Differential geometry with finite fields | Differential Geometry 7 | NJ Wildberger

With an algebraic approach to differential geometry, the possibility of working over finite fields emerges. This is another key advantage to following Newton, Euler and Lagrange when it comes to calculus!

In this lecture we introduce the basics of finite (prime) fields, where we work mod p for some fixed prime p, and show that our study of tangent conics to a cubic polynomial extends naturally, and leads to interesting combinatorial structures. There are many possible directions for investigation by interested amateurs who have understood this lecture.

After the basics of arithmetic over the field F_p, including a discussion of primitive roots and Fermat's theorem, we discuss polynomial arithmetic and illustrate tangent conics to a particular cubic over F_11. In particular Ghys' lovely observation about the disjointness of such tangent conics (for a cubic) can be illustrated completely here, and some additional patterns visibly emerge from the vertices of the various tangent conics.

One big difference here is that the sub-derivatives and the derivatives are NOT equivalent in general, and we must replace the usual Taylor expansions with one involving sub-derivatives. Some remarks about the useful distinction between polynomials and polynomial functions in this setting are made.

This lecture shows that the calculus is actually a much wider operational tool than is usually appreciated----finite calculus not only makes sense but is a rich source of both combinatorial and algebraic patterns---and questions for further investigations.

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[toni_canada]

Your classes are spectacular, thank you for giving us this new perspective on calculus and algebra, where everything makes more sense! Plus, it's applicable to other fields!

[TimTeatro]

I appreciate the errata notifications throughout this lecture series. Thanks very much for taking the time.

[YuvalRishuSanders]

Gorgeous lecture, thanks. Interesting point about the more fundamental nature of the "sub-derivatives": if they are indexed by natural numbers but are polynomials over finite fields, it makes no immediate sense why we should multiply them by an ill-defined function from natural numbers to the finite field!

[qazqwe]

Really a beautiful video!!
While watching this series, I really came to know the power of Algebra in all contexts.

[jpdemont]

I have loved and been able to follow all of the lectures in this series, but this one has (on first viewing) overwhelmed me. Thank goodness that I can watch it again. :-)

[njwildberger]

Indeed, quite correct, thanks for catching that. I have added a correction note.

[markovchainz]

This is a beautiful and fascinating application of this Lagrangian algebraic approach to DG. What I'm wondering is: what other applications are there for this approach? For example, most of these videos deal mainly with rational numbers and polynomials. Can we extend this treatment to, say, rational functions, or work over fields like the rational complex numbers? Interesting stuff nonetheless.

[Kontinuumshypothese]

At 21:20: If we have a primitive root a, then a^(p-1)=1 and for any element b of the field we have b=a^n for some n. This implies Which means that in this field the monomial x^m for m greater than or equal to p is identical to x^(m-(p-1)). Or, in other words, polynomials with degrees of p-1 or higher don't make sense here.

This means that the idea of dividing by the factorial k! doesn't sound so "dangerous" anymore - k will be small, so k! won't be 0.

[andrelucassen]

Regarding the subderivatives Dk(q) in Fp, when k=0, then k!=0!=1 I would say. This would mean that every time k is multiple of p the subderivative equalsD0(q)/0!= D0(q)/1 = D0(q).

[dehnsurgeon]

This should be in an algebraic geometry playlist.

[maxwang]

Beautiful but a bit above my level for now. Will park this one and revisit later.

[ericbischoff]

"Stick to subderivatives as being more fundamental objects" - very interesting remark! Thanks.

[WilliamTanksleyJr]

I know complex calc is complex - all kinds of stuff about residues and discs. Would be cool if Lagrange's approach were simpler!

[Kontinuumshypothese]

At about 5:00, shouldn't it be "any element is a^n (for some n)" instead of "a^p"?

[rickynave]

Around 26:30 you say that the subderivatives Dq D2q, ... are zero, but that D_p q =/= 0, but isn't that not quite accurate? Because just like in the Taylor series, these are the subderivatives around (I hesitate to use the word evaluate, but the Lagrange expression is fundamentally an identity which is still expanded about some abstract beta) beta, and so the "value" of the polynomial derivatives at beta are zero. How is this different than just taking the Taylor polynomial of y = x^4 about x = 0 so that y' = 4(x=0)^3 = 0, y" = 12(x=0)^2 = 0, y" ' = 24(x=0) = 0, y" " = 24 ? Or how the Taylor polynomial of y = sin(x) about x=0 has every odd derivative nonzero and every even derivative zero? These are just the derivatives evaluated at (or about) some point, which isn't the same as saying that y' " =24x = 0 everywhere or saying that y' = cos(x) = 1 everywhere.

[krishnachoudhary]

This is amazing! Really, why don't they teach this?

[juancamilorodriguezr]

Could you please give me some bibliography related to differential geomatry of finite Fields?
Thank you.

[josheisenthal]

Typo: when you corrected the expansion of the polynomial to use q's instead of p's (around 19mins), you should have also corrected 'deg p = n' to 'deg q = n'

(really enjoying the lectures - thank you!)

[mistershoujo]

Completely off topic, but how on Earth were you able to cohesively write on the board while it was slowly sliding up?!

[postbodzapism]

So if we can do differential geometry over finite fields, is it possible that we can combine general relativity with quantum mechanics?