2Once simplicial coupler is generated

PETL icomputing Rt is straightforward V

B E

we assume that all suplices are ordered 9according to their time of appearancelater greater index Ih particulara simplex is allowed to appear before all of itsboundary does As a result the boundary

matrixle's

upper angular

For a matrix AEheat K x k k we will call the height

of column L the larges dedex p with nonzero Agp

Then the computing of the PH is done bythe fearingalgorithmR

L Kthere is p di Llp LKd Lep to kill R LG a

Equivalently we right multiply R by elementaryupper triangular matrices with 1 on the diagonal and1 at Cp d

In the end 12 2 V with V upper triangular wt'son the diagonal i.e invertible and R will lure all alumswith different L Heine

Computing PH

for in

while

KesR Lspar of surplices with 0 columnsAs Kiso V KerR the dimensions ofZp are thesameas the dimensions of KerR p ofempty columns of R

of du pWhat happens as one adds a suplex of dimension pAs the Euler deeracteristies changes by 1 we have

them either a cycle in dineanu p appears or

a cycle in dimension p I disappearsIf a newly arriving suplex 2 generates a cyclethat means its boundary is a linear eearbination off

fof boundaries of simplices already thereThat is reducing9 by Lowery Lleads to an empty column

If this does hot happen a Cpc cycleis patched The suplex p LG is Ppttthe latest simplex that formed the HI

iii giant EEEEEEEand the empty columns 2 correspond

M 16to some hemology classes that are F.E.frborn with L and killed for not at Bp ia later point

I 2 3 y s 6 7 8 9 10Example o I l l

Z I I l

f y2 I l l3 so I I

I C u so 1 I10 5 4 s I

8 g 6 so I7 so

3 8 Iq soso so

Performan of this algorithm is Ikf whichmight be a problem If we if cerecentratedonly are homologies in dimension e p at west

pti signlines need to be generated

Some examples of software to compute PH

Example of application of PH to polling data

What happens to copulations of persistenthomology as the data sample is perturbed

If X X are E close in Hausdorff distance seuce

then fixes dca X and f nos d la X are e dose

in C warm

Question i given functions f f sup If g a EM

are the persistence diagrams dose and in what sense

Intuitively the haybars would surviveand who caresabout shot bars NNNBoltleneckdistai Given 2 collections of pointsin Cbd a bad CIR we define

W QQ iyzfgqu.jp l9 9 es

where g Cbd g Cbd I is thepairingof Q Q i e a Cokechen of pairs

9,9 qcQUD.glEQUI

that exactly one of each qEQ g EQ appears in

the pairing

Stability

In wards we couple long bars to Guybars and shot bars to diagonal

The The bottleneckdistanceW CPH.CH PH.tl EHf f'He

Root Consider Ie ffee3 ff'seThen Hf glee Face Fd Fo e

Lef p bid fulb d eye L b ofbarsstraddling b d

Recall f b d rkZCFDBCEDAZCFBTBdz.BZ

Notation 2 Fc Ze BCE Be

dz 2 FL i 2 e BCFL Ble

Talent ofbars within theis p

Bd E diSquare badi pcb dz

Zb iZ fZb Zz pcb d fo bz.dz

b bz rkz

rk nB rkz Brk TB

B CBz Z CZzStandard for linear subspaces UNW

rk Ij rk ark Unv rklenw

So rtz Bz

rk E nB rkz Brk TB

rk Z AB rkZznBz rk 2 in Bz rk 2ZAB

Recall that Z C Zz B CBz so ZzRBzc uand for any 4 linear subspaces Zarb 2 in Bzv uforming this quiver ranks are Syberwedeeler z B

So rkzfjz rkztfg rkffB.tkFBz70The only inclusions we used were B CBz Z CZz So

if we take 2fee Z and Bae Ble etc we

will see that the content of d e

pl Lbie bteJxEdiedztD is da

p b bzTx d dz E dDYE1

So if we create a bipartite i1

graph connecting g of which iare further than e from the

bleb b2b'EE

diagonal the conditions of Hall perfectmatching theoremare satisfied and we can couple all such pairs D

As it turned out one can use PH not to charalerizethe underlying topological space but also to characterizethe functions that generate the filtrationThe persistent diagrams are used to characterize the textureof the function f defining the filtration Xs ffes3

Notably material scientists became avid users of thetechnique

Example Characterization of amorphous solid States

This begets the general question whatcan be saidabout the behavior of the shot bars of genniefunctions on simplicial complexes

Say for the Veierstrap typefunction feat 2 coypu

1 If 1 what isthe structure of the PHO

i Recall that the filtration is tame

if the Pita content of any quadrantRbd is finite

i When Ptt is finite How fast its contentcan grow

Jitter

In general the numberof bars near the diagonalcan be arbitrarily large e g PH Qbd 3 L Ydb where

L is arbitrarilyfastgrowing function