Cheyne J Miller. Jan 11, 2018

Transcription

1 Cheyne J Miller St Joseph s College Jan 11, cmiller5@sjcny.edu 1

2 Why am I giving this talk? Studying for graduate school oral exams First undergraduate research students Given two sections of Linear teach Using Strang s book... incidence matrices! Personal biases Success via anecdotal evidence Topological Data Analysis This MAA Contributed Paper Session s title! cmiller5@sjcny.edu 2

3 Presentation Outline 3

4 Computing H (S 1 S 1, R): a good open cover Use open sets of R 2 to cover S 1. cmiller5@sjcny.edu 4

5 Computing H (S 1 S 1, R): a good open cover Use open sets of R 2 to cover S 1 and call the intersections of these open sets with S 1, U i. U 3 U 4 U 1 U 2 U 5 cmiller5@sjcny.edu 5

6 Computing H (S 1 S 1, R): a good open cover Label the intersections of the U i s, U ij := U i U j. U 12 U 14 U 23 U 45 U 13 U 15 cmiller5@sjcny.edu 6

7 U 45 7 Computing H (S 1 S 1, R): a good open cover Check that each U i and each U i U j is contractible.

8 Computing H (S 1 S 1, R): the nerve Convert your good open cover to its nerve. U 3 U 4 U 13 U 14 U 1 U 23 U 45 U 12 U 15 U 2 U 5 cmiller5@sjcny.edu 8

9 Computing H (S 1 S 1, R): the complex 0 C 0 δ 0 C 1 0, where C 0 is generated by the vertices (open sets, U i ) and C 1 is generated by the edges (intersections, U ij ). δ 0 0 C 0 = R 5 C 1 = R 6 0 f 1 f 2 f 3 f 4 f 5 δ 0 f 2 f 1 f 3 f 1 f 4 f 1 f 5 f 1 f 3 f 2 f 5 f 4 Each f C 0 has components f, f, f, f, f, f and each cmiller5@sjcny.edu 9

10 Computing H (S 1 S 1, R): the matrix, δ 0 0 C 0 δ 0 C 1 0 δ(f ) ij := f j f i f f 2 f f 2 f 3 f f f 4 = f 4 f 1 f 5 f 1 f f 3 f f 5 f 4 cmiller5@sjcny.edu 10

11 Computing H (S 1 S 1, R): the image of δ 0 0 C 0 δ 0 C 1 0 δ(f ) ij := f j f i f f 2 f f 2 f 3 f f f 4 = f 4 f 1 f 5 f 1 f f 3 f f 5 f 4 rank(δ 0 ) = 4 im(δ 0 ) = R 4 cmiller5@sjcny.edu 11

12 Computing H (S 1 S 1, R): the kernel of δ 0 0 C 0 δ 0 C 1 0 δ(f ) ij := f j f i f f 2 f f 2 f 3 f f f 4 = f 4 f 1 f 5 f 1 f f 3 f f 5 f 4 rank(δ 0 ) = 4 ker(δ 0 ) = R dim(c 0 ) rank(δ 0) = R 1 cmiller5@sjcny.edu 12

13 A formal definition: the cohomology groups Definition Given a cochain complex, i.e. a sequence of vector spaces with linear maps connecting them,... δp 2 C p 1 δ p 1 C p δ p C p+1 δ p+1..., where δ δ = 0, we define the p-th cohomology group, H p by H p := ker(δ p )/im(δ p 1 ). The idea In other words, we simply need to count the dimensions of the kernel and image of each matrix, δ. cmiller5@sjcny.edu 13

14 Computing H (S 1 S 1, R): The cohomology groups 0 δ 1 C 0 δ 0 C 1 δ 1 0 H 1 (S 1 S 1, R) = 0 δ(f ) ij := f j f i H 0 (S 1 S 1, R) = ker(δ 0 )/im(δ 1 ) = R 1 0 = R 1 H 1 (S 1 S 1, R) = ker(δ 1 )/im(δ 0 ) = R 6 4 = R 2 H 2 (S 1 S 1, R) = 0 cmiller5@sjcny.edu 14

15 Presentation Outline 15

16 Basic summary of cohomology computations 1 Fix a topological space, X. 2 Choose a (good) open cover. 3 Write down (and draw) the nerve of this cover. 4 Write out your complex. 5 Write out the boundary maps (matrices). 5 Compute the kernel and image subspaces of these matrices by the Gauss-Jordan algorithm and the Rank-Nullity theorem. 6 Compute the cohomology groups by subtracting dimension numbers. 7 State your computed cohomology groups. cmiller5@sjcny.edu 16

17 Linear Algebra Prerequisites and Applications Computing the matrix for a given linear map. The kernel and image subspaces for a linear map. Rank/Nullity Theorem Gauss-Jordan Reduction Dimension of the quotient of a finite-dimensional vector space by a subspace. cmiller5@sjcny.edu 17

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19 Some simple candidates for your space, X. The circle, S 1. The wedge of two circles, S 1 S 1. The sphere, S 2. The torus, S 1 S 1. (a nice independent project) cmiller5@sjcny.edu 19

20 Presentation Outline 20

21 An algorithm for computing cohomology theories for a topological space, X computes... under the right conditions... singular cohomology (Topology) de Rham cohomology (Differential Geometry) sheaf cohomology (-DeRham ; Algebraic Geometry)... perhaps more things the speaker doesn t fully comprehend. cmiller5@sjcny.edu 21

22 Interesting Applications is ideally used in... Topological Data Analysis... in particular, in Persistent Homology?? 22

23 Thank you! Questions? 23