Space curves, vector fields and strange surfaces. Simon Salamon

Transcription

1 Space curves, vector fields and strange surfaces Simon Salamon Lezione Lagrangiana, 26 May 2016

2 Curves in space 1 A curve is the path of a particle moving continuously in space. Its position at time t is represented by a triple (x(t), y(t), z(t)). The twisted cubic (x, y, z) = (3t, 3t 2, t 3 ) does not lie in any plane:

3 Shadows 2 Projection parallel to the axes defines three plane curves: y 3 = 27z 2, x 3 = 27z, x 2 = 3y The first, a semi-cubical parabola with a cusp point, has the most elementary arc length function s = 1 2 (4 + t2 ) 3/2.

4 Tangent vectors 3 The velocity at time t is represented by the vector ( dx dt, dy dt, dz ) = (3, 6t, 3t 2 ). dt Later: we shall pass from arrows to curves.

5 A flat ruled surface 4 The tangent lines sweep out a curved surface the discriminant of x 2 y 2 4y 3 4x 3 z +18xyz 27z 2 = 0, ζ 3 + xζ 2 + yζ + z. It has zero Gaussian curvature, so locally isometric to a plane. Every point of the curve is a cusp on the surface.

6 Digression: hyperbolic paraboloids 5 These illustrate payoffs for players #1 and #2 in a 2-player stag hunt : #1 #2 4, 4 0, 2 2, 0 1, 1 A mixed-strategy Nash equilibrium occurs where two horizontal rulings intersect.

7 A model space curve 6 For the helix (x, y, z) = (12 cos t, 12 sin t, 5t), arclength s from (12, 0, 0) equals t 0 dx 2 + dy 2 + dz 2 = 13 t. Theorem: Up to a rigid motion (rotation and translation R 3 ), any space curve is determined by two functions of arclength, the curvature κ(s) and the torsion τ(s). For the helix they are both constant: κ , τ

8 The Serret-Frenet formulae 7 T is the unit tangent vector N is a unit normal vector B = T N is the binormal vector Proposition: dt ds = κn dn ds = κt + τb db ds = τ N Or, with F = (T N B), (df )F 1 = ( 0 κ 0 κ 0 τ 0 τ 0 ) so(3).

9 Curvature and torsion 8 At a given point X (t) = (x(t), y(t), z(t)) of a space curve, κ(t) = X (t) X (t) X (t) 3. It equals 1/r(t), where r(t) is the radius of the best-fitting circle at X (t). The torsion τ(t) = (X (t) X (t)) X (t) X (t) X (t) 2 measures the extent that the curve does not lie in a plane. If V = X (t), the numerator can be written δ = det ( V V V 2 V V ).

10 Chirality 9 The sign of the torsion represents the chirality of the curve at the point in question: if τ > 0 it is right-handed like a screw. The blue helix (that part with τ(s) = log s > 0) is right-handed. The green climber plant is left-handed:

11 Assigning curvature and torsion 10 There is no restriction on the torsion, but the curvature should be non-negative (κ 0). One can then construct the space curve, unique up to a rigid motion. The competition curve has κ(s) = s cos s, τ(s) = log s starting from s = 1.

12 Vector fields 11 A vector field assigns an arrow at each point of space. They describe ODE s such as the Lorenz system: dx dt dy dt dz dt = σ(y x) = x(ρ z) y = xy βz. Based on a toy simplification of the Navier-Stokes equations to describe atmospheric convection: x, y are temperature variations, z is rate of convective overturning. Chaotic behaviour occurs for suitable values of the parameters, such as σ = 10, β = 8 3, and ρ = 28.

13 The three equilibria 12 L(x, y, z) = (10y 10x, 28x y xz, xy 3z). Our Lorenz field L vanishes at exactly 3 points: (0, 0, 0), (9, 9, 27), ( 9, 9, 27).

14 Lorenz knots 13 Picard s theorem: Let V be a C 1 vector field. Starting from any point X 0 R 3 where V 0, there exists a unique trajectory ( ε, ε) R 3 with X (0) = X 0 and X (t) = V (X (t)). Ghys theorem: L has infinitely many closed trajectories (knots), which correspond to those of the flow on the space of lattices SL(2, R) SL(2, Z) = S 3 \, where = {g2 3 27g3 2 }, generated by left translation by ( e t 0 0 e t ).

15 Ambidextrous trajectories 14 The L-trajectory through (7, 7, 28) weaves between right-handed segments (τ > 0, blue) and left-handed ones (τ < 0, red):

16 Vanishing torsion 15 Definition: The planar set of a vector field V is the zero set of δ = det ( V V V 2 V V ). It is the union of all points where a trajectory has vanishing torsion. For the Lorenz field, δ = x x x 3 y x 5 y x 2 y x 4 y 2 10x 6 y xy x 3 y y x 2 y xy x 2 z x 4 z + 560x 6 z xyz x 3 yz + 20x 5 yz y 2 z x 2 y 2 z 200x 4 y 2 z xy 3 z + 300x 3 y 3 z 1000y 4 z x 2 z x 4 z 2 10x 6 z xyz x 3 yz y 2 z x 2 y 2 z xy 3 z x 2 z 3 200x 4 z xyz x 3 yz y 2 z x 2 z 4. Note that this vanishes on the vertical axis x = y = 0.

17 The Lorenz stand 16 Almost all of the planar set consists of two sheets intimately folded together with self-intersection.

18 Upside down 17 Spikes point to a zero at S 3.

19 The three equilibria 18 The surfaces are coloured by the angle that L makes with grad τ.

20 The Rössler band 19 R = ( y z, x + 1 2y, 2 + xz 4z)... coloured by curvature.

21 Linear and quadratic fields 20 Lemma. The planar set of a vector field defined by a diagonalizable matrix is the union of 3 planes. In particular, V = (ax, by, cz) has δ = abc(b c)(c a)(a b)xyz. For the next slide, consider the family Z θ = (1, 1, 1) + (x 2, y 2, z 2 ) cos θ + (yz, zx, xy) sin θ. Its underlying 3-fold symmetry is expressed as Z θ ρ = ρ Z θ, where ρ is a rotation by 2π/3. We shall plot δ xyz for θ/π = 1 4, 3 8, 1, 4 3, 3 2, 5 6 :

22 A gallery of planar surfaces 21

23 A triply-periodic surface 22 V = (sin y, sin z, sin x). δ = 0

24 Questions How special are planar surfaces? Can one characterize them in terms of curvature? Compare: developable surfaces, Hessian curves of polynomials f R[x, y]. 2. Is there a normal form for a planar set near an equilibrium point? 3. Can one classify those vector fields for which δ 0? E.g. Z 3π/4 = (1 x 2 + yz, 1 y 2 + zx, 1 + z 2 xy), which has an S 1 symmetry and planar trajectories. 4. Can one classify the behaviour of planar sets of quadratic fields? How many connected components do they have?

25 Hilbert s 16th problem, part one 24 On the topology of algebraic curves and surfaces: investigation of the position of branches of curves, and the number and form of sheets of an algebraic surface in space. This subject links analysis, geometry, number theory, random matrix theory, computer science: A. Harnack, 1876 R. Courant, A. Stern, 1925 M. Kac, 1942 J. Nash, 1952 V. Arnold, D. Gudkov, V. Rokhlin 1, O. Viro, 1971 onwards B. Gross, J. Harris, 1981 F. Nazarov, M. Sodin, 2009 D. Gayet, 2011 P. Sarnak, I. Wigman, 2012 A. Lerario, E. Lundberg, Y. Fyodorov, σ on M 4 spin

26 From the special to the general 25 A generic algebraic curve in CP 2 has genus g = (n 1)(n 2). 2 A real algebraic curve in RP 2 of degree n has at most g + 1 components. If n = 2m, these are all null homotopic ( ovals ). The 11 ovals of a Harnack sextic may be nested in 3 ways: (P, N) = (10, 1) (6, 5) (2, 9) All satisfy Gudkov s conjecture: P N = m 2 mod 8.

27 Gaussian ensembles 26 If V (f ) is the zero set of f in RP m and N(f ) the number of connected components of V (f ), then how big is N(f ) for a random polynomial of degree n? [Sarnak] Let m = 2, and let {Y k } be an orthonormal basis of the space H n = {f : S 2 R : f = n(n + 1)f }. of spherical harmonics of degree n. Consider f = k= n with a n i.i.d. Gaussians with mean 0 and variance 1/(2n + 1). n a k Y k, Theorem (SW). E [N(f ) : f H n ] n 2, E [N(f ) : f W n ] n 2. In the latter context, a random plane curve is 4% Harnack.

28 Chladni plates 27

29 Spherical harmonics showing regions where f > 0 and f < 0, by S. Panasyuk

30 Ovals of a random harmonic curve of degree 100, by M. Nestasescu:

31 The ood cubic vector field and (sur)face 30 O = ( 2 x 3 + yz, 2 y 3 + zx, 2 z 3 + xy) δ = 0