Given Friday, February 22nd, 2019.
 Difference between General GRE and Subject GRE
 Content of Math section of general GRE (Quantitative portion):
 Euclidean Geometry, basic Algebra (exponents, factoring, etc), Arithmetic (Estimation, ratios), Data Analysis (Graphs, tables)
 Essentially everything before a Calculus class
 Content of Subject GRE:
 50% Calculus (Single and Multivariable, ODEs)
 25% Algebra (Linear Algebra, Abstract Algebra, ‘Number Theory’)
 25% Mixed upper division stuff (Real Analysis, Combinatorics, Probability, PointSet Topology)
 Purpose of the exam
 Who should take it, when, and why
Logistics and Technicalities
 Usually offered 3 times per year
 Around March, early September, and late October
 Grad apps are usually due in early December, so you have 23 chances
 Registration deadlines are usually 12 months before actual test date.
 Might have to travel to testing center: for San Diego, SDSU.
 Everything is done online through ets.org
 Cost:
 About $200 per exam
 Fee waivers available that cut it down to $100 per exam
 Easy to qualify, but requires getting form from website, getting some verification paperwork from financial aid
 Takes at least 6 weeks to process, if not longer. Start early!
 Structure:
 Multiple choice (5 choices)
 66 questions
 170 minutes (2h 50m)
 Leaving around 154 seconds (~2.5 minutes) per problem
 Final score: simply a sum of how many correct answers you have. Given as percentile based on cohort taking the exam on the same date.
 Exam Day
 Must bring photo ID. Don’t forget!!
 Not able to bring phone, backup, etc.
 Must have confirmation/registration number or printed copy of email they send you near the exam date.
 No restroom breaks!!

Studying
 Start early. Like really early, like at least 6 months before the exam.
 Spaced repetition:
 There are plenty of definitions, formulas and equations that are just worth memorizing outright:
 Examples: The definition of a connected space, $\frac{d}{dx} \sin(x)$, $\int x^n dx$, solutions to ODEs of the form $ay'' + by' + cy = 0$, etc
 But it’s also helpful to have some methods and proofs memorized
 Examples: Techniques for solving the nonlinear systems arising from Lagrange multipliers, the proof the differentiability implies continuity, etc
 Flashcard programs like Anki are great for this, also just solving problems and then revisiting them regularly is perfectly sufficient to get them into your working memory.
 A recommendation  just pick up Stewart’s Calculus and go through the entire thing. 50% of the exam is Calculus, so this pays off! Use the chapter reviews as a diagnostic, use a solutions manual or Wolfram to check your answers, and drill into sections that you’re weak in.

Speed
 With 2.5 minutes per question, being quick is absolutely vital.
 Quickly solving easy problems gives you more time on difficult problems, or time double check answers and/or be more careful on tricky problems.
 Benchmark: aim to be able to do most Calculus problems in <1 minute.
 Simple limits, derivatives, or integrals should be on the order of seconds at most.

Guessing:
 Blank answers are wrong answers, so blindly guessing has higher expected value than not answering at all.
 But note that because there are 5 choices, blindly guessing still has negative expected value!
 So to break even (zero expected value), you need to eliminate at least one incorrect choice from each question.

Choice elimination
 It is tempting to thoroughly solve each problem that you are given, but this is not always necessary.
 Example: if you’re asked to compute an indefinite integral, it may be faster to just take the derivative of all 5 answers to see which matches the integrand.
 Use the multiple choice format to your advantage – they often have hints for what constants might appear in the answer, or hint at a technique to use.
 Similar strategy to general GRE reading sections – look at answer choices before reading question, so you’ve already framed what you’re looking for
 If you eliminate 4 choices with 100% confidence, then the answer must be the remaining choice. No need to check!

Practice Exams
 There are 5 official exams out there (see reference section) with solutions.
 Take all of them! Time yourself! All include scoring rubric to compute your percentile, so score yourself too.
 Note: 3 hour exams can be mentally exhausting if you are not used to them. Practicing is essential to get your mental stamina up.
 Note that all the released exams are significantly easier than the actual exams.
 Aim to complete practice exams within 2 hours at around the 90th percentile.

Self Care
 Applies to every exam  taking care of your mind and body pays dividends!
 No allnighters, no cramming  getting an extra few hours of sleep is immensely more beneficial than trying get a few more formulas tucked into your shortterm memory.
 Eat relatively healthy before the exam. Be sure to have breakfast the morning of the exam too!
 Stay hydrated, drink water (but not too much – remember, no restroom breaks during the exam)
 Don’t stress too much  this exam isn’t a barrier, it is a chance to revisit a great deal of fun mathematics and show grad schools how far you’ve come and how much you’ve learned during your degree!
 Note: I think graduate schools mostly just want to see that you don’t totally tank this exam, because in many cases, grad students will be TAing the very courses covered in this exam. Also, remember that there are many grad programs that do not require the exam at all!
Some unsolicited advice: Calculus I takes up ~25% and Calculus II, III takes up another ~25%. Calculus I should be doable. The integrals will usually yield to tricks and not brute force e.g. rationalizing denominator, factorization, partial fractions, and even symmetry. Know your small angle approximations! Calculus II and III may seem scary, but it isn’t. Expect simple Lagrange multipliers, arc length of curves, Green’s theorem, divergence theorem. (Stokes’ is unlikely.) For complex analysis, the Cauchy integral formula and residue theorem are sufficient (~3 questions). For differential equations, refer to the chapter in Princeton Review (~3 questions).
Linear algebra takes up about 10%. Abstract algebra takes up another 510%. The scope is quite varied. Questions on the Jordan canonical form have appeared before, but don’t waste your time if you have not learnt it before. Focus on systems of linear equations. Be familiar with the invertibility conditions and consequences. Be familiar with the properties of det and tr and how to compute them. The abstract algebra questions will cover groups, rings and fields. Preparing for this is not like preparing for calculus – I feel that you should just memorize the basic definitions and results and let your skill do the rest in the exam.
The remaining ~35% of questions do not belong to any of the above categories. The topics covered include basic number theory (congruences, Fermat’s Little Theorem), plane geometry (high school level), polynomials, foundations (functions, relations, orders). My exam had a problem on CauchySchwarz.
A few other topics in the “Others” category: Combinatorics. Permutations, combinations, inclusionexclusion. Derangementlike questions may appear. Statistics. Mine had a question on the standard deviation of the addition of two normal distributions. Topology. Arguably the hardest. If you have taken a topology class then go through the basic definitions, results, and proofs of some of the results. The rest depends on your skill.
PreCalculus and Proof Fundamentals
Series and Sequences
 Common Series (geometric, harmonic, $p$)
 Convergence Tests (integral, ratio, root, etc)
Approximation and Optimization
 Linear approximation
$$f(p) \approx f(p) + f'(p)(xp) + f''(p)(xa)^2 + o(x^3)$$
$$f(\vector p) \approx f(\vector p) + \nabla f(\vector p)(\vector x  \vector a) + (\vector x  \vector p )^T H_f(p)(\vector x  \vector p) + o(\norm{\vector x  \vector p}^3)$$
 Single variable
 Multivariable
 Eigenvalues of Hessian
 Negative definite: Min
 Positive definite: Max
 Any equal to 0: Inconclusive
 Lagrange Multipliers
 $\nabla f(\vector x) = \lambda \nabla g(\vector x)$
 Separable, linear up to 2nd order, homogeneous and otherwise
 Systems of differential equations
 The Wronskian
 Fourier and Laplace Transforms
 Systems of Linear Equations
 Number of possible solutions
 Rowreducing algorithm / Gaussian Elimination / RREF
 Properties of determinant and trace
 Computing nullspace, rowspace, columnspace
 Finding eigenvalues and the eigenspace
 Jordan Canonical Form
 Conditions for invertibility
 Complex roots and branch cuts
 Complex limits and the complex derivative
 Cauchy Integral Formula
 The Residue Theorem
 Intermediate Value Theorem and Mean Value Theorem
 Least upper bound / Supremum and Greatest lower bound / infimum
 Epsilondelta proofs
 Uniform and pointwise continuity
 Metrics and Metric Spaces
 The CauchySchwarz Inequality
 Definitions of Sequences and Series
 Testing Convergence of a Series:
 Integral Test
 Ratio Test
 Root Test
 $p$ Test
 Cauchy Sequences
Tricks section"
 Definition of a Topology

 Topology: Arbitrary unions and finite intersections
 Connected
 Disconnected
 Totally Disconnected
 Weird topologies
 Prime decomposition
 Divisibility
 Modular congruences
 Euler’s Totient function
 Fermat’s Little Theorem
 The Chinese Remainder Theorem
 Groups
 Vocabulary: homomorphisms, orders, centralizer, normal subgroup, etc
 Division Rings, Integral Domains
 Classification of finite simple abelian groups
 e.g. count number of unique groups of order $n$
 The Cyclic, Symmetric, and Dihedral Groups
 Lagrange’s Theorem (orders of subgroups)
 The Sylow Theorems
 Rings
 Fields
 Unique Factorization Domains
 Principal Ideal Domains
 Division Rings
 The Chinese Remainder Theorem
 12fold counting method
 Stars and Bars
 Permutations
 Combinations
 Inclusions/Exclusions
 Derangements
 The Symmetric Groups
 Integer Partitions
 Common Distributions
 Bernoulli
 Binomial
 Geometric
 Exponential
 Mean / Expected Value / Variance / Standard Deviation
 Density functions
 The normal distribution
 Normal Rule: $68/95/99.7\%$ within $1/2/3 \sigma$
 Normal approximations (e.g. to binomial)
 Newton’s Method
 Euler’s Method
 Quadrature
 Commuting differentials and integrals:
$$\frac{d}{dx} \int_{a(x)}^{b(x)} f(x,t) dt = f(x, b(x))\frac{d}{dx}b(x)  f(x, a(x))\frac{d}{dx}a(x) + \int_{a(x)}^{b(x)} \frac{\partial}{\partial x} f(x, t) dt$$
 Need $f, f'$ to be continuous in both $x$ and $t$. Also need $a(x),b(x) \in C_1$.
 If $a,b$ are constant, boundary terms vanish.
 Recover the fundamental theorem with $a(x) = a, b(x) = b, f(x,t) = f(t)$.