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The Spectral Point of View in Geometry

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The Spectral Point of View in Geometry The Spectral Point of View in Geometry Masoud Khalkhali Western University, London, Canada Colloqium, Windsor, Feb. 2013 A brief history of spectrum Figure: Newton first used the word spectrum in print in 1671 in describing his experiments in optics Frauenhoffer lines, 1812 Figure: Sun's absorption spectrum; notice the black lines Hydrogen lines Figure: Hydrogen spectral lines in the visible range Black body radiation Figure: Black body spectrum Balmer's formula; Planck's radiation law I Balmer's formula for hydrogen lines (1885): 1 1 1 = R( − ); m = 2; n = 3; 4; 5; 6: λ m2 n2 Ritz-Rydberg combination principle; spectral terms and spectral lines. I Physicists understood that the spectral energy density function ρ(ν; T ) will be independent of the shape of the cavity and should only depend on its volume. I Planck's formula (1900): 8πhν3 1 ρ(ν; T ) = c3 ehν=kT − 1 Weyl's law: asymptotic distribution of eigenvalues • In 1910 H. A. Lorentz gave a series of lectures in G¨ottingenunder the title \old and new problems of physics". Weyl and Hilbert were in attendance. In particular he mentioned attempts to drive Planck's radiation formula in a mathematically satisfactory way and remarked: `It is here that there arises the mathematical problem to prove that the number of sufficiently high overtones which lie between ν and ν + dν is independent of the shape of the enclosure and is simply proportional to its volume. .......There is no doubt that it holds in general even for multiply connected spaces'. • Hilbert was not very optimistic to see a solution in his lifetime. His bright student Hermann Weyl solved this conjecture of Lorentz and Sommerfeld within a year and announced a proof in 1911! All he needed was Hilbert's theory of integral equations and compact operators developed by Hilbert and his students in 1900-1910. Figure: Hermann Weyl in G¨ottingen Figure: H. A. Lorentz Dirichlet eigenvalues and Weyl law • Let Ω ⊂ R2 be a compact connected domain with a piecewise smooth boundary. ( ∆u = λu uj@Ω = 0 0 < λ1 ≤ λ2 ≤ λ3 ≤ · · · ! 1 2 hui ; uj i = δij o.n. basis for L (Ω) • Weyl Law for planar domains Ω ⊂ R2 Area(Ω) N(λ) ∼ λ λ ! 1 4π where N(λ) is the eigenvalue counting function. • In general, for Ω ⊂ Rn !nVol(Ω) n N(λ) ∼ λ 2 λ ! 1 (2π)n One can hear the scalar curvature I (M; g) = closed Riemannian manifold. Laplacian on functions 4 = d ∗d : C 1(M) ! C 1(M) is an unbounded positive operator with pure point spectrum 0 ≤ λ1 ≤ λ2 ≤ · · · ! 1 I The spectrum contains a lot of geometric and topological information on M. In particular the dimension, volume, total scalar curvature, of M are fully determined by the spectrum of ∆. To see this we need: The heat engine −t4 I Let k(t; x; y) = kernel of e . Restrict to the diagonal: as t ! 0, we have (Minakshisundaram-Plejel; Seeley, MacKean-Singer, Gilkey,...) 1 k(t; x; x) ∼ (a (x; 4) + a (x; 4)t + a (x; ∆)t2 + ··· ) (4πt)m=2 0 1 2 I ai (x; 4): the Seeley-De Witt-Gilkey coefficients. I Functions ai (x; 4): expressed by universal polynomials in curvature tensor R and its covariant derivatives: a0(x; 4) = 1 1 a (x; 4) = S(x) scalar curvature 1 6 1 a (x; 4) = (2jR(x)j2 − 2jRic(x)j2 + 5jS(x)j2) 2 360 a3(x; 4) = ······ Short time asymptotics of the heat trace Z −t4 X −tλi Trace(e ) = e = k(t; x; x)dvolx M 1 −m X j ∼ (4πt) 2 aj t (t ! 0) j=0 So Z aj = aj (x; 4)dvolx ; M are manifestly spectral invariants. Z a0 = dvolx = Vol(M); =) Weyl's law M 1 Z a1 = S(x)dvolx ; total scalar curvature 6 M Abelian-Tauberian Theorem P1 −λnt Assume 1 e is convergent for all t > 0. TFAE: 1 X lim tr e−λnt = a; t!0+ 1 N(λ) a lim = λ!1 λr Γ(r + 1) Spectral Triples: (A; H; D) I A= involutive unital algebra, H = Hilbert space, π : A!L(H); D : H!H D has compact resolvent and all commutators [D; π(a)] are bounded. I An asymptotic expansion holds −tD2 X α Trace (e ) ∼ aαt (t ! 0) The metric dimension and dimension spectrum of (A; H; D) I Metric dimension= n (need not be an integer) if jDj−n 2 L1;1(H) 2 I Let 4 = D . Spectral zeta function −s −s=2 ζD (s) = Tr (jDj ) = Tr (∆ ); Re(s) >> 0: The set of poles of ζD (s) is the dimension spectrum of (A; H; D): For classical spectral triples defined on M, the top pole of ζD (s) is integer and is exactly equal to dim (M); also the dimension spectrum is inside R. In general both of these restrictions can fail. The Dixmier trace Let N 1;1 X L (H) := fT 2 K(H); µn(T ) = O (logN)g: 1 The Dixmier trace of an operator T 2 L1;1(H) measures the logarithmic divergence of its ordinary trace. Let PN µ (T ) σ (T ) = 1 n N logN lim σN (T ) may not exist and must be replaced by a carefully chosen `regularized limit' ! : Tr!(T ) := lim σN (T ) ! For operators T for which LimN!1 σN (T ) exit, the Dixmier trace is independent of the choice of ! and is equal to LimN!1 σN (T ): Weyl's law implies that for any elliptic s.a. differential operator of order 1 on M, jDj−n 2 L1;1(H) and −n Tr!(jDj ) = cnVol(M) Commutative examples of spectral triples Natural first order elliptic PDE's on M define spectral triples on A = C 1(M): 1. D = d + d ∗, H = L2(^T ∗M); A = C 1(M) acting on H by left multiplication. Index (D) is the Euler characteristic of M. Signature of M is the index of a closely related spectral triple. 2. D = Dirac operator on a compact Riemannian Spinc manifold, H = L2(M; S), L2-spinors on M. A = C 1(M) acts on L2(M; S) by multiplication. One checks that for any smooth function f , the commutator [D; f ] extends to a bounded operator on L2(M; S). Geodesic distance from spectral triples Now the geodesic distance d on M can be recovered from the following beautiful distance formula of Connes: d(p; q) = Supfjf (p) − f (q)j; k [D; f ] k≤ 1g; 8p; q 2 M: Compare with Riemannian formula: Z 1 p µ ν d(p; q) = Inff gµν dx dx ; c(0) = p; c(1) = qg; 8p; q 2 M 0 Example: a spectral triple on the Cantor set Let A = C(Λ) be the commutative algebra of continuous functions on a Cantor set Λ ⊂ R. Let Jk be the collection of bounded open intervals in R n Λ with lengths L = f`k gk≥1 `1 ≥ `2 ≥ `3 ≥ · · · ≥ `k ··· > 0: (1) Let E = fxk;±g be the set of the endpoints of Jk . Consider the Hilbert space H := `2(E) (2) There is an action of C(Λ) on H given by f · ξ(x) = f (x)ξ(x); 8f 2 C(Λ); 8ξ 2 H; 8x 2 E: A sign operator F is defined by choosing the closed subspace H^ ⊂ H given by H^ = fξ 2 H : ξ(xk;−) = ξ(xk;+); 8kg: Then F has eigenspaces H^ with eigenvalue +1 and H^? with eigenvalue −1, so that 0 1 F j = : Hk 1 0 The Dirac operator D = jDjF is defined as: ξ(xk;+) −1 ξ(xk;−) DjHk = `k · : ξ(xk;−) ξ(xk;+) The data (A; H; D) form a spectral triple. The zeta function satisfies −s Tr(jDj ) = 2ζL(s); where ζL(s) is the geometric zeta function of L = f`k gk≥1, defined as X s ζL(s) := `k : k −k For the classical middle-third Cantor set, we have `k = 3 with k−1 multiplicities mk = 2 , so that X 2 · 3−s Tr(jDj−s ) = 2ζ (s) = 2k 3−sk = : L 1 − 2 · 3−s k≥1 Thus the dimension spectrum of the spectral triple of a Cantor set has complex points! In fact, the set of poles of (26) is log 2 2πin + : log 3 log 3 n2Z Thus the dimension spectrum lies on a vertical line and it intersects the log 2 real axis in the point D = log 3 which is the Hausdoff dimension of the ternary Cantor set. Noncommutative Torus ∗ I Fix θ 2 R. Aθ = C -algebra generated by unitaries U and V satisfying VU = e2πiθUV : I Dense subalgebra of `smooth functions': 1 Aθ ⊂ Aθ; 1 a 2 Aθ iff X m n a = amnU V 2 where (amn) 2 S(Z ) is rapidly decreasing: 2 2 k sup (1 + m + n ) jamnj < 1 m;n for all k 2 N. I Differential operators on Aθ 1 1 δ1; δ2 : Aθ ! Aθ ; Infinitesimal generators of the action m n is:(m;n) m n 2 αs (U V ) = e U V s 2 R : 1 @ 1 @ Analogues of i @x ; i @y on 2-torus. I Canonical trace t : Aθ ! C on smooth elements: X m n t( am;nU V ) = a0;0: m;n2Z Complex structures on Aθ 2 I Let H0 = L (Aθ)= GNS completion of Aθ w.r.t. t. I Fix τ = τ1 + iτ2; τ2 = =(τ) > 0, and define ∗ @ := δ1 + τδ2;@ := δ1 +τδ ¯ 2: I Hilbert space of (1; 0)-forms: (1;0) P 1 H := completion of finite sums a@b, a; b 2 Aθ , w.r.t.
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