Sheaf. A sheaf is a functor from from the category of simplicial complexes. Example: σ F(σ), a vector space ρ σ L : F(ρ) F(σ), a linear map

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1 Sheaf A sheaf is a functor from from the category of simplicial complexes Example: σ F(σ), a vector space ρ σ L : F(ρ) F(σ), a linear map F(σ) is called a stalk. F(ρ σ) is called a restriction map Let X be a simplicial complex. For every σ, choose s σ F(σ). This assignment (s σ ) σ X of an element of F(σ) to every simplex is called a global section if these choices are compatible with the restriction maps. F X (X) =set of all global sections. /7

2 Example: The constant sheaf Let X be a simplicial complex The constant sheaf G X : σ G ρ σ id : G G Suppose (s σ ) σ X is a global section. Then if s τ = g G, then s σ = g for all σ X. The the (group/vector space/...) of global sections in isomorphic to G. /7

3 Example: The skyscraper sheaf The skyscraper sheaf G τ { G if σ = τ σ 0 otherwise. σ σ identity map ρ σ zero map if ρ σ. Suppose (s σ ) σ X is a global section. Then s σ = 0 for all σ τ, since s σ F(σ) =0 If dimτ >0, then ρ τ and F(ρ τ) =0. Thus s τ = 0. Thus the (group/vector space/...) of global sections {0}. If dimτ >0, let s τ = g. Thus in this case, the (group/vector space/...) of global sections G. 3/7

4 Direct sum If F and G are sheaves, then F G is a sheaf. Example: σ F(σ) G(σ) ρ σ F(ρ σ) G(ρ σ) τ X G τ σ τ X G τ = G ρ σ { identity map ρ = σ G τ (ρ σ) = τ X zero map ρ σ Thus the (group/vector space/...) of global sections G since we can assign any element of G to a vertex v. v X 4/7

5 sheaf

6 simplices v v v v v v v v v v v v v v v 7

7 atachmentdiagram v v v v v v v v v v v v v v v 8

8 sheaf stalk of on 9

9 sheaf restriction 0

10 sheaf =

11 globalsection

12 3 X X

13 Sheaf Cohomology and its Interpretation SIMPLEX Program 05 This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

14 Global sections, revisited The space of global sections alone is insufficient to detect redundancy or possible faults, but another invariant works It's based on the idea that we can rewrite the basic condition(s) for a global section s of a sheaf v e v (v e) s(v) = (v e) s(v) (e) (v e) (v) (v e) + (v e) s(v) - (v e) s(v) = 0 (v) - (v e) s(v) + (v e) s(v) = 0 ( (a b) is the restriction map connecting cell a to a cell b in a sheaf ) 4

15 Global sections, revisited The space of global sections alone is insufficient to detect redundancy or possible faults, but another invariant works It's based on the idea that we can rewrite the basic condition(s) for a global section s of a sheaf (+ (v e) - (v e)) s(v) =0 s(v) (v e) s(v) = (v e) s(v) + (v e) s(v) - (v e) s(v) = 0 - (v e) s(v) + (v e) s(v) = 0 ( (a b) is the restriction map connecting cell a to a cell b in a sheaf ) 5

16 Recall: A queue as a sheaf Contents of the shift register at each timestep N = 3 shown ( ) ( ) ℝ ℝ3 ( ) ( ) ℝ ℝ ℝ ( ) ℝ3 ( ) ℝ ℝ3 ( ) 00 00

17 Recall: A single timestep Contents of the shift register at each timestep N = 3 shown ( ) ( ) (9,) (,9,) (,9) (,,9) ( ) ( ) ( ) (,) (5,,) ( ) (5,) (,5 ( ) 00 00

18 Rewriting using matrices Same section, but the condition for verifying that it's a section is now written linear algebraically ( ) ( ) (,9,) (,9) (,,9) ( ) (,) (5,,) ( ) =

19 The cochain complex Motivation: Sections being in the kernel of matrix suggests a higher dimensional construction exists! Goal: build the cochain complex for a sheaf k- d ;) k k C (X; ) k+ d d k+ C (X; ) k+ C (X; From this, sheaf cohomology will be defined as k k k- H (X; ) = ker d / image d much the same as homology (but the chain complex goes up in dimension instead of down) 9

20 Generalizing up in dimension Global sections lie in the kernel of a particular matrix We gather the domain and range from stalks over vertices and edges... These are the cochain spaces C (X; ) = (a) k a is a k-simplex An element of Ck(X; ) is called a cochain, and specifies a datum from the stalk at each k-simplex (The direct sum operator forms a new vector space by concatenating the bases of its operands)

21 The cochain complex k The coboundary map dk : Ck(X; ) Ck+(X; ) is given by the block matrix d = [bi:aj] (aj bi) Row i 0, +, or - depending on the relative orientation of aj and bi Column j

22 The cochain complex We've obtained the cochain complex k- d ;) k C (X; ) k d k+ d k+ C (X; ) Ck+(X; Cohomology is defined as Hk(X; ) = ker dk / image dk- All the cochains that are consistent in dimension k... that weren't already present in dimension k - 3

23 Cohomology facts H0(X; ) is the space of global sections of H (X; ) usually has to do with oriented, non-collapsible data loops Nontrivial H (X; ℤ) k H (X; ) is a functor: sheaf morphisms induce linear maps between cohomology spaces 4

24 Cohomology versus homology Homologies of different chain complexes: Chain complex: simplices and their boundaries Ck+(X) Ck(X) k Ck-(X) k- Transposing the boundary maps yields the cochain complex: functions on simplices T k+ k+ T k+ kt k-t Ck+(X) Ck(X) Ck-(X) With ℝ linear algebra, homology* of both of these carry identical information for a wide class of spaces * we call the homology of a cochain complex cohomology 5

25 Cohomology versus homology Homologies of different chain complexes: Transposing the boundary maps yields the cochain complex: functions on simplices T k+ Ck+(X) T k+ Ck(X) kt Ck-(X) k-t The coboundary maps work like discrete derivatives and compute differences between functions on higher dimensional simplices 6

26 Sheaf cohomology versus homology Homologies of different chain complexes: Transposing the boundary maps yields the cochain complex: functions on simplices T k+ X; ) Ck+(X) T k+ Ck(X) kt Ck-(X) k-t Sheaf cochain complex: also functions on simplices, but they are generalized! d k+ k k+ C (X; ) d 7 k- k C (X; ) d k- C (X;

27 Weather Loop a simple model Sensors/ Questions Rain? (R) News (N) X Weather Website (W) X Humidity % (H) Clouds? (L) X X Rooftop Camera (C) Twitter (T) Sun? (S) X X X X Make sim plicial co mplex Question: Can misleading globalized information be detected? 8

28 h"ps://youtu.be/dafvrtdfcs4?list=plsekr_gm4hwlvftjx0wuuevo65uhvbpra

29 fab AA The basis of (A) is the set of elements of A frf A A Rf B B f BB Rf 9

30 h"ps://youtu.be/dafvrtdfcs4?list=plsekr_gm4hwlvftjx0wuuevo65uhvbpra

31 Switching sheaves It's possible to construct a sheaf that represents the truth table of a logic circuit Each vertex is a logic gate, each edge is a wire A B pr Sheafify and pr AB C and C Quiescent* logic sheaf Logic circuit *Quiescent = steady state, prn = projection onto nth component 9

32 Switching sheaves Vectorify everything about a quiescent logic sheaf, and you obtain a switching sheaf pr [ ] = pr Vectorify! and A vector space whose basis is the set of ordered pairs = = Tensor product Quiescent* logic sheaf *Quiescent = steady state, prn = projection onto nth component 0

33 Switching sheaves Vectorify everything about a quiescent logic sheaf, and you obtain a switching sheaf pr ( ) pr Vectorify! and Quiescent* logic sheaf ( ) ( ) Switching sheaf *Quiescent = steady state, prn = projection onto nth component

34 Global sections of switching sheaves In the case of a 3 input gate, the global sections are spanned by all simultaneous combinations of inputs (a,a) (b,b) a b c a b C a B c a B C A b c A b C A B c A B C (c,c) 8 = 56 sections in total

35 Global sections of switching sheaves Global sections consist of simultaneous inputs to each gate, but consistency is checked via tensor contractions There is an inherent model of uncertainty ow taf da When we instead consider a logically equivalent circuit, the situation changes 4 4 3

36 Global sections of switching sheaves The space of global sections is (a,a) now 6 dimensional some sections were lost! (b,b) a b + c d a b +C d a B+ c d A b+ c d A B+c D A B+C D 4 AND (d,d) (c,c) 4 Recall that the space of global 4 4 sections is a subspace of 4

37 Global sections of switching sheaves The space of global sections is (a,a) now 6 dimensional some sections were lost! a b + c d a b +C d a B+ c d A b+ c d A B+c D A B+C D (b,b) 4 AND (d,d) All local sections on the upstream gate are represented 5 (c,c) 4

38 Global sections of switching sheaves The space of global sections is (a,a) now 6 dimensional some sections were lost! a b + c d a b +C d a B+ c d A b+ c d A B+c D A B+C D (b,b) 4 AND (d,d) All local sections supported on the downstream gate are there too 6 (c,c) 4

39 Global sections of switching sheaves (a,a) (b,b) No quiescent logic states were actually lost, but the sections of this sheaf represent sets of simultaneous data at each gate that might be in transition! 4 AND (d,d) (c,c) 4 7

40 Higher cohomology spaces Switching sheaves are written over -dimensional spaces, so they could have nontrivial -cohomology Nontrivial -cohomology classes consist of directed loops that store data Since we just found that logic value transitions are permitted, this means that -cohomology can detect glitches 8

41 Glitch generator: cohomology D A H0(X;ℱ) is generated by F A+C+D e a+c+d E C A+a+C+c+d e+d E E Indication that there's a race condition possible H(X;ℱ) ℤ H detects the race condition 9 Hazard transition state

42 Example: fip-fop C B A T Q C A B T 0 0 Q Transition out of the hazard state to the hold state causes a race condition Hazard! Set Reset Hold This is what traditional analysis gives... 5 possible states 30

43 Flip-fop cohomology C B A H(X;F) ℤ H0(X;F) ℤ7 T Race condition detected! Generated by: a B c a b c+a B C A B c a b c+a b c a b C These states describe the A b C possible transitions out of the A B C hazard state something that takes a bit more trouble to obtain traditionally Q States from the truth table 3

44 Bonus: Cosheaf homology

45 Cosheaf homology The globality of cosheaf sections concentrates in top dimension, which may vary over the base space No particular degree of cosheaf homology holds global sections if the model varies in dimension But what is clear is that numerical instabilities can arise if certain nontrivial homology classes exist These can obscure actual solutions, but can look very real resulting in confusion There are many open questions 33

46 Wave propagation as cosheaf Δu + ku = 0 with Dirichlet boundary conditions Narrow feed channel 34 Open aperture

47 Wave propagation as cosheaf Solving Δu + ku = 0 (single frequency wave propagation) on a cell complex with Dirichlet boundary conditions M([0, ),ℂ) ℂ Narrow feed channel Open aperture pr ev0 ℂ ev0 (an integral transform) M(S,ℂ) (an integral transform) M((-,0],ℂ) M(X,ℂ) = Space of complex measures on X 35

48 Wave propagation as cosheaf Solving Δu + ku = 0 (single frequency wave propagation) on a cell complex with Dirichlet boundary conditions δ0 ℂ M([0, ),ℂ) ℂ Narrow feed channel Open aperture pr ev0 ℂ ev0 (an integral transform) M(S,ℂ) (an integral transform) M((-,0],ℂ) ℂ δ 0 M(X,ℂ) = Space of complex measures on X 36

49 Wave propagation cosheaf homology The global sections indeed get spread across dimension Here's the chain complex: Dimension Dimension M(S,ℂ) ℂ ℂ M((-,0],ℂ) ℂ M([0, ),ℂ) Dimension 0 ℂ Global sections are parameterized by a subspace of these 37

50 Further reading... Louis Billera, Homology of Smooth Splines: Generic Triangulations and a Conjecture of Strang, Trans. Amer. Math. Soc., Vol. 30, No., Nov 998. Justin Curry, Sheaves, Cosheaves, and Applications Inverse problems in geometric graphs using internal measurements, Asynchronous logic circuits and sheaf obstructions, Electronic Notes in Theoretical Computer Science (0), pp Pierre Schapira, Sheaf theory for partial differential equations, Proc. Int. Congress Math., Kyoto, Japan,

51 How do we Deal with Noisy Data? SIMPLEX Program 05 This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

52 Inexact matching Sections of a sheaf are great: they globalize information across data sources But they seem to require exact matches between data sources, which is undesirable Camera Images Camera Images What if instead we want matches that are approximate, to a certain tolerance? 4

53 Relaxing matching requirements Sections require exact matches... Sheaf: ℝ 00 F Data: id id ℝ 00 F 00 F 00 F ℝ 00 F All equal It's a section 5

54 Relaxing matching requirements any inconsistency cannot be tolerated Sheaf: ℝ 99 F Data: id id ℝ 99 F 0 F 00 F ℝ 00 F Not a section 6

55 Relaxing matching requirements so relax the matching condition to handle errors Sheaf: ℝ 99 F Data: id ℝ 99 F 0 F 00 F id ℝ 00 F Variance({99,0,00}) < 0% it's a pseudosection 7

56 Relaxing matching requirements so it can do more than just check consistency Sheaf: ℝ 50 F Data: id {Hot, Not hot} Not hot Hot Hot {Burn hazard, Not burn hazard} Burn hazard If any([not hot, Hot, Hot]==Hot) Perhaps you should not touch! 8

57 Consistency structures Given a sheaf on a simplicial complex X, one also needs a consistency structure: Assign to each non-vertex k-simplex a, a function Ca : (a) +k {0,} A pseudosection p (a) satisfies Ca(p(a), (v0 a)p(v0),, (vk a)p(vk)) = everywhere it's defined, assuming a = (v0,, vk). The consistency structure Ca returns whenever the data at a are consistent 9

58 Pseudosections are sections just of a different sheaf Theorem: Pseudosections of a sheaf over an abstract simplicial complex X are sections of another sheaf over the barycentric subdivision of X Conclusion: at least theoretically, it suffices to work with sheaves 0

59 Mathematical dependency tree Sheaf cohomology Cellular sheaves de Rham cohomology (Stokes' theorem) Homology Linear algebra Lectures 3, 4 Simplicial Complexes Sheaves Calculus Manifolds Lectures 7, 8 CW complexes Abstract Simplicial Complexes Topology Lectures 5, 6 Lecture Set theory

60 Tutorial objectives What are sheaves? Encoding existing data into sheaves Sections, cohomology Practice analyzing sheaves in software Sheafification Data analytic capabilities enabled by sheaves The local to global viewpoint Persistence, local homology Interpret this analysis into the context of the data 30

61 What's next? There are many open questions remaining some will be addressed by DARPA SIMPLEX, but not all Focus: heterogeneity, hypothesis generation Wide open areas with little coverage in the literature: Persistence for sheaves Duality relationships between sheaves and cosheaves Sheaf computations (cohomological or otherwise) Seriously addressing large or varied datasets 3

62 The biggest engineering problems are usually based on fun math problems

63 Mathematician's view of the world Applications are merely corollaries of great theorems Physics Differential equations Linear algebra Data processing Numerical analysis Dynamical systems Control and modeling Computational theory Computer hardware and software Logic 33

64 Applications lead to pure math Physics Differential equations Data processing Linear algebra Numerical analysis Control and modeling Dynamical systems Computer hardware and software Computational theory Logic 34

65 Further reading... Herbert Edelsbrunner and John Harer, Persistent homology: A survey, Surveys on Discrete and Computational Geometry. Twenty Years Later, 57 8 (J. E. Goodman, J. Pach, and R. Pollack, eds.), Contemporary Mathematics 453, Amer. Math. Soc., Providence, Rhode Island, 008. Robert Ghrist, Barcodes: The persistent topology of data, Bull. Amer. Math. Soc., Vol. 45, No., January 008., Pseudosections of consistency structures, AU-CASMathStats Technical Report No Multipath-dominant, pulsed doppler analysis of rotating blades, IET Radar Sonar and Navigation, Volume 7, Issue 3, March 03, pp and Robert Ghrist Topological localization via signals of opportunity, IEEE Trans. Sig. Proc, Vol. 60, No. 5, May 0. Shmuel Weinberger, What is persistent homology? Notices of the Amer. Math. Soc., Vol. 58, No., January 0. 35

66 The End! Preprints available from my website: 36