This post will mainly be about how to crash study for the "additional topics".

As a background, the topics I had to study from scratch are

topology, real analysis, complex analysis, and a little bit of abstract algebra.

I used to be an IMO silver medalist, but my major is not math, so I didn't take any classes in these.

I am a very fast problem solver so I had no problem with calculus/linear algebra stuff.

Before I start, some general advice on the exam

1)The problems got a lot harder recently.

The penalty for guessing wrong has disappeared, so the problems have become a lot harder than the practice tests. The top score between 2014-2017 was a 940, I was able to top that simply because the problems got harder.

2)The problems are different for different people even on the same day.

I found this out from this forum. Someone was talking about a problem involving something, which only some people had seen. I did not see this problem on the test, so I knew that my test was a different one, even on the same day.

3)The main source to study from is the Princeton Review book.

The questions are a lot easier than the recent tests, but it is a very good book to brush up or learn your weaknesses in a very short time.

+ The problem sets you absolutely need to do are the 5 official practice tests. (See the link below. those are the ones that say "0568" "1268". Means they were the 2005, 2012 test problems )

+ Rambo's test is nice.

+ I also did the REA tests. They were harder and I think are a pretty good representation of the tests these days. Note that REA has some weird problems that are unlikely to be helpful, so you should ignore these problems.

4)Every problem set(even the REA) you need is here

https://math.stackexchange.com/question ... oks-advice

5)This test is mostly about solving problems fast. Problems are computationally extensive so you need to use tricks or you are unable to solve them in time.

6)This is why timing your practice exams is essential to getting a good score.

In the real test you will have less time as the problems are a lot harder these days.

Always time your practice tests!

7)Most of your score will be correlated to your ability to solve calculus and linear algebra problems fast. I see the additional topics section as an easy way to get points. The problems are very stereotyped, they do not require calculations but rely more on intuition.

None of this has anything to do with math research, so don’t be too disappointed even if you don’t get a good score. Personally I think it’s a really bad test and a way for ETS to make money.

9)Because there are only 5 answers, sometimes it is possible to solve the problems faster by checking all 5 answers.

DISCLAIMER: I don’t claim this is an exhaustive list. I have tried to include the stuff I think is important.

The link viewtopic.php?t=3303 has a lot of good suggestions for texts

for non-math students i think worth checking out.

A. Topology

This was the section that gave me the most trouble.

For me the Princeton review + practice tests were not enough, so I had to look through the Munkres topology book.

Luckily you don’t have to go through that much. The test is stereotyped and only tests on a few things.

Problems are only from “Point set topology” (Part I- General Topology of Munkres) and from that book only up to Chapter 3. Connected and compactness.

But you don’t need to know everything even till chapter 3. Stuff like Metric topology, limit point compactness, local compactness doesn’t come up.

Stick to what I say below and I think you will largely be safe.

Given a topology, you need to be able to tell if a set is A)open B)closed C)connected(path connected) D)compact

You also need to be able to tell if a function is A)continuous B)open C)homeomorphic

You need to have a good intuition behind all these concepts and also the concept of limit points, boundary, closure. The concept of Hausdorff also seems to come up in practice tests.

This is more difficult than it sounds. You need to be able to answer questions like

Can a closed set be open?

What are the open sets in a lower limit topology with basis [a,b)

What are the closed sets in the discrete topology?

The topologies given are all subsets of R^n.

You need to know cold what kind of sets are open/closed/connected/compact in the discrete topology, standard topology, lower limit topology, k-topology, finite complement topologies. This will give you a good intuition. I recommend going through the munkres book.

For example, the only connected sets in [a,b) lower limit topology are singletons and the null set. You need to be able to do stuff like this.

Also pay attention to the weird counterexamples, especially the topologist’s sine curve.

Final tips – these are useful criterions in solving problems.

1) To test if a subset O is open, prove that for all o in O exists a neighborhood of o(open set containing o) that is contained in O.

2) To test if a set O is closed : 1)if it contains all its limit points 2)if its complement is open.

3) To test if a set is connected > test using the definition.

4) To test if a set is compact > for the standard topology this is equivalent to the set being closed and bounded. (Heine-borel theorem)

5) Another way to approach testing if a set is open/closed is by using the basis topology. Also useful in tests.

Note that topological subspaces do not preserve connected or compactness

Continuous functions preserve 1)connected sets, 2)compact sets 3)convergent sequences

Homeomorphisms preserve connected, compact, open, closed sets

B. Abstract algebra

http://www.math.colostate.edu/~pries/467/Judson12.pdf

This link I found helpful. Don't study everything but after trying out two or three practice tests you will know which parts are relevant.

Get a sheet of paper and put down the relationships between

Binary structures, semigroups, monoids, groups,

Rings, ring with unity, division ring, field, commutative ring, integral domain, and fields.

If you can draw the relationship between all of them, you can memorize their definitions.

Know the examples cold – (Z, +), (Q*,X)

Nonabelian groups -permutation group, alternating group, dihedral group

Cyclic groups. The examples of A) Quaternion B)Klein 4 group is especially relevant. Find out why.

Other topics you need to know cold are

Finite abelian groups – how many are there of order 600?

As a tip if its p_1^e_1*p_2^e_2…p_k^e_k the answer is given by the product partition(e_1)*..partition(e_k)

Sylow thm (with p^k)

Lagrange’s theorem, cosets.

Normal subgroups and ideal of ring is related. This is a very popular topic.

Determining whether something is a group/ring homomorphism. Pay attention to the frobenius homomorphism.

Many of the problems ask if something is a group, ring, or a field. You should be able to check the definitions.

One tip – a quick way to check whether X is a group is by checking whether for a,b in X a*b^-1 is also in X. Similar criterion like this exists for a lot of other concepts(like homomorphisms), and are useful for the test.

C. Real Analysis

Also conceptually difficult.

You should be able to apply/understand epsilon-delta methods.

Riemann integrability, Lebesgue integrability, Weierstrass thm (compactness), pseudocompactness, dominated convergence theorem, uniform limit theorem

Pointwise convergence vs uniform convergence,

Lipschitz continuit

Know the hierarchy – continuously differentiable< lipschitz< uniformly continuous < continuous.

Any Lipschitz continuous function on compact set is differentiable almost anywhere.

The weierstrass function is continuous everywhere but differentiable nowhere. Important counterexample to keep in mind.

D. Complex analysis

By far the most stereotyped section. Only two types of questions.

A) Calculating a complex line integral for a closed curve

You need to be able to apply the residue theorem.

To calculate a residue, you can either do a taylor expansion, or

You can use the formula for the residue that’s given by the limit (it’s in Princeton review)

B) When is a function differentiable in the complex domain? > Cauchy Riemann condition

Memorize this along with the green theorem in calculus, and the integrating factor of differential equations. I kept confusing the three so worth memorizing the three at the same time.

E. Combinatorics

Enumeration problems mostly.

A graph theory problem may come up.

I knew this stuff so I really can’t give you much advice on this. If you’ve never done combinatorics probably not worth doing in depth. Just practice on enumeration problems a lot.

F. Algorithms(Programming Problems)

I think there is one programming problem every test.

If you have done programming you can solve this, if you haven’t probably not worth the time to learn for one problem.

*I do think programming is something everyone should learn, but not if you are learning it for this test.

G. Logic theory/Set theory

These should be easy to solve.

Basic stuff like p>q is equivalent to not q > not p

You also need to know what the negation of statements like “for some X, there exists Y so that Z”. will be.

Also know how to compare Cardinal numbers. You should be able to compare Z^Z, R^Z, R^Q, R^R, 2^R.

Other tricks I found useful for the calculus/linear algebra section.

1) checking if a point of a multivariable function is a 1)local maximum 2)local minimum 3)saddle point.

In general you calculate whether the hessian is positive definite or negative definite

For the tests often it’s a bivariate (n=2) function.

In that case you only need to see determinant of the hessian. If this is <0 it’s a saddle point, else it’s a local maximum or minimum

2) The distance between point (a,b,c) and plane dx+ey+fz+g=0

Is given by (ad+be+cf+g) / sqrt(d^2+e^2+f^2). huge time saver.

3) For two n*n matrices AB=I implies BA=I. This looks false but is true so you should know this.

4) Is a Diagonalizable matrixe invertible? > No. All 4 combinations of (diagonalizable?,invertible?) are possible.

5) The gaussian integral of e^(-x^2) from -infinity ~infinity is given by sqrt(pi)

You should know to calculate this using polar coordinates. Look it up.

6) Learn how to use vector cross products to calculate area of triangle in three dimensional coordinates.

I hope this is helpful!