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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.

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    $\begingroup$ A good next textbook for you might be Differential Topology by Guillemin and Pollack. $\endgroup$ – Neal Jul 15 '20 at 14:43
  • $\begingroup$ What are the prerequisites? $\endgroup$ – abc1455 Jul 15 '20 at 14:49
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    $\begingroup$ @abc1455 how much does your "complex analysis" course do? For example, does it cover open mapping theorem and Riemann mapping theorem? (in other words, we are trying to gauge how much topology and differential geometry is thrown into the mix). You probably also want some exposure to algebraic topology, e.g., how far can you get into reading Bott&Tu? $\endgroup$ – user10354138 Jul 15 '20 at 14:58
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For control theory, given your background, you should start with

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

  2. 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.

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    $\begingroup$ I mostly agree here, but I think that a better path might be to take "curves and surfaces" first, so that when you get to more general differential geometry, you have the touchstone of "how it was done for surfaces" to refer back to. Barrett O'Neil's book does a nice job of setting you up for this, I find, although opinions will differ, of course. $\endgroup$ – John Hughes Jul 15 '20 at 16:00
  • $\begingroup$ @JohnHughes: It depends how much time/money (to pay for the class) one has. Yes, knowing how differential geometry works for curves and surfaces is useful for developing intuition (and, as a bonus probably proving Gauss-Bonnet which the more advanced class is likely to skip), but the drawback is spending a month or two on staff never used in control theory. $\endgroup$ – Moishe Kohan Jul 15 '20 at 16:54
  • $\begingroup$ @MoisheKohan: Do you have a recommendation for point-set topology? $\endgroup$ – abc1455 Jul 15 '20 at 17:25
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    $\begingroup$ @abc1455: Munkres' book is quite standard, but many other books would do just fine. What you need from it is: Open sets, closed sets, boundary, subspace topology, continuity, compactness, connected sets, Hausdorfness. $\endgroup$ – Moishe Kohan Jul 15 '20 at 17:34
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    $\begingroup$ Another (freely available) general topology textbook is "Topology without tears," topologywithouttears.net/topbook.pdf. It is slower than Munkres' though. $\endgroup$ – Moishe Kohan Jul 15 '20 at 18:53

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