Advice for the Analysis Qual

tips + advice for the analysis qualifying exam at UW Madison.

First and foremost, I strongly implore you to highly familiarize yourself with the following webpages:

Everything you (objectively) need to know can be found on those pages. The goal of this webpage is to supplement already existing references for analysis qual preparation with my own suggestions/experiences.

Both the real and complex exams are structured similarly: the first three questions are “basic” undergraduate analysis (MATH 521/522), then three questions come from the first semester of measure theory/real analysis (MATH 721), and the final three questions cover topics in complex analysis (MATH 722) or second semester real analysis (MATH 725). Several people very generously advised/helped me prepare for this exam, so I would like to “give back” by sharing what I believed helped me perform well.

General Advice

There are a lot of definitions and important theorems that you need to know for either version of the exam, and several of them have nuianced details that are easy to conveniently forget under the pressure of taking an exam. I ended up using a software called Anki to make digital flashcards of definitions/theorems. Physical flashcards work as well. If you need ideas on what to write on flashcards, another graduate student, Allison Byars, compiled a list of what definitions/theorems she made flash cards of when preparing for the real version. Moreover, I found it immensely helpful to also include notes on what kinds of problems a particular theorem is often used for. For example, consider problem 2 from the January 2023 exam:

Can one find a bounded sequence of real numbers \(\{x_n\}\), \(n \in \mathbb{Z}\) that satisfies \(x_n = \sin n + 0.5 x_{n-1} + 0.4 \sin(x_{n+1})\) for every \(n \in \mathbb{Z}\)?

At first glance, this problem looks super intimidating and one might be tempted to do a nasty brute force computation, but this is actually a straight-forward application of the Banach fixed-point theorem - recall that the space of real-valued bounded sequences \(l^\infty(\mathbb{R})\) is Banach, and define the operator

\[T : l^\infty(\mathbb R) \rightarrow l^\infty(\mathbb R) : \{x_n\} \mapsto \{ \sin n + 0.5 x_{n-1} + 0.4 \sin(x_{n+1}) \}.\]

After a couple of straightforward computations, we see \(T\) is well-defined (i.e., \(Tx_n \in l^\infty(\mathbb R)\)) and is also a contraction map. By the Banach fixed-point theorem, there exists a unique sequence for which \(T x_n = x_n\), proving the result. So, on the flashcard for the Banach fixed-point theorem, I noted that this is a useful for any questions involving the existence of a function or sequence satisfying some kind of a recursion; sometimes it is phrased in an obvious way like this problem was, other times it is not (see exam 8.2020.1).

Obviously, that was not a complete solution, which brings me to my next point: a significant part of the exam is assessing your ability to not only come up with an “idea” for the solution, but to also write a complete, coherent, self-contained proof. I recommmend fully writing out detailed solutions to problems rather than just scribbling notes on the “general idea” riddled with (probably incorrectly used) jargon. It took me maybe a minute to type up the “gist” of how to solve the problem, but if you work out the computation for showing \(T\) is a contraction, then you might notice a small problem: how does one estimate \(0.4 \| \sin x_{n+1} - \sin y_{n+1}\|_\infty\) in a way that introduces \(\| x_{n+1} - y_{n+1} \|_\infty\)?

For many of the problems, one can find solutions/hints on the World Wide Web. There is also a lovely document containing several solutions to questions from the real analysis exam written by Max Bacharach and Jacob Denson. Solutions are a helpful reference if you are genuinely stuck, but the worst thing you can do is being in the routine of just solution hunting and attempting to memorize everything rather than affording yourself an opportunity to conjure up ideas on your own. Sometimes questions are repeated; for example, proving the continuity of \(f(r) = m(E \cap (E+r))\) where \(E\subset \mathbb R\) is measurable and \(m(E)<\infty\), showed up a few times and on the August 2023 exam!

However, it is extremely foolish to rely on memorizing solutions to problems. There are simply too many different questions that could be asked, slight modifications to the question often changes the proof, and there is also the possibility that majority of the questions that appear are new to you. Ignoring the issue of other people’s solutions (infinitely) often will not be 100% correct or complete, your time is better spent understanding why certain techniques work and when to apply them.

For example, if you go through the old quals, then you will realize fairly quickly that the exam writing committee seems to have a neverending supply of exercises that call for using the Lebesgue Differentiation Theorem itself, or replicating the proof of it to prove a similar result for some other integral operator. The same goes for problems applying the Banach fixed-point and Arzelà-Ascoli theorems. You truly are better off learning when to recognize applications of these results, rather than memorizing the solutions to a small subset of potential questions.

Complex Analysis Advice

You can read the aforementioned linked documents about the real analysis qualifying exam. The SEP that happens during the summer also usually covers the real version of the qual. For complex analysis:

  • Know the residue theorem and contour integration! I’m pretty confident that the set of exams that didn’t ask something involving either has (counting) measure zero. Read through your favorite undergraduate textbook in complex analysis if you struggle with identifying contours.
  • The exam writers really like to ask questions that use Schwarz’s Lemma or Rouche’s Theorem.
    • If you want to use the Schwarz-Pick Lemma, you should rederive it using Schwarz Lemma; if \(f : \mathbb{D} \rightarrow \mathbb{D}\) with \(f(a)=b\), then look at \(g = \phi_{-b} \circ f \circ \phi_a\) where \(\phi_{-b} = (z+b)/(1+\overline{b} z)\) and \(\phi_{a} = (z-a)/(1-\overline{a} z)\) are, of course, automorphisms of the unit disc. Clearly \(g(0)=0\). Then the result follows Schwarz Lemma.
    • Make sure to know all versions of Rouche’s theorem (in reality they can be obtained by taking linear combinations of your functions).
    • Sometimes, you have to apply Rouche’s theorem more than once - try to use functions whose zeros are easy to identify when choosing your “comparison” functions.
    • The problems on Schwarz’s Lemma are usually asking about the existence of holomorphic functions on the unit disk that satisfy some random properties. Schwarz’s Lemma says, “yeah, sure, it could happen!” But there’s a catch: you need to be able to identify said holomorphic function. You can basically just use the conditions imposed on your function along with all of those charming theorems about analytic maps of the unit disc (Blaschke products, etc…) to construct the example, or prove that such a function cannot exist by contradiction. For example, see problems 8.2020.9C, 8.2018.7C, and 1.2018.8C.
  • Know Cauchy’s integral formula! Including the version for \(n\)-th order derivatives. It’s used for proving, like, 90% of the theorems in complex analysis. When in doubt, see if you can use this formula to prove what you want to show.
  • There will often be questions about proving some result about a seemingly random region. Usually, you just need to define a conformal map from the region to the unit disk and then the problem becomes super straight-forward to answer. For example, see problems 1.2023.7C and 8.2018.8C.
  • Know the maximum modulus principle and its proof. You often need it for proving an intermediary result as part of the actual thing you want to show.
  • Similar to the real analysis qualifying exam, a lot of the problems asked are supposed to be “corollaries” of the major named theorems. For example, 8.2023.9C is supposed to be an obvious consequence of Hurwitz’s theorem (… which I did not realize while in the exam. Oops.).

Final Words

The analysis qual is objectively hard. You need to know a lot of information, and to an extent there is an element of getting lucky with what questions the writers decide to ask that year. I would like to emphasize that it is not the end of the world if you don’t pass it your first time; many people have to retake it several times, including some of our colleagues doing some great work in analysis, PDEs, and probability. This exam is not an absolute metric of your ability to do analysis or math.

Regardless of where in the world you are, there are some extremely unpleasant people who will try making you feel bad because you didn’t already know of XYZ theorem, or you didn’t pass the exam on your first try, or you didn’t get as high of a score on it as they did, or you only passed one qual while they passed both analysis and ABC qual. Please don’t let them get to you; majority of the time, they are projecting their own insecurities by acting like a math bully.

As a concluding remark, I would like to illuminate the fact that most of this advice is not unique to the analysis qualifying exam: part of becoming an “expert” in any subfield of mathematics involves familiarizing yourself with and eventually mastering the tricks of the trade. I promise this stuff gets easier with time and (perhaps exorbitant amount of) practice.