Sheaf Cohomology and its Interpretation

Transcription

1 Sheaf Cohomology and its Interpretation SIMPLEX Program 5 This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4. International License.

2 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

3 Session objectives What do global features of fused data look like? What new do the other sheaf invariants tell you? 3

4 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) = (v) - (v e) s(v) + (v e) s(v) = ( (a b) is the restriction map connecting cell a to a cell b in a sheaf ) 4

5 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) = s(v) (v e) s(v) = (v e) s(v) + (v e) s(v) - (v e) s(v) = - (v e) s(v) + (v e) s(v) = ( (a b) is the restriction map connecting cell a to a cell b in a sheaf ) 5

6 Recall: A queue as a sheaf Contents of the shift register at each timestep N = 3 shown ( ) ( ) ℝ ℝ3 ( ) ( ) ℝ ℝ3 ℝ ( ) ℝ3 ( ) 6 ℝ ℝ3 ( )

7 Recall: A single timestep Contents of the shift register at each timestep N = 3 shown ( ) ( ) (9,) (,9,) (,9) (,,9) ( ) ( ) ( ) (,) (5,,) ( ) 7 (5,) (,5 ( )

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

9 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

10 Notational interlude Homology Cohomology Ck(X) : chain space Ck(X) : cochain space k : boundary map dk : coboundary map Hk(X) : cohomology space Hk(X) : homology space Dimensions go up in cochain complex Dimensions go down in chain complex

11 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)

12 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, +, or - depending on the relative orientation of aj and bi Column j

13 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

14 Cohomology facts H(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

15 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

16 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

17 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;

18 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

19 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

20 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

21 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

22 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

23 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

24 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

25 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

26 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

27 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

28 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

29 Glitch generator: cohomology D A H(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

30 Example: fip-fop C B A T Q C A B T 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 3

31 Flip-fop cohomology C B A H(X;F) ℤ H(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

32 Bonus: Cosheaf homology

33 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

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

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

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

37 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((-,],ℂ) ℂ M([, ),ℂ) Dimension ℂ Global sections are parameterized by a subspace of these 37

38 Next up... Interactive session: Computing Homology and Cohomology Next and final lecture: How do we Deal with Noisy Data? 38

39 Further reading... Louis Billera, Homology of Smooth Splines: Generic Triangulations and a Conjecture of Strang, Trans. Amer. Math. Soc., Vol. 3, 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 (), pp Pierre Schapira, Sheaf theory for partial differential equations, Proc. Int. Congress Math., Kyoto, Japan,