I am considering taking a PhD position in stochastic stability and control within the mechanical engineering department at a university which has offered me a lucrative RA. In discussing my topic with my potential advisor he gave me a paper to read this week. After I read the paper it was clear that I need to develop a background in pure mathematics (I have taken an introduction to proofs course as well as additional math classes in probability and nonlinear differential equations). From the paper I made a list of unfamiliar concepts. I am going to list them below. I am asking for help in identifying the subjects which these concepts span and books which might cover said topics. And as a bonus, if someone could give me a shoet explanation of any of the concepts or how they are related that would be very helpful. The list is below: 1) SO(3) (rotation group SO(3)) - the set of all rotation in R3? 2) Tangent Bundle (used in the context of Tangent bundle on SO(3)) - set of all tangent vectors to SO(3)? 3) unwinding phenomena - in the context of continuous feedback control, im guess this is a type of instability? 4) frobenius norm - I read a chapter about various norms, but what makes this one special vs an L2 norm (or any other norm for that matter), for example. My understanding is that norms are a measure of how similar two matricies are, but how is this measure significant? 5) Ito formula - I know this is from stochastic calculus. Wikipedia says this is the "stochastic chain rule" 6) LMI theory - I believe this is "linear matrix inequalities theory", but what does the theory state? 7) Lie Algebra and/or Lie groups - I dont know anything about Lie algebra 8) wong zakai correction - all I know is that this is related to SDEs 9) class κ function - I know this is used to check stability of a function in control theory, but other than that I am clueless sadly 10) Gain matrix - again, from control theory, but I know nothing about it 11) martingale/supermartingale - from stochastic calculus. I know this is basically a model for which the probability of a current event cannot be determined from the outcomes of past events 12) manifolds - basically a region which can be approximated as euclidean over small dimensions, but is not euclidean globally. I am not sure how these are used in applied math or control theory 13) morse-lyapunov function - I know what a lyapunov function is (used for stability analysis in nonlinear DEs) 14) LaSalles theorem - Ive never seen this before but wikipedia says it is another tool in stability analysis 15) Lie Group Variational Integrator - no idea, never seen before Obviously I have a lot to learn. From this list I can see that these topics cover control theory and stochastic calculus, but it is my understanding that before learning stochastic calculus one should learn functional analysis or variational calculus. My background is in aeronautics, specifically fluid mechanics. In providing this list my hope is that someone who has a better mathematical background than myself can point me to materials or subjects so I may get up to speed on my topic. I also know nothing about lie algebra and am not sure where that fits-in in terms of stochastic stability and control.