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I'm just starting my graduate studies in Analysis and PDE's and am a bit lost about what topics should I cover in order to do a good Phd program. I`ve already done the usual undergrad courses, plus Real and complex analysis (graduate level), functional analysis and measure theory.

So, if you guys can recommend me which courses I should do, (I can get my university to open new courses as needed), and which books I should study, it'd make me really happy

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Some interesting courses that can be done with a standard PDE course: (with exemplary lecture notes so you can have a look into these)

And some Analysis courses:

  • Fourier Analysis
    1. Laplace Transform
    2. Fourier Series
    3. Fourier Transform
    4. Schwartz Functions
  • Distribution Theory
    1. Distributions
    2. Tempered Distributions
    3. Distributions with compact support
  • Dynamical Systems
    1. Linear Systems and Stability
    2. Nonlinear Systems and Stability
    3. Bifurcation Theory
    4. Chaos Theory
  • Differential Forms
    1. Differential Forms: Definition
    2. Hodge Star Operator
    3. Lemma of Poincare
    4. Stokes' Theorem
  • Nonlinear Functional Analysis
    1. Analysis in Banach Spaces
    2. Brouwer Mapping Degree
    3. Leray-Schauder Mapping Degree
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  • $\begingroup$ Nice, do you have any ideas about analysis courses also? The lecture notes are really welcomed! $\endgroup$ – David Stolnicki Feb 12 '17 at 21:24
  • $\begingroup$ @DavidStolnicki I updated some analysis courses. I hope it helps you. $\endgroup$ – Fritz Feb 12 '17 at 21:42
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Graduate level course in Complex Analysis, Real Analysis and PDE's which usually cover the following textbooks:

  1. Complex Analysis by Lars Ahlfors
  2. Complex Analysis by Elias M. Stein & Rami Shakarchi
  3. Real Analysis: Measure Theory, Integration, and Hilbert Spaces by Elias M. Stein & Rami Shakarchi
  4. Real Analysis: Modern Techniques and Their Applications by Gerald B. Folland
  5. Partial Differential Equations by Lawrence C. Evans

Then a graduate level course in Functional Analysis.

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    $\begingroup$ +1 for Evans, fantastic book. If you want to go more into Functional Analysis you can also try 'Functional Analysis, Sobolev Spaces and Partial Differential Equations' by H. Brezis $\endgroup$ – Fritz Feb 12 '17 at 17:23
  • $\begingroup$ That was the book I used for functional analysis. Great book! $\endgroup$ – David Stolnicki Feb 12 '17 at 17:36
  • $\begingroup$ I've gone through most of those books, any idea about more advanced courses? $\endgroup$ – David Stolnicki Feb 13 '17 at 1:39
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Fixed Point Theory is an important part of analysis to cover. And if you want to mix analysis with a little bit geometry, you MUST check the two brilliant books by I. Chavel: eigenvalues in riemannian geometry and isoperimetric inequalities. They do reveal beatiful applications of PDE's to geometric problems.

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