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Stream: event: MIT Categories Seminar

Topic: October 29: Conal Elliott's talk

Paolo Perrone (Oct 27 2020 at 14:56):

Hello all,
Here's the thread for Conal Elliott's talk, "Efficient automatic differentiation made easy via category theory".

When: Thursday October 29th, 12 noon EDT (Boston time)

Zoom meeting:
https://mit.zoom.us/j/280120646
Meeting ID: 280 120 646

Youtube live stream:
https://youtu.be/2StYHN-wX64

Paolo Perrone (Oct 29 2020 at 15:55):

Hello all! We start in 5 minutes.

Paolo Perrone (Oct 29 2020 at 16:57):

Paper: https://arxiv.org/abs/1804.00746

Paolo Perrone (Oct 29 2020 at 16:58):

http://conal.net/

Conal Elliott (Oct 29 2020 at 18:28):

Paper with abstract and related work: http://conal.net/papers/essence-of-ad/

Paolo Perrone (Oct 29 2020 at 21:12):

Video here: https://youtu.be/17gfCTnw6uE

Todd Trimble (Oct 30 2020 at 01:07):

I was a bit confused by Conal's remarks on dual numbers (around the 73 minute mark). For instance, that they only work for scalars or 1-dimensional differentiation. (There were a few more remarks but let me start here.) On the contrary, if $k$ is a field and $A$ is any augmented commutative $k$-algebra (i.e., comes equipped with a $k$-algebra map $\varepsilon: A \to k$, which you can think of as a point in $Spec(A)$), then the evident augmented algebra $k[x]/x^2 \to k$ represents derivations $d$ on $A$, in the sense that augmented algebra maps $A \to k[x]/x^2$ are in natural bijection with linear maps $d: A \to k$ such that $d(ab) = d(a)\varepsilon(b) + \varepsilon(a)d(b)$, i.e., with tangent vectors at the "point" $\varepsilon$. Here the underlying $k$-algebra of $A$ could be for example the algebra of smooth functions on a manifold of any dimension. So I'm not clear on the remark about 1-dimensionality. Better yet, you can think of $Spec(k[x]/x^2)$ as representing the tangent bundle functor.