From 309f3110ffeef86a781381a95b1bd3111a86e3dc Mon Sep 17 00:00:00 2001 From: Dimitri Lozeve Date: Thu, 12 Nov 2020 18:20:30 +0100 Subject: [PATCH] Update post --- bib/bibliography.bib | 29 +++++++++++++++++ posts/lie-theory.org | 77 ++++++++++++++++++++++++++++++++++++++++++++ 2 files changed, 106 insertions(+) diff --git a/bib/bibliography.bib b/bib/bibliography.bib index cbc4aad..a0d1878 100644 --- a/bib/bibliography.bib +++ b/bib/bibliography.bib @@ -357,3 +357,32 @@ doi = {10.1109/ARITH48897.2020.00016}, url = {https://doi.org/10.1109/ARITH48897.2020.00016}, } + +@book{riehlCategoryTheoryContext2017, + langid = {english}, + location = {{United States}}, + title = {Category Theory in Context}, + isbn = {978-0-486-82080-4}, + publisher = {{Dover Publications : Made available through hoopla}}, + date = {2017}, + author = {Riehl, Emily}, + file = {/home/dimitri/Nextcloud/Zotero/storage/H2XLYX3I/Riehl - 2017 - Category theory in context.pdf}, + note = {OCLC: 1098977147} +} + +@article{fongSevenSketchesCompositionality2018, + archivePrefix = {arXiv}, + eprinttype = {arxiv}, + eprint = {1803.05316}, + primaryClass = {math}, + title = {Seven {{Sketches}} in {{Compositionality}}: {{An Invitation}} to {{Applied Category Theory}}}, + url = {http://arxiv.org/abs/1803.05316}, + shorttitle = {Seven {{Sketches}} in {{Compositionality}}}, + abstract = {This book is an invitation to discover advanced topics in category theory through concrete, real-world examples. It aims to give a tour: a gentle, quick introduction to guide later exploration. The tour takes place over seven sketches, each pairing an evocative application, such as databases, electric circuits, or dynamical systems, with the exploration of a categorical structure, such as adjoint functors, enriched categories, or toposes. No prior knowledge of category theory is assumed. A feedback form for typos, comments, questions, and suggestions is available here: https://docs.google.com/document/d/160G9OFcP5DWT8Stn7TxdVx83DJnnf7d5GML0\_FOD5Wg/edit}, + urldate = {2019-04-29}, + date = {2018-03-14}, + keywords = {Mathematics - Category Theory,18-01}, + author = {Fong, Brendan and Spivak, David I.}, + file = {/home/dimitri/Nextcloud/Zotero/storage/WBUCWRPK/Fong and Spivak - 2018 - Seven Sketches in Compositionality An Invitation .pdf;/home/dimitri/Nextcloud/Zotero/storage/MT7MPULY/1803.html} +} + diff --git a/posts/lie-theory.org b/posts/lie-theory.org index 231c1b1..d3c8f97 100644 --- a/posts/lie-theory.org +++ b/posts/lie-theory.org @@ -9,3 +9,80 @@ expression: according to the answers to [[https://english.stackexchange.com/q/25 it might date back to Horace's [[https://en.wikipedia.org/wiki/Ars_Poetica_(Horace)][/Ars Poetica/]] ("prodesse et delectare"). I like the idea that books (and ideas!) should be both instructive and enjoyable...] + +While exploring [[./quaternions.html][quaternions]] and the theory behind them, I noticed an +interesting pattern: in the exposition of +cite:sola2017_quater_kinem_error_state_kalman_filter, quaternions and +rotations matrices had exactly the same properties, and the derivation +of these properties was rigorously identical (bar some minor notation +changes). + +This is expected because in this specific case, these are just two +representations of the same underlying object: rotations. However, +from a purely mathematical and abstract point of view, it cannot be a +coincidence that you can imbue two different types of objects with +exactly the same properties. + +Indeed, this is not a coincidence: the important structure that is +common to the set of rotation matrices and to the set of quaternions +is that of a /Lie group/. + +* Why would that be interesting? + +From a mathematical point of view, seeing a common structure like this +should raise alarm signals in our heads. Is there a deeper concept at +play here? If we can find that two objects are two examples of the +same abstract structure, maybe we'll also be able to identify that +structure elsewhere, maybe where it's less obvious. And then, if we +prove interesting theorems on the abstract structure, we'll +essentially get the same theorems on every example of this structure, +and /for free!/ (i.e. without any additional work!)[fn:structure] + +[fn:structure]{-} When you push that idea to its extremes, you get +[[https://en.wikipedia.org/wiki/Category_theory][category theory]], which is just the study of (abstract) structure. This +in a fun rabbit hole to get into, and if you're interested, I +recommend the amazing [[https://www.math3ma.com/][math3ma]] blog, or +cite:riehlCategoryTheoryContext2017 for a complete and approachable +treatment. cite:fongSevenSketchesCompositionality2018 gives an +interesting perspective on why category theory is interesting in the +real world. + + +We can think of it as a kind of factorization: instead of doing the +same thing over and over, we can basically do it /once/ and recall the +general result whenever it is needed, as one would define a function +and call it later in a piece of software. + +* Important structure + +Continuing on the example of rotations, what common properties can we +identify? + +1. Quaternions and rotation matrices can be multiplied together (to + compose rotations), have an identity element, along with nice + properties. +2. Quaternions and rotation matrices can be differentiated, and we can + map them with usual vectors. + +These two group of properties actually correspond to common +mathematical structures: a /group/ and a /differentiable manifold/. + +You're probably already familiar with [[https://en.wikipedia.org/wiki/Group_(mathematics)][groups]], but let's recall the +basic properties: +- It's a set $G$ equipped with a binary operation $\cdot$. +- The group is closed under the operation: for any element $x,y$ in G, + $x \cdot y$ is always in $G$. +- The operation is associative: $x \cdot (y \cdot z) = (x \cdot y) + \cdot z$. +- There is a special element $e$ of $G$ (called the /identity + element/), such that $x \cdot e = e \cdot x$ for all $x \in G$. +- For every element $x$ of $G$, there is a unique element of $G$ + denoted $x^{-1}$ such that $x \cdot x^{-1} = x^{-1} \cdot x = e$. + +A [[https://en.wikipedia.org/wiki/Differentiable_manifold][differentiable manifold]] is a more complex beast. Although the +definition is more complex, we can loosely imagine it as a surface (in +higher dimension) on which we can compute derivatives at every +point. This means that there is a tangent hyperplane at each point, +which is a nice vector space where our derivatives will live. + +* References