Dirac Operators on Fractal Manifolds, Noncommutative Geometry and Intrinsic Geodesic Metrics

Transcription

1 Dirac Operators on Fractal Manifolds, Noncommutative Geometry and Intrinsic Geodesic Metrics Michel L. Lapidus University of California, Riverside Department of Mathematics lapidus/ April 5, 2014 AMS Special Session on Topics in Spectral Geometry and Global Analysis 2014 AMS Western Spring Sectional Meeting University of New Mexico, Albuquerque Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

2 Contents 1 Spectral Geometry. 2 Fractal Sets Built on Curves. 3 Measurable Riemannian Geometry, K H. 4 Spectral Geometry of K H. 5 Conclusions and Future Directions. Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

3 Objective Construct a Dirac operator and spectral triple for K H in order to recover its geometry. inf γ q Noncommutative Geometry (NCG) p < γ, Z γ > 1 2 ds (Classical). sup{ f (p) f (q) : [D, π(f )] 1} (A. Connes). Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

4 Spectral Geometry The use of methods from noncommutative geometry to construct geometries for sets using the operator-theoretic data contained in a Dirac operator and its associated spectral triple for the set. Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

5 Spectral Triples T = (A, H, D) is called a spectral triple if A is a unital C -algebra. H is a Hilbert space which carries a unital representation π of A. D is an unbounded self-adjoint operator on H such that Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

6 Spectral Triples T = (A, H, D) is called a spectral triple if A is a unital C -algebra. H is a Hilbert space which carries a unital representation π of A. D is an unbounded self-adjoint operator on H such that 1 [D, π(a)] is densely defined and extends to a bounded operator on H, for all a in a dense subalgebra A 0 of A. 2 (I + D 2 ) 1 is compact. Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

7 Theorem (Gelfand-Naimark) A is a unital commutative C -algebra A = C(X ) for some compact Hausdorff space X. Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

8 Theorem (Gelfand-Naimark) A is a unital commutative C -algebra A = C(X ) for some compact Hausdorff space X. Remarks: 1 X is recovered as the set of pure states of A. 2 Commutative topologies/geometries X commutative C -algebras (C(X )). 3 Noncommutative topologies/geometries noncommutative C -algebras. (Modulo Morita equivalence.) Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

9 Noncommutative Geometry M is a (smooth) compact spin Riemannian manifold. A = C(M) (the algebra of continuous functions on M). H = the Hilbert space of L 2 -spinors on M. D is the Dirac operator on M (constructed from the spin connection). (A, H, D) is a spectral triple. Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

10 Noncommutative Geometry M is a (smooth) compact spin Riemannian manifold. A = C(M) (the algebra of continuous functions on M). H = the Hilbert space of L 2 -spinors on M. D is the Dirac operator on M (constructed from the spin connection). (A, H, D) is a spectral triple. Theorem (A. Connes) One can recover the geometry of M from the data contained in the spectral triple ( A, H, D ). Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

11 Let d g be the geodesic distance function on M (geodesic metric). Then, we have the following results: Theorem (A. Connes) d g (p, q) = sup{ a(p) a(q) : [D, a] 1, a A 0 }. Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

12 Let d g be the geodesic distance function on M (geodesic metric). Then, we have the following results: Theorem (A. Connes) d g (p, q) = sup{ a(p) a(q) : [D, a] 1, a A 0 }. Theorem (M. Wodzicki/A. Connes; H. Weyl s Asymptotic Formula) For every f C(M), we have fdv = c(d)tr w (f D d ), M where dv is the Riemannian volume (measure) of M, d is the dimension of M and c(d) = 2 (d d 2 ) π d 2 Γ( d 2 + 1). Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

13 Remarks: 1 Tr w is the Dixmier trace. (It captures the semi-classical spectral asymptotics.) 2 See also Guillemin/Manin/Wodzicki ( noncommutative residue ). Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

14 Remarks: 1 Tr w is the Dixmier trace. (It captures the semi-classical spectral asymptotics.) 2 See also Guillemin/Manin/Wodzicki ( noncommutative residue ). 3 Unpublished work of Connes and Sullivan for certain Julia sets (cf. Connes NCG book). 4 Research program (1994, 1997) of M. L. Lapidus to develop noncommutative fractal geometry. (Construction of volume measures on fractals via the Dixmier trace; spectral dimension. (See also Kigami & Lapidus, CMP, 1993 & 2001.) Search for suitable Connes metrics, Dirac operators and spectral triples on fractals. Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

15 Fractal Sets Built on Curves Christensen, Ivan, Lapidus, Advances in Math. (2008), Constructed spectral triples for: Circle. Interval. Curve on a compact Hausdorff space. Finite sum of curves. Parametrized graphs. Infinite trees. Sierpinski Gasket (SG). Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

16 Results for the Sierpinski Gasket On SG: The distance function induced by the spectral triple (A, H, D) coincides with the geodesic metric on the (Euclidean) gasket. The spectral dimension of the triple coincides with the Hausdorff and Minkowski dimension ( log 3 log 2). Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

17 Results for the Sierpinski Gasket On SG: The distance function induced by the spectral triple (A, H, D) coincides with the geodesic metric on the (Euclidean) gasket. The spectral dimension of the triple coincides with the Hausdorff and Minkowski dimension ( log 3 log 2). Remarks: The spectral dimension of the triple is given by D St = inf{p > 0 : Tr(I + D 2 ) p 2 < }. Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

18 Michel L. Lapidus (UC 4.pdf Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

19 The normalized Hausdorff measure of the gasket is recovered via the Dixmier trace. More precisely, using an analog for fractals of Connes theorem on a compact spin manifold. The geodesic metric recovered in that work is induced (from R 2 on SG) by the Euclidean metric. Later on, in the work of Lapidus and Sarhad, we will obtain a more natural geodesic metric (on the harmonic gasket, HG) from the point of view of analysis on fractals. Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

20 Measurable Riemanian Geometry The energy on the Sierpinski gasket (SG), K, is defined by ( ) 5 m ( ) E(u, v) = lim (u(p) u(q)) v(p) v(q), m 3 p q p, q V m where K is approximated by an increasing sequence of finite graphs {V m } m=0. (The notation p q means that p, q V m are neighbors in the graph V m.) Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

21 Measurable Riemanian Geometry The energy on the Sierpinski gasket (SG), K, is defined by ( ) 5 m ( ) E(u, v) = lim (u(p) u(q)) v(p) v(q), m 3 p q p, q V m where K is approximated by an increasing sequence of finite graphs {V m } m=0. (The notation p q means that p, q V m are neighbors in the graph V m.) Theorem (Kusuoka, 1989) There exists, Z, ν, such that E(u, v) = K ( u, Z v)dν. Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

22 Remark: Note that, Z, ν are measurable analogs to gradient, Riemannian metric, Riemannian volume. (, Z, ν) is a measurable Riemanian structure. (At ν-almost every point of SG, there is a one-dimensional tangent space; cf. the picture of the Harmonic Gasket (HG), denoted by K H.) Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

23 Remark: Note that, Z, ν are measurable analogs to gradient, Riemannian metric, Riemannian volume. (, Z, ν) is a measurable Riemanian structure. (At ν-almost every point of SG, there is a one-dimensional tangent space; cf. the picture of the Harmonic Gasket (HG), denoted by K H.) J. Kigami (2008) extended the above structure to a measurable Riemannian geometry. Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

24 Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

25 Let (( Φ(x) = 1 h1 (x) ) 1 )) h 2 (x) ( h 3 (x) 1 x K, where Φ : K Φ(K) is a homeomorphism. Φ is a harmonic coordinate chart for K. Then, Φ(K) := K H. Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

26 Let (( Φ(x) = 1 h1 (x) ) 1 )) h 2 (x) ( h 3 (x) 1 x K, where Φ : K Φ(K) is a homeomorphism. Φ is a harmonic coordinate chart for K. Then, Φ(K) := K H. Here, K = SG, and for each j = 1, 2, 3, h j is the harmonic function with boundary values given by h j (v k ) = δ jk (Kronecker delta), for k = 1, 2, 3, with v 1, v 2, v 3 being the vertices of the equilateral triangle V 0. Note that 3 i=1 h i = 1, so that Φ(K) = K H R 2. Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

27 Φ Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

28 Theorem (Kigami, 2008) For all p, q K H, there exists a C 1 path in K H, denoted by Γ and connecting p and q, which is the shortest of all paths in K H connecting p and q. Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

29 Theorem (Kigami, 2008) For all p, q K H, there exists a C 1 path in K H, denoted by Γ and connecting p and q, which is the shortest of all paths in K H connecting p and q. This induces (on K H ) Kigami s geodesic distance function d (p, q), which can be given (for p, q K H ) by d (p, q) = q p ( γ, Z γ) 1 2 dν. Remark: The above geodesic path Γ is not unique, in general. Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

30 Joint Work With Jonathan Sarhad Main Goals: Construct a spectral triple on the harmonic gasket K H which induces the natural geodesic metric (Kigami s distance) on K H : Evaluate its spectral dimension in terms of geometric data. Construct a variety of such spectral triples having the same property. Generalize this construction to include all the fractals built on curves of [CIL, Adv. Math., 2008], as well as a broader class of such fractals (including the SG,K, and the HG, K H ). Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

31 Spectral Geometry of K H Theorem (Lapidus Sarhad, 2012) Two methods for constructing a spectral triple for K H : Countable sum of curve triples. Countable sum of circle triples. Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

32 Curve Triples Construction Each cell boundary τ w of K H can be identified as follows: Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

33 Proposition 1: The triple ST τw, s = (C(K H ), H w, s, Dw,s) t associated to the curve τ w, s is a spectral triple such that: 1 The spectrum of the Dirac operator D t w, s is given by {( ) (2k + 1) 2α w, s } : k Z. (α w, s = arclength of the corresponding geodesic segment.) Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

34 Proposition 1: The triple ST τw, s = (C(K H ), H w, s, Dw,s) t associated to the curve τ w, s is a spectral triple such that: 1 The spectrum of the Dirac operator D t w, s is given by {( ) (2k + 1) 2α w, s } : k Z. (α w, s = arclength of the corresponding geodesic segment.) 2 The metric induced by ST τw, s on τ w, s coincides with the restriction of Kigami s geodesic distance (geodesic metric) on K H. 3 The spectral dimension is 1. Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

35 Let H w = D w = π w = s=l, R, B s=l, R, B s=l, R, B H w, s, D t w, s, π w, s, and let ST τw = (C(K H ), H w, D w ) be the triple associated to the w-cell τ w. Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

36 Proposition 2: ST τw = (C(K H ), H w, D w ) is a spectral triple such that: 1 The spectrum of the Dirac operator D w is given by {( ) } (2k + 1)π : k Z, s {L, R, B}. 2α w, s (α w,s = arclength of the corresponding geodesic segment.) Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

37 Proposition 2: ST τw = (C(K H ), H w, D w ) is a spectral triple such that: 1 The spectrum of the Dirac operator D w is given by {( ) } (2k + 1)π : k Z, s {L, R, B}. 2α w, s (α w,s = arclength of the corresponding geodesic segment.) 2 The metric d w induced by ST τw, s on τ w, s coincides with the R 2 -induced arclength metric on τ w. 3 The spectral dimension is 1. Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

38 Spectral Triple on K H Via a Countable Sum of the ST τw Triples Let H Kw = n N w =n H w, and π KH = D KH = be the curve triple associated to K H. n N w =n n N w =n π w, D w, ST KH = (C(K H ), H KH, D KH ) Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

39 Given a Lipschitz function f on K H, let Lip g (f ) be the Lipschitz seminorm with respect to Kigami s geodesic distance on K H. Lemma (Lapidus Sarhad, 2012) For all f in the domain of D KH, we have D KH f, KH = Lip g (f ). Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

40 Theorem (Lapidus Sarhad, 2012) ST KH = (C(K H ), H K, D KH ) is a spectral triple such that: The spectrum of the Dirac operator D KH n N w =n is given by {( ) } (2k + 1)π : k Z, s {L, R, B}. (2α w, s ) (α w, s = arclength of the corresponding geodesic segment.) Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

41 Theorem (Lapidus Sarhad, 2012) ST KH = (C(K H ), H K, D KH ) is a spectral triple such that: The spectrum of the Dirac operator D KH n N w =n is given by {( ) } (2k + 1)π : k Z, s {L, R, B}. (2α w, s ) (α w, s = arclength of the corresponding geodesic segment.) q d spec (p, q) = inf γ p ( γ, Z γ) 1 2 dν, where γ range over all continuous paths connecting p, q K H. Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

42 Theorem (Lapidus Sarhad, 2012 continued) The spectral dimension of K H with respect to ST KH is the infimum of all p > 0 such that w, s, k ( ) (2k + 1)π p <. 2α w, s Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

43 Circle Triples The spectral triple for a complete cell boundary comes directly from the circle triple. Let GT τw = (C(K H ), h w, D w ) be the circle triple describing τ w. Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

44 Proposition 3: For each cell w, the circle triple GT τw is defined much as before, as a countable direct sum of spectral triples. GT τw is a spectral triple for τ w such that 1 The spectrum of the Dirac operator D w is given by {( ) } (2k + 1π) : k Z. 2α w Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

45 Proposition 3: For each cell w, the circle triple GT τw is defined much as before, as a countable direct sum of spectral triples. GT τw is a spectral triple for τ w such that 1 The spectrum of the Dirac operator D w is given by {( ) } (2k + 1π) : k Z. 2α w 2 The metric d w induced on τ w coincides with the R 2 -induced arclength metric on τ w. 3 The spectral dimension of τ w is 1. Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

46 Spectral Triple on K H Via a Countable Sum of Circle Triples Let h KH π KH = n N w =n h w, = n N w =n π w, D KH and let = n N w =n D w, GI KH = (C(K H ), h KH, D KH ) be the circle triple associated to K H. Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

47 Lemma For all f in the domain of D KH, we have D KH f, KH = Lip g (f ). Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

48 Theorem (Lapidus Sarhad 2012) Same as Theorem 1, except that the spectrum of the Dirac operators D KH is given by n N w =n and the spectral dimension is inf { p > 0 : w, k {( ) (2k + 1)π 2α w ( (2k + 1)π 2α w } : k Z ) p < }. Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

49 Theorem (Lapidus Sarhad 2012) Same as Theorem 1, except that the spectrum of the Dirac operators D KH is given by n N w =n and the spectral dimension is inf { p > 0 : w, k {( ) (2k + 1)π 2α w ( (2k + 1)π 2α w } : k Z ) p < }. Remark: As in Theorem 1, this last expression (for the spectral dimension) can be easily simplified. Furthermore, α w denotes the length of the corresponding geodesic segment. Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

50 Let ST = ST KH GIKH. Theorem ST is a spectral triple for K H and the metric induced by ST on K H coincides with Kigami s geodesic distance. Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

51 Let ST = ST KH GIKH. Theorem ST is a spectral triple for K H and the metric induced by ST on K H coincides with Kigami s geodesic distance. Corollary ST, ST KH and GI KH all recover Kigami s geodesic metric. Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

52 Fractal Sets Built on Curves in R n Let R = j N R j, where R j is a curve in R n. Suppose the following conditions on R hold: 1 R is a pre-compact set in R n. 2 R j is a rectifiable C 1 curve. Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

53 Fractal Sets Built on Curves in R n Let R = j N R j, where R j is a curve in R n. Suppose the following conditions on R hold: 1 R is a pre-compact set in R n. 2 R j is a rectifiable C 1 curve. 3 The lengths of the R j s go to zero at a geometric rate. 4 For any two points in R (the closure of R in R n ), there exists a rectifiable piecewise-c 1 path γ, connecting the points, whose length, L(γ), is the minimum of the lengths of all paths connecting them. Moreover, γ can be given as a (possibly infinite) concatenation of the R j s. Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

54 For p, q R and γ as in (4), define the geodesic distance d geo by d geo (p, q) = L(γ). Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

55 For p, q R and γ as in (4), define the geodesic distance d geo by d geo (p, q) = L(γ). Theorem (Lapidus Sarhad, 2010) If R satisfies the axioms (1) (4), then the countable sum of R j -curve triples is a spectral triple for R and if d R is the distance induced by the spectral triple, then d R = d geo. Moreover, the spectral dimension can be computed in terms of the lengths of the curves. Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

56 Remarks: In [LS, 2012], the above axioms and the corresponding theorem have been refined and extended (to include curves in length spaces rather than in Euclidean spaces). Examples of fractals satisfying these axioms include the Euclidean Sierpinski gasket (SG), K, and the harmonic gasket (HG), K H. Other examples include all the fractals built on curves considered in [CIL, Adv. Math., 2008] (SG, infinite trees, certain Cayley graphs,...). Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

57 Conclusions and Future Research Directions The fact that there are multiple Dirac operators inducing the same geometry gives rise to questions of the Dirac operator vs a Dirac operator, as well as what suitable equivalences of Dirac operators (and spectral triples) there may be. The self-affinity (rather than self-similarity) of K H has posed quite a challenge in computing the spectral dimensions of the Dirac operators. It is still unknown exactly how to do this. Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

58 Further Research Directions Construct a Dirac operator on K H globally from the measurable metric Z and gradient. Determine the asymptotics of the spectra of the Dirac operators in order to compute the spectral dimensions, as well as construct (via the Dixmier trace) a volume form from the spectral triples. In particular, we conjecture that we would then recover the natural Hausdorff measure with respect to the geodesic metric on the harmonic gasket K H (thereby generalizing the results of [CIL, Adv. Math., 2008] for the Euclidean gasket K). Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

59 Conclusions and Future Research Directions Determine the relationship between the Dirac operators and the Kusuoka Laplacian associated to K H. There is a notion of effective resistance metric on K. It will be interesting to show that the square root of this metric coincides with, or is at least equivalent to, Kigami s geodesic distance on K H. Develop aspects of geometric analysis (or global analysis) on fractals, using some of the above constructions. Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

60 References Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

61 [1] Giovanni Alberti, Geometric measure theory, Encyc. Math. Phy., 2, , (2005). [2] Erik Christensen, Christina Ivan and Michel. L. Lapidus, Dirac operators and spectral triples for some fractal sets built on curves, Adv. Math., 217, No. 1, 42 78, (2008). [3] Alain Connes, Compact metric spaces, Fredholm modules, and hyperfiniteness, Ergodic Theory and Dynamical Systems, 9, (1989). [4] Alain Connes, Noncommutative Geometry, Academic Press, San Diego, (1994). [5] Herbert Federer, Geometric Measure Theory, Springer, New York, (1969). [6] Arpita Gosh, Stephen Boyd and Amin Saberi, Minimizing effective resistance of a graph, Siam. Rev., 50, No. 1, 37 66, (2008). Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

62 [7] Jun Kigami, Harmonic metric and Dirichlet form on the Sierpinski gasket, in: Asymptotic Problems in Probability Theory: Stochastic Models and Diffusion on Fractals (Sanda/Kyoto 1990) (Elworthy and N. Ikeda, eds.), Pitman Res. Math, 283, pp , (1993). [8] Jun Kigami, Analysis on Fractals, Cambridge Univ. Press, Cambridge, [9] Jun Kigami, Measurable Riemannian geometry on the Sierpinski gasket: The Kusuoka measure and the Gaussian heat kernel estimate, Math. Ann., 340, No. 4, , (2008). [10] Jun Kigami and Michel L. Lapidus, Weyl s problem for the spectral distribution of Laplacians on p.c.f self-similar sets, Commun. Math. Phys., 158, , (1993). [11] Jun Kigami and Michel L. Lapidus, Self-similarity of volume measures for Laplacians on p.c.f self-similar fractals, Commun. Math. Phys. 217, , (2001). Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

63 [12] S. Kusuoka, Lecture on diffusion processes on nested fractals, Springer Lecture Notes Math., 1567, Springer, Berlin, pp , (1993). [13] S. Kusuoka, Dirichlet forms on fractals and products of random matrices, Publ. Res. Inst. Math. Sci., 25, , (1989). [14] Michel L. Lapidus, Vibrations of fractal drums, the Riemann hypothesis, waves in fractal media, and the Weyl Berry conjecture, in: Ord. and Part. Diff. Eqts. (B. D. Sleeman and R. J. Davis, eds.), IV, Proc. Twelfth Internat. Conf. (Dundee, Scotland, UK, June 1992), Pitman Research Notes in Math. Series, 289, pp , (1993). [15] Michel L. Lapidus, Analysis on fractals, Laplacians on self similar sets, noncommutative geometry and spectral dimensions, Topological Methods in Nonlinear Analysis, 4, , (1994). (Special issue dedicated to Jean Leray.) [16] Michel L. Lapidus, Towards a noncommutative fractal geometry? Laplacians and volume measures on fractals, in: Harmonic Analysis Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

64 and Nonlinear Differential Equations, Contemp. Math., 208, Amer. Math. Soc., Providence, R.I., pp , (1997). [17] Michel L. Lapidus, In Search of the Riemann Zeros: Strings, Fractal Membranes and Noncommutative Spacetimes, Amer. Math. Soc., Providence, R.I., [18] Michel L. Lapidus and Jonathan J. Sarhad, Dirac operators and geodesic metric on the harmonic Sierpinski gasket and other fractal sets, Journal of Noncommutative Geometry, 2014, in press. (E-print: arxiv: v1 [math.mg], 2012; IHES preprint, IHES/M/12/32, 2012.) [19] Peter Peterson, Riemannian Geometry, Springer, New. York, (1998). [20] Michael Reed and Barry Simon, Methods of Modern Mathematical Physics, vol. I, Functional Analysis, rev. and enl. edn., Academic Press, (1980). [21] Marc A. Rieffel, Metrics on states from actions of compact groups, Doc. Math., 3, , (1998). Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

65 [22] Marc A. Rieffel, Metrics on state spaces, Doc. Math., 4, , (1999). [23] Yuri I. Manin, Topics in Noncommutative Geometry, Princeton University Press, Princeton, (1991). [24] Robert S. Strichartz, Differential Equations on Fractals: a Tutorial, Princeton University Press, Princeton, (2006). [25] Alexander Teplyaev, Energy and Laplacian on the Sierpinski Gasket, in: Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot (M.L. Lapidus and M. van Frankenhuijsen, eds.), Proc. Sympos. Pure Math., 72, Part 1, Amer. Math. Soc., Providence, R.I., pp , (2004). [26] Alexander Teplyaev, Harmonic coordinates on fractals with finitely ramified cell structure, Canadian J. Math., 60, , (2008). [27] Joseph C. Varilly, Hector Figueroa and Jose M. Garcia Bondia, Elements of Noncommutative Geometry, Birkhäuser, Boston, (2001). Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

66 [28] M. Wodzicki, Local invariants of spectral asymmetry, Invent. Math. 75, (1984), [29] M. Wodzicki, Noncommutative residue, Part I: Fundamentals, in: K-Theory, Arithmetic and Geometry (Moscow, ) Lecture Notes in Math., vol Springer-Verlag, 1987, pp Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43

67 Remark: For additional references regarding noncommutative fractal geometry, see, for example, references [2], [15], [16], [17] and [18], as well as the relevant references therein. Michel L. Lapidus (UC Riverside) Dirac Operators on Fractal Manifolds... April 5, / 43