2, 2013

Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics

The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions

Table of contents

The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions

Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics

The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions

General quintic equation

This is what we are going to solve for x: x5 + a1x4 + a2x3 + a3x2 + a4x + a5 = 0 where a1, a2, . . . , a5 are any complex numbers.

Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics

The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions

General quintic equation

This is what we are going to solve for x: x5 + a1x4 + a2x3 + a3x2 + a4x + a5 = 0 where a1, a2, . . . , a5 are any complex numbers. What do we mean by solve?

Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics

The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions

Solutions of polynomials in radicals

◮ Solving an equation in radicals means using only +, −, ·, / and

root extraction (finitely many times).

◮ We can solve quadratic, cubic and quartic equations in

radicals.

Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics

The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions

Solutions of polynomials in radicals

◮ Solving an equation in radicals means using only +, −, ·, / and

root extraction (finitely many times).

◮ We can solve quadratic, cubic and quartic equations in

radicals.

◮ Famously, we cannot solve the quintic equation in radicals

(Abel & Ruffini : 1823, 1799).

◮ Galois developed the most important tools: Galois group

permutes roots. Sn in general.

◮ We must explore other means of solving the quintic.

Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics

The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions

Tschirnhaus reduction of the quintic

◮ General quintic: x5 + a1x4 + a2x3 + a3x2 + a4x + a5 = 0. ◮ Well known substitution y = x − a1/5 eliminates degree 4

term.

Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics

The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions

Tschirnhaus reduction of the quintic

◮ General quintic: x5 + a1x4 + a2x3 + a3x2 + a4x + a5 = 0. ◮ Well known substitution y = x − a1/5 eliminates degree 4

term.

◮ In fact if we allow y to be quadratic in x we can also

eliminate degree 3 term.

◮ Thus reduce general quintic to canonical form:

y5 + 5αy2 + 5βy + γ = 0.

Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics

The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions

Geometry of the roots

◮ Vector of roots of quintic is a point in C5 and so defines point

in (complex) projective space P4.

◮ If quintic is in canonical form then roots satisfy

yi = y2

i = 0 and so point in P4 lies on quadric surface. ◮ Quadric surface is doubly-ruled surface. This might be useful

so let’s remind ourselves what this means.

Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics

The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions

The quadric surface looks like this

Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics

The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions

What are they?

◮ A quadric surface is a 2-dimensional space that is solution of a

quadratic equation.

◮ ax2 + by2 + cz2 + dxy + eyz + fzx + gx + hy + iz + j = 0

Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics

The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions

What are they?

◮ A quadric surface is a 2-dimensional space that is solution of a

quadratic equation.

◮ ax2 + by2 + cz2 + dxy + eyz + fzx + gx + hy + iz + j = 0 ◮ Over R these are ellipsoids, paraboloids, hyperboloids (if

non-singular).

◮ Much simpler over C and in projective case: all non-singular

quadric surfaces can be put in form XY = ZW for homogeneous coordinates [X, Y , Z, W ] for P3.

Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics

The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions

Double ruling explicitly

Given: Q = {[X, Y , Z, W ] | XY = ZW } ⊂ P3 We have bijection: Q → P1 × P1 [X, Y , Z, W ] → ([X, Z], [X, W ]) with inverse: ([λ0, λ1], [µ0, µ1]) → [λ0µ0, λ1µ1, λ1µ0, λ0µ1] Fibres over each factor are lines in Q

Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics

The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions

A family of lines

Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics

The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions

The complex projective line is a sphere

The projective line adds just one point, namely ∞, to the affine line: P1 = C ∪ {∞}. In homogeneous coordinates ∞ = [1, 0]. Stereographic projection gives natural bijection S2 → C ∪ {∞} thus have natural bijection S2 → P1.

Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics

The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions

The complex projective line is a sphere

The projective line adds just one point, namely ∞, to the affine line: P1 = C ∪ {∞}. In homogeneous coordinates ∞ = [1, 0]. Stereographic projection gives natural bijection S2 → C ∪ {∞} thus have natural bijection S2 → P1. For sake of definiteness, we give coordinate representation of S2 → P1: (x, y, z) → [x + iy, 1 − z] with inverse: [u, v] → 1 |u|2 + |v|2 (u¯ v + ¯ uv, −i(u¯ v − ¯ uv), |u|2 − |v|2)

Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics

The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions

Stereographic projection looks like this

Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics

The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions

Roots not naturally ordered

◮ Roots unordered so actually just get S5-orbit on quadric

surface.

◮ S5 action is just permutation of coordinates so is linear and so

is induced from action on the two families of lines in Q.

◮ Odd permutations interchange two families so restrict to A5.

Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics

The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions

Roots not naturally ordered

◮ Roots unordered so actually just get S5-orbit on quadric

surface.

◮ S5 action is just permutation of coordinates so is linear and so

is induced from action on the two families of lines in Q.

◮ Odd permutations interchange two families so restrict to A5. ◮ Thus well-defined points in quotients: Z ± ∈ P1/A5.

Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics

The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions

Roots not naturally ordered

◮ Roots unordered so actually just get S5-orbit on quadric

surface.

◮ S5 action is just permutation of coordinates so is linear and so

is induced from action on the two families of lines in Q.

◮ Odd permutations interchange two families so restrict to A5. ◮ Thus well-defined points in quotients: Z ± ∈ P1/A5. ◮ In fact P1/A5 ≃ P1 = C ∪ {∞} so we can regard the point in

the quotient as a possibly-infinite complex number.

◮ Does not depend on ordering of roots so must be expressible

as rational function of α, β, γ (almost).

Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics

The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions

Icosahedral invariant

y5 + 5αy2 + 5βy + γ = 0

◮ When α = 0, β = 1 we have:

∇2 = γ4 + 256 Z ± = 1 2 · 1728[2 · 1728 + 207γ4 + γ8 ± γ2(81 + γ4)∇]

Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics

The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions

Icosahedral invariant

y5 + 5αy2 + 5βy + γ = 0

◮ When α = 0, β = 1 we have:

∇2 = γ4 + 256 Z ± = 1 2 · 1728[2 · 1728 + 207γ4 + γ8 ± γ2(81 + γ4)∇]

◮ Too messy to state explicitly like this for general α, β, γ but

still tractable when stated appropriately.

◮ If we knew point above Z we could solve the quintic. Thus

good idea to consider means of inverting this quotient.

Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics

The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions

The quotient by A5

◮ We will get tighter grip on P1 if we introduce a bit more

geometry.

◮ Thus introduce A5-invariant hermitian metric on C2 (can

always do this by averaging).

◮ This reduces group of symmetries from PSL(2, C) to

PSU(2) = SO(3). In other words it turns P1 into a round sphere where it makes sense to measure distances, angles etc.

◮ A5 action has generically free orbit but three non-free orbits

with stabilizers of order 2, 3, 5. These are locations of vertices, edge midpoints and face centres of an icosahedron!

Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics

The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions

The icosahedron

Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics

The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions

The icosahedron has symmetry group A5

Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics

The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions

The icosahedron on its circumsphere

Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics

The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions

A picture of the quotient map

Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics

The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions

Inverting the icosahedral quotient

◮ Riemann mapping theorem (aka uniformization theorem) tells

us that any simply connected (non-empty) complex domain that is not C can be biholomorphically mapped to upper half space H = {w ∈ C | Im(w) > 0}.

◮ Thus given a triangle, there is thus a map from H to its

interior and it can be made unique by adding simple boundary conditions (H has small automorphism group PSL(2, R)).

◮ It is easy to construct this map fairly explicitly for Euclidean

• triangles. The key parameters defining the map are the angles
• f the triangle.

Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics

The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions

Inverting the icosahedral quotient (continued)

◮ With just a little more work, the theory can be extended to

spherical triangles and the angles are all that is required.

◮ The map can be expressed in terms of the (analytic

continuation of) Gauss’s hypergeometric series:

2F1(a, b; c; w) =

• n≥0

a(a + 1) · · · (a + n − 1) · b(b + 1) · · · (b + n − 1) c(c + 1) · · · (c + n − 1) wn n!

◮ This is exactly what we need to invert out quotient map and

from our icosahedral construction we can read off the angles are π/2, π/3, π/5. These determine the values of a, b, c.

Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics

The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions

In summary

◮ Reduce general quintic to form y5 + 5αy2 + 5βy + γ = 0. ◮ Consider vector of roots in P4 and note it lies in quadric

surface Q.

◮ In fact we get an S5-orbit in Q rather than a point since roots

do not have natural ordering.

◮ Because of double-ruling Q ≃ P1 × P1 we get A5-orbits in

each P1.

◮ Since P1/A5 ≃ C ∪ {∞} get (possibly-infinite) invariant Z. ◮ We can naturally identify P1 with circumsphere of an

icosahedron and use hypergeometric functions to invert the quotient map P1 → P1/A5.

Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics

The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions

A formula for the quintic

Given y5 + 5y + γ = 0, calculate: ∇ =

• γ4 + 256

Z = 1 2 · 1728[2 · 1728 + 207γ4 + γ8 − γ2(81 + γ4)∇] z =

2F1(31 60, 11 60; 6 5; Z −1)

(1728Z)1/52F1(19

60, − 1 60; 4 5; Z −1)

Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics

The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions

A formula for the quintic (continued)

Let: B(z) = −1 − z − 7(z2 − z3 + z5 + z6) + z7 − z8 D(z) = −1 + 2z + 5z2 + 5z4 − 2z5 − z6 f (z) = z(z10 + 11z5 − 1) H(z) = −(z20 + 1) + 228(z15 − z5) − 494z10 T(z) = (z30 + 1) + 522(z25 − z5) − 10005(z20 + z10) (The zeros of these are the locations of vertices, face centres of a cube and vertices, edge midpoints and face centres of an icosahedron.)

Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics

The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions

A formula for the quintic (still continued!)

Then: y = −γ · B(z)f (z) H(z) − 7γ2 + 9∇ 2γ(γ4 + 648) · B(z)D(z)T(z) H(z)f (z)2 is a root! Replacing z with exp(2πνi/5)z for ν = 1, 2, 3, 4 provides all the

• ther roots.

Also similar (substantially more complicated) formulae for general α, β.

Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics

The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions

Who they are

◮ SIG = Susquehanna International Group. ◮ ‘A world-leading technology-empowered financial trading firm’.

Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics

The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions

Who they are

◮ SIG = Susquehanna International Group. ◮ ‘A world-leading technology-empowered financial trading firm’. ◮ c.1500 people; 13 offices; c.280 in Dublin (European HQ).

Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics

The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions

Who they are

◮ SIG = Susquehanna International Group. ◮ ‘A world-leading technology-empowered financial trading firm’. ◮ c.1500 people; 13 offices; c.280 in Dublin (European HQ). ◮ Founded 1987 by group of options traders; HQ in Philadelphia. ◮ Culture of poker.

I worked in SIG, Dublin for 5 years and highly recommend them as an employer for someone interested in working in finance / trading.

Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics

The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions

What they do

◮ Proprietary trading. ◮ Securities traded include: spot equities, equity options,

commodities, index futures, ETFs, convertible bonds, ...

Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics

The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions

What they do

◮ Proprietary trading. ◮ Securities traded include: spot equities, equity options,

commodities, index futures, ETFs, convertible bonds, ...

◮ Each office trades several ‘strategies’ – invented/overseen by

traders; range from manual trading to computer assisted trading to fully automatic.

◮ Market making, liquidity provision, arbitrage, statistical

arbitrage, ...

Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics

The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions

Assistant Trader role

◮ AT = Assistant Trader. ◮ Key role: traders are raison d’ˆ

etre of SIG.

Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics

The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions

Assistant Trader role

◮ AT = Assistant Trader. ◮ Key role: traders are raison d’ˆ

etre of SIG.

◮ Relevant skills include: excellent numeracy, intuition for

probability, game theory, high attention to detail.

Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics

The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions

Assistant Trader role

◮ AT = Assistant Trader. ◮ Key role: traders are raison d’ˆ

etre of SIG.

◮ Relevant skills include: excellent numeracy, intuition for

probability, game theory, high attention to detail.

◮ Basic computer skills necessary, simple coding skills a bonus

(but not necessary).

Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics

The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions

Assistant Trader role

◮ AT = Assistant Trader. ◮ Key role: traders are raison d’ˆ

etre of SIG.

◮ Relevant skills include: excellent numeracy, intuition for

probability, game theory, high attention to detail.

◮ Basic computer skills necessary, simple coding skills a bonus

(but not necessary).

◮ Financial knowledge not necessary. SIG provides extensive and

well respected training.

◮ Well compensated and leads to role of trader: extremely well

compensated.

Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics

The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions

Questions

Feel free to ask me anything about either the icosahedral of the quintic or about career opportunities at SIG.

Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics