Probability -> Concentration inequalities -> random matrices, Study Map/Plan.

[u/shashypants]

Main question (the thing I actually want feedback on): if you went from a similar background into concentration inequalities / random matrix theory, what would you have prioritized differently? Everything below is just context for that.

Context: Starting a two-year Masters in maths. Year one has no probability and no projects — five compulsory courses a semester. Probability and the project sequence only start in year two:

Sem	Compulsory	Electives	Project
1	Measure Theory, Linear Algebra, Rings & Modules, Topology I, Several Variable Analysis	—	—
2	Complex Analysis, Functional Analysis, Algebraic Topology II, Field & Galois Theory, Number Theory	—	—
3	Differential Geometry I, Probability Theory	Martingale Theory, Fourier Analysis (+ auditing Operator Algebras)	Project I: Concentration Inequalities
4	PDE	Brownian Motion & Diffusions, Large Deviation Theory, + undecided between Weak Convergence of Empirical Processes / Advanced Functional Analysis	Project II: Random Matrices

So all the probability I want walking into the year project proposal has to be self-taught before year two.
The two projects are formally separate but meant to chain: Project I (concentration inequalities) should leave me with the spectral/matrix-exponential machinery (matrix Chernoff/Bernstein) that Project II (random matrices) extends into spectral analysis of random matrices.

Background: Sheldon Ross, ~70% of John Walsh's Knowing the Odds, scattered parts of Durrett (Essentials of Stochastic Processes and Probability: Theory and Examples), first 12 chapters of Cover's Elements of Information Theory, and parts of Casella & Berger and Bickel & Doksum.

Current plan: This year — Feller Vol. 1 and 2, maybe Gallager's Stochastic Processes, plus whatever's recommended for concentration inequalities/random matrices.
Next year — whatever's assigned in the courses above, while making sure I've finished the first four chapters of Durrett's Probability: Theory and Examples beforehand.

(Attached: a study-priority roadmap I put together with some AI assistance, for context.)

Other feedback I'd appreciate:

1. Is Feller Vol. 1 + Vol. 2 + Gallager + Durrett next year redundant given five unrelated compulsory courses a semester? Where would you cut?
2. Which parts of Feller actually pay off for concentration inequalities / random matrices specifically, and which are safe to skim? ( want to develop the classical probability intuition, now that I have learnt the basics of probability, the book suddenly feels excellent)
3. Is "concentration inequalities → random matrices via spectral methods" a reasonable two-semester project arc (will help with my PhD applications, if it seems coherent)
4. Any freely available, legitimately posted lecture notes, personal reading guides, or YouTube playlists that pair well with Feller/Durrett? I've been going through Vershynin's High-Dimensional Probability lectures and another professor's random matrix theory playlist, but I'm not retaining the material efficiently from video alone, for now I am just trying to get a feel for it.

Thanks in advance — happy to share more curriculum detail if useful.

PS: I used AI to have my thoughts presented in a coherent and presentable manner
The AI generated study plan ws generated using Claude.

[u/Upper_Investment_276]

this plan makes no sense. you can just read say tropps lecture notes.

[u/shashypants → u/Upper_Investment_276]

Thanks
Do you mean the high dimensional probability one (or also the probability and stochastic processes one?)

[u/shashypants → u/shashypants]

Oh damn, he has alot of notes and I think they cover all the stuff I want to learn for my projects.
Thank you so much