I need help in studying advanced mathematics. I'm an engineering student. My college courses didn't include advanced mathematical topics, e.g., abstract algebra and topology. I only had mathematics enough to understand linear dynamical systems and stability analysis using lyapunov method. Three months ago, I started reading a reference in non-linear dynamics . I was startled and couldn't understand most of the words used in the proofs. I then had to take a basic course in real analysis which made me become very fascinated with mathematics. I then finished basic courses in complex analysis , discrete mathematics and now I am currently studying abstract algebra.
I enjoy studying mathematics but I want first to focus on mathematics that can be most useful to me in control theory . I came to the conclusion that it might be differential topology as deals with differentiable functions and manifolds and contains theorems such as Poincare-Hopf theorem, Morse Theory and Brouwer's fixed-point theorem which I think can be used in the study of dynamical systems.
Can someone help me by giving me a step by step list of mathematical topics that I need to study with recommendation for the references.
My current knowledge in real analysis is till Taylor series ,derivative and mean value theorem. In complex analysis, I know till residue theorem . no real study in algebraic topology just heard about some basic concepts such as homotopy groups , CW complex and euler characteristic.
 A: For control theory, given your background, you should start with

*

*General (point-set) topology. You can find textbook recommendations here. McCleary's book is the fastest, Morris' book is the slowest.


*Then differential geometry/Riemannian geometry (it will serve you better than a differential topology class): Make sure to take a more advanced Differential Geometry class, not "Curves and surfaces:" The advanced class should cover differentiable manifolds, connections, Riemannian metrics.
My favorite for Riemannian Geometry is do Carmo's "Riemannian Geometry," mostly chapters 0 through 4. It is also the fastest.
Another good option is
Abraham, Ralph; Marsden, Jerrold E., Foundations of mechanics. 2nd ed., rev., enl., and reset. With the assistance of Tudor Ratiu and Richard Cushman, Reading, Massachusetts: The Benjamin/Cummings Publishing Company, Inc., Advanced Book Program. m-XVI, XXII, 806 p. $ 36.50 (1978). ZBL0393.70001.
They will cover differentiable manifolds, forms, Frobenius theorem, basics of Riemannian geometry...
Frequently (but not always) these are covered in a differential topology class (Guillemin and Pollack while a really good book, will not help you here). This will clear most of the language problems you are currently facing. But, critically, you need an advisor to point you in the right direction since there will be no general-purpose courses helping with control theory beyond that point.
Edit. What you really need for CT is "sub-Finslerian differential geometry," which is a combination of the theory of general (typically non-integrable) distributions and Finsler metrics defined on such distributions. A Riemannian Geometry class will teach you about Riemannian metrics. The formalism of Finsler metrics is similar, but the technicalities are much harder. For sub-Riemannian geometries, take a look here:
Montgomery, Richard, A tour of subriemannian geometries, their geodesics and applications, Mathematical Surveys and Monographs 91. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-1391-9/hbk). xx, 259 p. (2002). ZBL1044.53022.
For Finsler geometry, the standard reference is
Bao, D.; Chern, S.-S.; Shen, Z., An introduction to Riemann-Finsler geometry, Graduate Texts in Mathematics. 200. New York, NY: Springer. xx, 431 p. (2000). ZBL0954.53001.
For sub-Finslerian geometry, you have to read research papers, there are no textbook treatments.
