This thesis addresses an advanced topic at the intersection of differential geometry and control theory - sub-Riemannian geometry on orthogonal Lie groups. Specifically, it focuses on the analysis of groups with Euclidean signature (3,0) and groups with Minkowski signatures (2,1) and (3,1), and systematically compares the specific differences between these geometric structures.

The student was exceptionally diligent and capable of independent study. In a short period of time, he mastered advanced material that would normally take several semesters to cover. Furthermore, he successfully applied the theoretical framework he acquired to a detailed examination of the aforementioned geometric structures with potential applications. I see the main original contribution of the work in the detailed description of sub-Riemannian geodesics in Minkowski signatures. The author precisely describes their properties, calculation, and potential complications associated with the loss of global optimality (cut locus).  

Last but not least, the high linguistic and stylistic quality of the text must be emphasized. The work smoothly guides the reader through the fundamentals of differential, symplectic, and sub-Riemannian geometry and can therefore serve as an excellent framework for educational materials. It also demonstrates how AI can be helpful and used effectively in the creation of a thesis, both for a quicker understanding of a complex topic and, for example, for creating illustrative images.

I definitely recommend grading the thesis with an A. (excellent)

As far as I understand, the aim of the work is to come close to current research problems in geometric
control theory, handling invariant sub-Riemannian structures on Lie groups and focusing on the
orthogonal groups and algebras. This is a hot topic with diverse striking applications now.
Duy Long Nguyen has done good job in cutting all the stories short and blending intuitive description of
concepts and properties with concise formulations. The resulting text might serve as a good and quick
introduction for someone, who is already rather advanced in geometric analysis or other not too far
areas of Mathematics, but lacks the knowledge in the geometric control theory business and the roles of
the (known or hidden) symmetries there. Maybe the author has served exactly this for himself when
working out the thesis. I can imagine that nowadays technologies allow gaining quick insight in rather
advanced topics, chatting about the books and other resources difficult to read in detail in the old
traditional way. I am sure that a reader of the thesis could enjoy it as perfect guideline and combine it
with questioning further details online aiming at quick digesting the quite advanced topic and get to
qualified use of it in diverse applications.
In my opinion, the text is composed in a neat and clean way, with good balance of general glances and
more detailed expositions, although it should not be considered as a complete monograph or textbook
introducing all details and proofs, etc. But clearly this was not the aim.
I find it rather impressive, how the main line of exposition, starting with elementary concepts from
differential geometry, global analysis, etc., aiming at the core of Hamiltonian calculus for invariant sub-
Riemannian problems, appears in a lean and clean form, making the story well readable, say, for
mathematically educated engineers or computer scientists. This also includes the typographical style,
including nice and relevant pictures, perhaps drawn with standard Python libraries.
I will not comment here on the content of the individual six chapters in detail. This has been done by the
author himself in a good way, too. And all details and further relations are properly referenced to well
chosen literature.
To conclude my opinion easily, I would have appreciated explicit identification of the potential audience and the technical and didactical aims of the work. I mean, if the aim has been rather (self-)didactical, I find the result very good. If the aim was to deliver new techniques and results, then going back to all the elements was not necessary and more time and space could have been devoted to the questions touched in the last chapter.
Moreover, I am also curious about the actual approach of the author in terms of his communication with AI based resources of information. My feeling is that before we had such resources available, such a convincing and readable short introduction could be achieved by real masters only, based on long years of practice. Thus, I would like to include this issue in the discussion when defending the thesis. I should like to stress, that I do not think that a careful and proper use of the AI powers diminishes the value of the work. On the contrary, we should expect such tools should be fully exploited, once available.
My suggestion towards the marking is to assign either the mark A or B, depending on the author’s presentation or answers during the defence discussion.