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Edinburgh Lectures on Geometry, Analysis and Physics

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Edinburgh Lectures on Geometry, Analysis and Physics Edinburgh Lectures on Geometry, Analysis and Physics Sir Michael Atiyah Notes by Thomas K¨oppe Contents Preface iii 1 From Euclidean 3-space to complex matrices1 1.1 Introduction.............................1 1.2 Euclidean geometry and projective space.............2 1.3 From points to polynomials....................2 1.4 Some physics: hyperbolic geometry................4 1.5 The hyperbolic conjecture......................5 1.6 The Minkowski space conjecture.................6 1.7 The normalised determinant....................8 1.8 Relation to analysis and physics.................. 10 1.9 Mysterious links with physics................... 12 List of conjectures............................ 13 Bibliography............................... 13 2 Vector bundles over algebraic curves and counting rational points 14 2.1 Introduction............................. 14 2.2 Review of classical theory..................... 14 2.3 Analogy with number theory................... 17 2.4 Relation between homology and counting rational points.... 17 2.5 The approach via Morse theory.................. 20 2.6 Counting rational points...................... 26 2.7 Comparison of equivariant Morse theory and counting rational points................................ 31 2.8 Relation to physics......................... 32 2.9 Finite-dimensional approximations................ 33 2.10 Relation of 휁-functions for finite fields and Riemann's 휁-function 34 2.11 Arithmetic algebraic geometry (Arakelov theory)........ 35 2.12 Other question........................... 36 Bibliography............................... 36 ii Preface These lecture notes are based on a set of six lectures that I gave in Edinburgh in 2008/2009 and they cover some topics in the interface between Geometry and Physics. They involve some unsolved problems and conjectures and I hope they may stimulate readers to investigate them. I am very grateful to Thomas K¨oppe for writing up and polishing the lectures, turning them into intelligible text, while keeping their informal nature. This involved a substantial effort at times in competition with the demands of a Ph.D thesis. Unusually for such lecture notes I found little to alter in them. Michael Atiyah Edinburgh, September 2010 iii lecture series 1 From Euclidean 3-space to complex matrices December 8 and 15, 2008 1.1 Introduction 3 We will formulate an elementary conjecture for n distinct points in R , which is unsolved for n ≥ 5, and for which we have computer evidence for n ≤ 30. The conjecture would have been understood 200 years ago (by Gauss). What is the future for this conjecture? ∙ A counter-example may be found for large n. ∙ Someone (perhaps from the audience?) gives a proof. ∙ It remains a conjecture for 300 years (like Fermat). To formulate the conjecture, we recall some basic concepts from Euclidean and hyperbolic geometry and from Special Relativity. 1 2 1.2 Euclidean geometry and projective space 2 {︀ 2 2 2 2 }︀ The two-dimensional sphere S = (x; y; z) 2 R : x + y + z = 1 is \the 1 same as" the complex projective line CP = C t f1g, on which we have homogeneous coordinates [u1 : u2]. Stereographic projection through a \north 2 2 pole" N 2 S identifies S n fNg with C, and it extends to an identification of 2 1 S with CP by sending N to 1. Exercise 1.2.1. Suppose we have two stereographic projections from two 0 1 1 \north poles" N and N . Show that these give a map CP ! CP which is a complex linear transformation au + b u0 = , where a; b; c; d 2 and ad − bc 6= 0. cu + d C 1 Hint: Start by considering stereographic projection from S to R first. 1.3 From points to polynomials 3 We will now associate to each set of n distinct points in R a set of n complex polynomials (defined up to scaling). 3 The case n = 2. Given two points x1; x2 2 R with x1 6= x2, define x2 − x1 2 f(x1; x2) := 2 S , kx2 − x1k which gives a unit vector in the direction from x1 to x2. Under the identification 2 ∼ 1 1 S = CP , f associates to each pair (x1; x2) a point in CP . Exchanging x1 2 and x2 is just the antipodal map x 7! −x on S . 3 The general case. Given n (ordered) points x1; : : : ; xn 2 R , we obtain 1 n(n − 1) points in CP by defining xj − xi 2 ∼ 1 uij := 2 S = CP for all i 6= j. (1.1) kxj − xik For each i = 1; : : : ; n we define a polynomial 훽i 2 C[z] with roots uij (j 6= i): Y 훽i(z) = (z − uij) (1.2) j6=i The polynomials 훽i are determined by their roots up to scaling. We make th the convention that if for some j we have uij = 1, then we omit the j factor, so that 훽i drops one degree. In fact, a more invariant picture arises if 2 Edinburgh Lectures on Geometry, Analysis and Physics 3 instead we consider the associated homogeneous polynomials Bi 2 C[Z0;Z1] Q (︀ )︀ given by Bi(Z0;Z1) = j VijZ0 − UijZ1 , where [Uij : Vij] = [uij : 1], so 훽i(z) = Bi(z; 1). We are now ready to state the simplest version of the conjecture: 3 Conjecture 1.3.1 (Euclidean conjecture). For all sets (x1; : : : ; xn) ⊂ R of n distinct points, the n polynomials 훽1(z); : : : ; 훽n(z) are linearly independent over C. Remark 1.3.2. The condition of linear independence of the polynomials 훽i is independent of the choice of stereographic projection in Equation (1.1) by Exercise 1.2.1. 3 Example (n = 3). Suppose x1; x2; x3 are distinct points in R . They are 2 3 automatically co-planar, so that x1; x2; x3 2 R ⊂ R . So the points uij lie in 1 2 ∼ 1 some great circle S ⊂ S = CP . We can choose the north pole N for the stereographic projection in Equation (1.1) either such that all uij lie in the equator, in which case juijj = 1 and 1 uji = −uij, or such that all uij lie on a meridian, in which case uij 2 RP and uji = −1 uij. Let us stick with the first convention, so that all uij lie on the equator and we have juijj = 1 and uji = −uij. This defines three quadratics 훽1(z) = (z − u12)(z − u13) = (z − u12)(z − u13) 훽2(z) = (z − u21)(z − u23) = (z + u12)(z − u23) 훽3(z) = (z − u31)(z − u32) = (z + u13)(z + u23) In this case we can prove Conjecture 1.3.1 in two ways: ∙ By geometric methods: Represent quadratics by lines in a plane, then linear dependence of the 훽i is the same as concurrence. ∙ By algebraic methods: Compute the determinant of the (3 × 3)-matrix of coefficients of the 훽i and show that it has non-vanishing determinant. For the case n = 4, there exists a proof using computer algebra. For n ≥ 5, no proof is known, even for co-planar points (i.e. real polynomials). A proof will be rewarded with a bottle of champagne or equivalent. The easiest point of departure is to consider four points in a plane. 3 4 1.4 Some physics: hyperbolic geometry 2 3 Consider again the 2-sphere S ⊂ R , and add a fourth variable t (for \time"): x2 + y2 + z2 − R2t2 = 0 (1.3) This is the metric of Minkowski space-time. Here R is the speed of light, and Equation (1.3) defines a light cone. Our original 2-sphere is the base of the light cone, the \celestial sphere" of an observer. + 2 ∼ 1 The (proper, orthochronous) Lorentz group SO (3; 1) acts on S = CP as a group of complex projective transformations SL(2; C) ±1 = P SL(2; C). The Euclidean version of this picture is the following: The rotation group 3 ∼ 2 ∼ 1 of R , SO(3), acts as SU(2) ±1 =: P SU(2) = PU(2) on S = CP preserving the metric given by Equation (1.3) (\rigid motion"). We can also see this as the 2 projectivisation of the action of SU(2) or U(2) on C , and the projectivisation map ∼ SU(2) P SU(2) = SO(3) is a double cover. This map is the restriction to the maximal compact subgroup ∼ + of the double cover SL(2; C) P SL(2; C) = SO (3; 1). ∼ + We have two different representations of SL(2; C) = SO](3; 1) (double 3;1 cover): It acts on real 4-dimensional space-time R by proper, orthochronous 2 Lorentz transformations, and it acts on complex 2-dimensional space C (whose elements we call spinors). The fundamental link between these two representa- 2 × ∼ 1 tions is via projective spinors: A (projectivised) point in (C n f0g) C = CP corresponds to a point on the base of the light cone, S2. Consider the hyperboloid given by x2 + y2 + z2 − R2t2 = −m2. Denote the interior of the base of the light cone by Hm. The metric induced on Hm has constant negative curvature, and indeed it turns Hm into a model of hyperbolic 3-space with curvature −1=m2. The Lorentz group SO+(3; 1) acts transitively on hyperbolic 3-space H3 by isometries, and it acts by SL(2; C) ±1 on the 2-sphere at infinity. 4 Edinburgh Lectures on Geometry, Analysis and Physics 5 1.5 The hyperbolic conjecture. 3 Given n distinct, ordered points in H , define the point uij as the intersection 2 of the oriented geodesic joining xi to xj with the S at infinity. We define n polynomials 훽1; : : : ; 훽n, where 훽i has roots uij, as before in Equation (1.2) (but note that in hyperbolic space we no longer have a notion of \antipodal points"). This brings us to the second, stronger version of the conjecture: 3 Conjecture 1.5.1 (Hyperbolic conjecture).
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