Thread by @robertghrist

why learn differential forms & stokes' theorem?
besides the usual applications in physics, there are some lovely novel applications in data science...
it's time for the grand finale from the calculus BLUE project...
1/15

let's learn a thing or two about differential forms & time series data...
2/15

let's say you have a pair of perioid times series x_i(t) and x_j(t) -- sine waves out of phase -- and you want to know which is leading and which is lagging. here's a fun way to think about it: plot the parametric curve in the x_i-x_j plane. what do you see?
3/15

ahha... the *oriented* area in the x_i-x_j plane indicates leadership. and this is measured by what? by the area 2-form dx_i ^ dx_j
4/15

it gets better: for more than two signals, working with all the different 2-forms in the signal space can tell you the phase ordering of the sine waves. that's pretty neat to think about...
5/15

but hang on: what if you don't have sine waves? what if the signals are more... involved? how do you make sense of oriented projected area in the plane? this seems like it could get messy...
6/16

ah, hello stokes' theorem! don't worry about the 2-forms and projected area at all! work in the full N-dimensional signal space and instead integrate over the parametrized loop the 1-form (x_i dx_j - x_j dx_i)/2. by stokes's theorem, this gives the oriented projected area.
7/15

now, think... for each pair (x_i, x_j) of cyclic signals, you integrate the 1-form and get a value [A_ij]. together, these give the "lead matrix" of the system. A_ij>0 suggests that x_i leads x_j.
i wonder what this matrix might be good for?
8/15

now, why did i say "cyclic" instead of "periodic"? ah, well, because of the change of variables theorem, the integral of that 1-form over a loop is independent of the time-parametrization. in stokes' theorem, we have a result for "topolgical time series"
9/15

i think this is very significant. think of the "business cycle" in econ: cyclic, though not strictly periodic. cyclic time series also arise in EEG and fMRI data, as noted by yuliy baryshnikov and @chadgiusti and others. and recall the ability to discretize...
10/15

i learned about the lead-lag idea for time series from the amazing yuliy baryshnikov. you can find his 2016 paper with schlafly on it here: ieeexplore.ieee.org/document/77984…
11/15

my student, darrick lee @yldarrick , together with chad giusti @chadgiusti, have far-reaching extensions of these ideas to chen integrals and cochain models for path spaces in the context of applications to data: see arxiv.org/abs/1811.03558
12/15

there are so many open questions: this lead matrix [A_ij] is something like an antisymmetric covariance matrix for cyclic time series... what can you do with it? what happen when you start applying PCA & other techniques from data science? there's so much more...
13/15

this is how i've ended calculus BLUE & my course for freshmen engineers at penn @PennEngineers : they've been very brave and have come so far -- far enough to see some real applications of some really cool math to data.
we're all tired & happy...
14/15

if you'd like to see a brief writeup of these and other applications of differential forms & stokes' theorem to data, see the paper by yuliy baryshnikov & myself, "Stokes's Theorem, Data, and the Polar Ice Caps" in the @maanow Monthly: tandfonline.com/doi/abs/10.108…
15/15

CODA: if you want to see full versions of the videos, or use these lectures to supplement your class / self-learning / review, then check out calculus BLUE volume 4, chapters 16.5 and 18.5 starting here: