Add references and improve text

bib/bibliography.bib

@@ -386,3 +386,61 @@
   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}
 }
 
+@article{sola2018_micro_lie_theor_state_estim_robot,
+  author =	 {Sol{\`a}, Joan and Deray, Jeremie and Atchuthan,
+                  Dinesh},
+  title =	 {A Micro Lie Theory for State Estimation in Robotics},
+  journal =	 {CoRR},
+  year =	 {2018},
+  url =		 {http://arxiv.org/abs/1812.01537v7},
+  abstract =	 {A Lie group is an old mathematical abstract object
+                  dating back to the XIX century, when mathematician
+                  Sophus Lie laid the foundations of the theory of
+                  continuous transformation groups. As it often
+                  happens, its usage has spread over diverse areas of
+                  science and technology many years later. In
+                  robotics, we are recently experiencing an important
+                  trend in its usage, at least in the fields of
+                  estimation, and particularly in motion estimation
+                  for navigation. Yet for a vast majority of
+                  roboticians, Lie groups are highly abstract
+                  constructions and therefore difficult to understand
+                  and to use. This may be due to the fact that most of
+                  the literature on Lie theory is written by and for
+                  mathematicians and physicists, who might be more
+                  used than us to the deep abstractions this theory
+                  deals with.  In estimation for robotics it is often
+                  not necessary to exploit the full capacity of the
+                  theory, and therefore an effort of selection of
+                  materials is required. In this paper, we will walk
+                  through the most basic principles of the Lie theory,
+                  with the aim of conveying clear and useful ideas,
+                  and leave a significant corpus of the Lie theory
+                  behind. Even with this mutilation, the material
+                  included here has proven to be extremely useful in
+                  modern estimation algorithms for robotics,
+                  especially in the fields of SLAM, visual odometry,
+                  and the like.  Alongside this micro Lie theory, we
+                  provide a chapter with a few application examples,
+                  and a vast reference of formulas for the major Lie
+                  groups used in robotics, including most jacobian
+                  matrices and the way to easily manipulate them. We
+                  also present a new C++ template-only library
+                  implementing all the functionality described here.},
+  archivePrefix ={arXiv},
+  eprint =	 {1812.01537},
+  primaryClass = {cs.RO},
+}
+
+@book{stillwell2008_naive_lie_theor,
+  author =	 {John Stillwell},
+  title =	 {Naive Lie Theory},
+  year =	 2008,
+  publisher =	 {Springer New York},
+  url =		 {https://doi.org/10.1007/978-0-387-78214-0},
+  DATE_ADDED =	 {Sat Oct 31 17:40:56 2020},
+  doi =		 {10.1007/978-0-387-78214-0},
+  pages =	 {nil},
+  series =	 {Undergraduate Texts in Mathematics},
+}
+

posts/lie-theory.org

@@ -112,12 +112,27 @@ 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.
 
-And /that's all!/ What we have defined so far is a /Lie
+You can think of the manifold as a tablecloth that has a weird shape,
+all kinds of curvatures, but no edges or spikes. The idea here is that
+we can define an /atlas/, i.e. a local approximation of the manifold
+as a plane. The name is telling: they're called atlases because they
+play the exact same role as maps. The Earth is not flat, it is a
+sphere with all kinds of deformations (mountains, canyons, oceans),
+but we can have maps that represent a small area with a very good
+precision. Similarly, atlases are the vector spaces that provide the
+best linear approximation of a small region around a point on the
+manifold.
+
+So we know what a group and a differential manifold are. As it turns
+out, that's all we need to know! What we have defined so far is a /Lie
 group/[fn:lie], i.e. a group that is also a differentiable
-manifold. To take the example of rotation matrices:
+manifold. The tangent vector space at the identity element is called
+the /Lie algebra/.
+
+To take the example of rotation matrices:
 - We can combine them (i.e. by matrix multiplication): they form a
   group.
-- if we have a function $R : \mathbb{R} \rightarrow
+- If we have a function $R : \mathbb{R} \rightarrow
   \mathrm{GL}_3(\mathbb{R})$ defining a trajectory (e.g. the
   successive attitudes of a object in space), we can find derivatives
   of this trajectory! They would represent instantaneous orientation
@@ -130,6 +145,41 @@ general theory for differential equations.
 
 * Interesting properties of Lie groups
 
+For a complete overview of Lie theory, there are a lot of interesting
+material that you can find online.[fn:princeton_companion] I
+especially recommend the tutorial by
+cite:sola2018_micro_lie_theor_state_estim_robot: just enough maths to
+understand what is going on, but without losing track of the
+applications. There is also a [[https://www.youtube.com/watch?v=QR1p0Rabuww][video tutorial]] for the [[https://www.iros2020.org/][IROS2020]]
+conference[fn::More specifically for the workshop on [[https://sites.google.com/view/iros2020-geometric-methods/][Bringing
+geometric methods to robot learning, optimization and control]].]. For a
+more complete treatment, cite:stillwell2008_naive_lie_theor is
+great.[fn::{-} John Stillwell is one of the best textbook writers. All
+his books are extremely clear and a pleasure to read. You generally
+read a book because you're interested in learning the topic; you begin
+learning a topic just because Stillwell wrote a book on it.]
+
+[fn:princeton_companion] {-} There is also a chapter on Lie theory in
+the amazing /Princeton Companion to Mathematics/
+[[citep:gowersPrincetonCompanionMathematics2010][::, §II.48]].
+
+
+The Lie group is a set of elements. At every element, there is a
+tangent vector space. The tangent vector space at the identity is
+called the Lie algebra, and as we will see, it plays a special role.
+
+TODO
+
 * Applications
 
+Lie theory is useful because it gives strong theoretical guarantees
+whenever we need to linearize something. If you have a system evolving
+on a complex geometric structure (for example, the space of rotations,
+which is definitely not linear), but you need to use a linear
+operation (if you need uncertainties, or you have differential
+equations), you have to approximate somehow.
+
+Using the Lie structure of the underlying space, you immediately get a
+principled way of defining derivatives, random variables, and so on.
+
 * References