Mathematics Subject GRE Workshop

Agenda

  • Description of Mathematics Subject GRE
  • Topics it covers
  • Exam logistics
  • Recommended resources
  • Study techniques/tips
  • Review of topics + sample problems

What is the Mathematics Subject GRE?

  • Different from the Math section of the General GRE
  • Required of graduate student applicants to many Math Ph.D. programs
  • Tests a breadth of undergraduate topics

Topics

  • Calculus (50%)
    • Single Variable
    • Multivariable
    • Differential Equations

calculus

“Algebra” (25%)

  • Linear Algebra
  • Abstract Algebra
  • Number Theory

algebra

Mixed Topics (25%)

  • Real Analysis
  • Logic / Set Theory
  • Discrete Mathematics
  • Point-Set Topology
  • Complex Analysis
  • Combinatorics
  • Probability

Logistics

  • Multiple choice, 5 choices
  • 66 questions, 170 minutes
  • No downside to guessing
  • Only offered 3x/year
  • Need to register ~2 months in advance

References

Garrity, All the Mathematics You Missed (But Need to Know for Graduate School)

Good high-level overview of undergrad topics.

The Princeton Review, Cracking the Math GRE Subject Test

“Calculus: The Greatest Hits”, good breadth.

Shallow treatment of Algebra, Real Analysis, Topology, Number Theory.

Five Official Practice Exams (with Solutions)

  • GR 1268
  • GR 0568
  • GR 9367
  • GR 8767
  • GR 9768

All old and significantly easier than exams in recent years.

Aim for 90th percentile in $< 2$ hours.

General Tips

Math-Specific Tips

  • Focus on lower div
  • For Calculus, focus on speed: median $\leq 1$ minute
  • Drill a lot of problems
    • Seriously, a lot.
    • Seriously.
  • Should memorize formulas and definitions
    • No time to rederive!
  • Save actual exams as diagnostic tools

Study Tips

  • Start early
    • Steady practice paced over 3-9 months is 100x more effective than 1 month of cramming
  • Speed is important
  • Spaced repetition, e.g. Anki
  • Replicate exam conditions
  • Build mental stamina
    • i.e. 2-3 hours of uninterrupted problem solving
  • Self care!!
    • Sleep
    • Eat right

Single Variable Calculus

Differential

  • Computing limits
  • Showing continuity
  • Computing derivatives
  • Rolle’s Theorem
  • Mean Value Theorem
  • Extreme Value Theorem
  • Implicit Differentiation
  • Related Rates
  • Optimization
  • Computing Taylor expansions
  • Computing linear approximations

Integral

  • Riemann sum definition of the integral
  • The fundamental theorem of Calculus (both forms)
  • Computing antiderivatives
    • $u\dash$substitutions
    • Partial fraction decomposition
    • Trigonometric Substitution
    • Integration by parts
    • Specific integrands
  • Computing definite integrals
  • Solids of revolution
  • Series (see real analysis section)

Computing Limits

  • Tools for finding $\lim_{x\to a} f(x)$, in order of difficulty:
    • Plug in: equal to $f(a)$ if $f \in C^0(N_\varepsilon(a))$
    • Algebraic Manipulation
    • L’Hopital’s Rule (only for indeterminate forms $\frac 0 0, \frac \infty \infty$)
      • For $\lim f(x)^{g(x)} = 1^\infty, \infty^0, 0^0$, let $L = \lim f^g \implies \ln L = \lim g \ln f$
    • Squeeze theorem
    • Take Taylor expansion at $a$
    • Monotonic + bounded (for sequences)

Use Simple Techniques

When possible, of course.

$$\frac{a}{b+\sqrt{c}} = \frac{a}{b+\sqrt{c}} \left( \frac {b-\sqrt c} {b-\sqrt c} \right) = \frac {a(b-\sqrt c)} {b^2 - c} \\ \frac{1}{ax^2 + bx + c} = \frac{1}{(x-r_1)(x-r_2)} = \frac{A}{x-r_1} + \frac{B}{x-r_2}$$

The Fundamental Theorems of Calculus

$$\begin{align*} \frac{d}{dx} \int_a^x f(t)~dt &= f(x) \\ \\ \int_a^b \dd{}{x} f(x)~dx &= f(b) - f(a) \\ \end{align*}$$

First form is usually skimmed over, but very important!

FTC Alternative Forms

$$\frac{\partial}{\partial x} \int_{a(x)}^{b(x)} g(t) dt = g(b(x))b'(x) - g(a(x))a'(x)$$

Commuting $D$ and $I$

Commuting a derivative with an integral

$$\frac{d}{dx} \int_{a(x)}^{b(x)} f(x,t) dt = \int_{a(x)}^{b(x)} \frac{\partial}{\partial x} f(x, t) dt \\ + f(x, b(x))\frac{d}{dx}b(x) - f(x, a(x))\frac{d}{dx}a(x)$$

(Derived from chain rule)

Set $$ a(x) = a, b(x) = b, f(x,t) = f(t) \implies \dd{}{x} f(t) = 0, $$ then commute to derive the FTC.

Applications of Integrals

  • Solids of Revolution
    • Disks: $A = \int \pi r(t)^2 ~dt$
    • Cylinders: $A = \int 2\pi r(t)h(t) ~dt$
  • Arc Lengths
    • $ds = \sqrt{dx^2 + dy^2},\qquad L = \int ~ds$

Series

There are 6 major tests at our disposal:

  • Comparison Test
    • $a_n < b_n \and \sum b_n < \infty \implies \sum a_n < \infty$
    • $b_n < a_n \and \sum b_n = \infty \implies \sum a_n = \infty$
    • You should know some examples of series that converge and diverge to compare to.
  • Ratio Test
    $$R =\lim_{n\to\infty} \abs{\frac{a_{n+1}}{a_n}}$$
    • $R < 1$: absolutely convergent
    • $R > 1$: divergent
    • $R = 1$: inconclusive

More Series

  • Root Test
    $$R = \limsup_{n \to \infty} \sqrt[n]{\abs{a_n}}$$
    • $R < 1$: convergent
    • $R > 1$: divergent
    • $R = 1$: inconclusive
  • Integral Test
    $$f(n) = a_n \implies \sum a_n < \infty \iff \int_1^\infty f(x) dx < \infty$$

More Series

  • Limit Test
    $$\lim_{n\to\infty}\frac{a_n}{b_n} = L < \infty \implies \sum a_n < \infty \iff \sum b_n < \infty$$
  • Alternating Series Test
    $$a_n \downarrow 0 \implies \sum (-1)^n a_n < \infty$$

Advanced Series

  • Cauchy Criteria:
    • Let $s_k = \sum_{i=1}^k a_i$ be the $k\dash$th partial sum, then
      $$\sum a_i \text{ converges } \iff \theset{s_k} \text{ is a Cauchy sequence},$$
  • Weierstrass $M$ Test:
    $$\sum_{n=1}^\infty \abs{\norm{f_n}_\infty} < \infty \implies \\ \exists f\in C^0 \suchthat \sum_{n=1}^\infty f_n \rightrightarrows f$$
    • i.e. define $M_k = \sup\theset{f_k(x)}$ and require that $\sum \abs{M_k} < \infty$
    • “Absolute convergence in the sup norms implies uniform convergence”

Multivariable Calculus

General Concepts

  • Vectors, div, grad, curl
  • Equations of lines, planes, parameterized curves
    • And finding intersections
  • Multivariable Taylor series
    • Computing linear approximations
  • Multivariable optimization
    • Lagrange Multipliers
  • Arc lengths of curves
  • Line/surface/flux integrals
  • Green’s Theorem
  • The divergence theorem
  • Stoke’s Theorem

Geometry in $\RR^3$

Lines

$$Ax + By + C = 0,~ \vector x = \vector p + t\vector v,\\ \vector x \in L \iff \inner{\vector x-\vector p}{\vector n} = 0$$

Planes

$$A x + B y + C z + D = 0,~ \vector x(t,s) = \vector p + t\vector v_1 + s\vector v_2 \\ \vector x \in P \iff \inner{\vector x - \vector p}{\vector n} = 0$$

Distances to lines/planes: project onto orthogonal complement.

Tangent Planes/Linear Approximations

Let $S \subseteq \RR^3$ be a surface. Generally need a point $\vector{p} \in S$ and a normal $\vector{n}$.

Key Insight: The gradient of a function is normal to its level sets.

$$\text{Case 1: } S = \theset{[x,y,z] \in \RR^3 \mid f(x,y, z) = 0}$$

i.e. it is the zero set of some function $f:\RR^3 \to \RR$

  • $\nabla f$ is a vector that is normal to the zero level set.
  • So just write the equation for a tangent plane $\inner{\vector n}{\vector x - \vector p_0}$.

Tangent Planes/Linear Approximations

$$\text{Case 2: } S \text{ is given by } z = g(x,y)$$

  • Let $f(x, y, z) = g(x,y) - z$, then

    $$\vector p \in S \iff \vector p \in \theset{[x,y,z] \in \RR^3 \mid f(x,y, z) = 0}.\\$$

  • Then $\nabla f$ is normal to level sets, compute $\nabla f = [\dd{}{x}g, \dd{}{y}g, -1]$

  • Proceed as in previous case.

Optimization

Single variable: solve $\dd{}{x} f(x) = 0$ to find critical points $c_i$ then check min/max by computing $\dd{^2}{x^2}f(c_i)$.

Multivariable: solve $\nabla f(\vector x) = 0$ for critical points $\vector c_i$, then check min/max by computing the determinant of the Hessian:

$$H_f({ \mathbf { a } }) = \left[ \begin{array} { c c c } { \frac { \partial ^ { 2 } f } { \partial x _ { 1 } \partial x _ { 1 } } ( \mathbf { a } ) } & { \dots } & { \frac { \partial ^ { 2 } f } { \partial x _ { 1 } \partial x _ { n } } ( \mathbf { a } ) } \\ { \vdots } & { \ddots } & { \vdots } \\ { \frac { \partial ^ { 2 } f } { \partial x _ { n } \partial x _ { 1 } } ( \mathbf { a } ) } & { \cdots } & { \frac { \partial ^ { 2 } f } { \partial x _ { n } \partial x _ { n } } ( \mathbf { a } ) } \end{array} \right].$$

Optimization

Lagrange Multipliers:

$$\text{Optimize } f(\mathbf x) \text{ subject to } g(\mathbf x) = c \\ \implies \nabla \vector f = \lambda \nabla \vector g$$

  • Generally a system of nonlinear equations
    • But there are a few common tricks to help solve.

Multivariable Chain Rule

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Multivariable Chain Rule

To get any one derivative, sum over all possible paths to it:

$$\begin{align*} \left(\dd{z}{x}\right)_y &= \left(\dd{z}{x}\right)_{u,y,v} \\ & + \left(\dd{z}{v}\right)_{x,y,u} \left(\dd{v}{x}\right)_y \\ & + \left(\dd{z}{u}\right)_{x,y,v} \left(\dd{u}{x}\right)_{v,y} \\ & + \left(\dd{z}{u}\right)_{x,y,v} \left(\dd{u}{v}\right)_{x,y} \left(\dd{v}{x}\right)_y \end{align*}$$

Subscripts denote variables held constant while differentiating.

Linear Approximation

Just use Taylor expansions.

Single variable case:

$$f(x) = f(p) + f'(p)(x-p) \\+ f''(p)(x-a)^2 + O(x^3)$$

Multivariable case:

$$f(\vector x) = f(\vector p) + \nabla f(\vector p)(\vector x - \vector a) \\+ (\vector x - \vector p )^T H_f(p)(\vector x - \vector p) + O(\norm{\vector x - \vector p}_2^3)$$

Linear Algebra

Big Theorems

  • Rank Nullity:

    $$\abs{\ker(A)} + \abs{\im(A)} = \abs{\mathrm{domain}(A)}$$

  • Fundamental Subspace Theorems

    $$\im(A) \perp \ker(A^T), \qquad \ker(A) \perp \im (A^T)$$

  • Compute

    • Determinant, trace, inverse, subspaces, eigenvalues, etc
    • Know properties too!
  • Definitions

    • Vector space, subspace, singular, consistent system, etc

Fundamental Spaces

  • Finding bases for various spaces of $A$:
    • $\mathrm{rowspace} A/ \im A^T \subseteq \RR^n$

      • Reduce to RREF, and take nonzero rows of $\mathrm{RREF}(A)$.
    • $\mathrm{colspace} A / \im A \subseteq \RR^m$:

      • Reduce to RREF, and take columns with pivots from original $A$.

Fundamental Spaces

  • $\mathrm{nullspace}(A) / \ker A$:

    • Reduce to RREF, zero rows are free variables, convert back to equations and pull free variables out as scalar multipliers.
  • Eigenspace:

    • Recall the equation:
      $$\lambda \in \spec(A) \iff \exists \vector v_\lambda \suchthat A\vector v_\lambda = \lambda \vector v_\lambda$$
    • For each $\lambda \in \spec(A)$, compute $\ker (\lambda I - A)$

Big List of Equivalent Properties

Let $A$ be an $n\times n$ matrix representing a linear map $L: V \to W$

TFAE:

  • $A$ is invertible and has a unique inverse $A^{-1}$
  • $A^T$ is invertible
  • $\det(A) \neq 0$
  • The linear system $A\vector{x} = \vector{b}$ has a unique solution for every $b\ \in \RR^m$
  • The homogeneous system $A\vector{x} = 0$ has only the trivial solution $\vector{x} = 0$
  • $\rank(A) = \dim(W) = n$
    • i.e. $A$ is full rank
  • $\mathrm{nullity}(A) \definedas \dim(\mathrm{nullspace}(A)) = \dim(\ker L) = 0$

Big List of Equivalent Properties

  • $A = \prod_{i=1}^k E_i$ for some finite $k$, where each $E_i$ is an elementary matrix.
  • $A$ is row-equivalent to the identity matrix $I_n$
  • $A$ has exactly $n$ pivots
  • The columns of $A$ are a basis for $W \cong \RR^n$
    • i.e. $\mathrm{colspace}(A) = \RR^n$
  • The rows of $A$ are a basis for $V \cong \RR^n$
    • i.e. $\mathrm{rowspace}~(A) = \RR^n$
  • $\left(\mathrm{colspace}~(A)\right)^\perp = \left(\mathrm{rowspace}~(A^T)\right)^\perp = \theset{\vector 0}$
  • Zero is not an eigenvalue of $A$.
  • $A$ has $n$ linearly independent eigenvectors

Various Other Topics

  • Quadratic forms
  • Projection operators
  • Least Squares
  • Diagonalizability, similarity
  • Canonical forms
  • Decompositions ($QR, VDV^{-1}, SVD,$ etc)

Ordinary Differential Equations

Easy IVPs

  • Should be able to immediately write solutions to any initial value problem of the form
    $$\sum_{i=0}^n \alpha_i y^{(i)}(x) = f(x)$$
    • Just write the characteristic polynomial.

Easy IVPs

  • Example: A second order homogeneous equation
    $$ay'' + by' + cy = 0 \mapsto ax^2 + bx + c = 0$$
    • Two distinct roots:
      $$y(x) = c_1 e^{r_1 x} + c_2 e^{r_2 x}$$
    • One real root:
      $$y(x) = c_1 e^{rx} + c_2 x e^{rx}$$
    • Complex conjugates $\alpha \pm \beta i$:
      $$y(x) = e^{\alpha x}(c_1 \cos \beta x + c_2 \sin \beta x)$$

More Easy IVPs

  • The Logistic Equation

    $$\frac{dP}{dt} = r\left( 1 - \frac{P}{C} \right)P \implies P(t) = \frac{P_0}{\frac{P_0}{C} + e^{-rt}(1 - \frac{P_0}{C})}$$

  • Separable

    $$\frac{dy}{dx} = f(x)g(y) \implies \int \frac{1}{g(y)}dy = \int f(x) dx + C$$

More Easy IVPs

  • Systems of ODEs
    $$\vector{x}'(t) = A\vector{x}(t) + \vector{b}(t) \implies \vector{x}(t) = \sum_{i=1}^n c_i e^{\lambda_i t}~\vector{v}_i$$

    for each eigenvalue/eigenvector pair $(\lambda_i, \vector v_i)$.

Less Common Topics

  • Integrating factors
  • Change of Variables
  • Inhomogeneous ODEs (need a particular solution)
    • Variation of parameters
    • Annihilators
    • Undetermined coefficients
    • Reduction of Order
    • Laplace Transforms
    • Series solutions
  • Special ODEs
    • Exact
    • Bernoulli
    • Cauchy-Euler

Topics: Number Theory

Definitions

  • The fundamental theorem of arithmetic:

    $$n\in\mathbb Z \implies n = \prod_{i=1}^n p_i^{k_i}, \quad p_i \text{ prime}$$

  • Divisibility and modular congruence:

    $$x\mid y \iff y = 0 \mod x \iff \exists c \suchthat y = xc$$

  • Useful fact:

    $$x = 0 \mod n \iff x = 0 \mod p_i^{k_i} ~\forall i$$

    (Follows from the Chinese remainder theorem since all of the $p_i^{k_i}$ are coprime)

Definitions

  • GCD, LCM
    $$xy = \gcd{(x,y)}~\mathrm{lcm}{(x,y)} \\ d\mid x \and d\mid y \implies d \mid \gcd(x,y) \\ \quad \and \gcd(x,y) = d\gcd(\frac x d, \frac y d)$$
    • Also works for $\mathrm{lcm}(x,y)$
    • Computing $\gcd(x,y)$:
      • Take prime factorization of $x$ and $y$,
      • Take only the distinct primes they have in common,
      • Take the minimum exponent appearing

The Euclidean Algorithm

Computes GCD, can also be used to find modular inverses:

$$\begin{aligned} a & = q _ { 0 } b + r _ { 0 } \\ b & = q _ { 1 } r _ { 0 } + r _ { 1 } \\ r _ { 0 } & = q _ { 2 } r _ { 1 } + r _ { 2 } \\ r _ { 1 } & = q _ { 3 } r _ { 2 } + r _ { 3 } \\ & \vdots \\r_k &= q_{k+2}r_{k+1} + \mathbf{r_{k+2}} \\ r_{k+1} &= q_{k+3}r_{k+2} + 0 \end{aligned}$$

Back-substitute to write $ax+by = \mathbf{r_{k+2}} = \gcd(a,b)$.

(Also works for polynomials!)

Definitions

  • Coprime

    $$a\text{ is coprime to } b \iff \gcd(a,b) = 1$$

  • Euler’s Totient Funtion

    $$\phi(a) = \abs{\theset{x \in \NN \suchthat x \leq a \and \gcd(x,a) = 1}}$$

    • Computing $\phi$:
      $$\gcd(a,b) = 1 \implies \phi(ab) = \phi(a)\phi(b) \\ \phi(p^k) = p^k - p^{k-1}$$
    • Just take the prime factorization and apply these.

Definitions

Know some group and ring theoretic properties of $\ZZ/n\ZZ$

  • $\ZZ/n\ZZ$ is a field $\iff n$ is prime.
    • So we can solve equations with inverses: $ax = b \mod n \iff x = a^{-1}b \mod n$
  • But there will always be some units; in general,
    $$\abs{(\ZZ/n\ZZ)^{\times}} = \phi(n)$$

    and is cyclic when $n=1,2,4,p^k, 2p^k$

Chinese Remainder Theorem

The system

$$\begin{array} { c } { x \equiv a _ { 1 } \quad \left( \bmod m _ { 1 } \right) } \\ { x \equiv a _ { 2 } \quad \left( \bmod m _ { 2 } \right) } \\ { \vdots } \\ { x \equiv a _ { r } \quad \left( \bmod m _ { r } \right) } \end{array}$$

has a unique solution $x \mod \prod m_i$ iff $\gcd(m_i, m_j) = 1$ for each pair $i,j$.

Chinese Remainder Theorem

The solution is given by

$$x = \sum_{j=1}^r a_j \frac{\prod_i m_i}{m_j} \left( \left[ \frac{\prod_i m_i}{m_j} \right]^{-1}_{\mod m_j}\right)$$

Seems symbolically complex, but actually an easy algorithm to carry out by hand.

Chinese Remainder Theorem

Ring-theoretic interpretation: let $N = \prod n_i$, then

$$\gcd(i,j) = 1 ~~\forall (i,j) \implies \ZZ_N \cong \bigoplus \ZZ_{n_i}$$

Theorems

  • Fermat’s Little Theorem and Euler’s Theorem

    $$a^p = a \mod p \\ p \not\mid a \implies a^{p-1} = 1 \mod p \\ \text{and in general, } \\ a^{\phi(p)} = 1 \mod p$$

  • Wilson’s Theorem

    $$n \text{ is prime } \iff (n-1)! = -1 \mod n$$

Advanced Topics

  • Mobius Inversion
  • Quadratic residues
  • The Legendre/Jacobi Symbols
  • Quadratic Reciprocity

Topics: Abstract Algebra

Definitions

  • Group, ring, subgroup, ideal, homomorphism, etc
  • Order, Center, Centralizer, orbits, stabilizers
  • Common groups: $S_n, A_n, C_n, D_{2n}, \ZZ_n,$ etc

Structure

  • Structure of $S_n$

    • e.g. Every element is a product of disjoint cycles, and the order is the lcm of the order of the cycles.
    • Generated by (e.g.) transpositions
    • Cycle types
    • Inversions
    • Conjugacy classes
    • Sign of a permutation
  • Structure of $\ZZ_n$

    $$\ZZ_{pq} = \ZZ_p \oplus \ZZ_q \iff (p,q) = 1$$

Basics

Group Axioms

  • Closure: $a,b \in G \implies ab \in G$
  • Identity: $\exists e\in G \mid a\in G \implies ae = ea = a$
  • Associativity: $a,b,c \in G \implies (ab)c = a(bc)$
  • Inverses: $a\in G \implies \exists b \in G \mid ab =ba = e$

One step subgroup test:

$$H \leq G \iff a,b \in H \implies a b^{-1} \in H \\$$

Useful Theorems

Cauchy’s Theorem

  • If $\abs{G} = n = \prod p_i^{k_i}$, then for each $i$ there exists a subgroup $H$ of order $p_i$.

The Sylow Theorems

  • If $\abs{G} = n = \prod p_i^{k_i}$, for each $i$ and each $1 \leq k_j \leq k_i$ then there exists a subgroup $H_{i,j}$ for all orders $p_i^{k_j}$.
    • Note: partial converse to Cauchy’s theorem.

Classification of Abelian Groups

Suppose $\abs{G} = n = \prod_{i=1}^m p_i^{k_i}$

$$G \cong \bigoplus_{i=1}^n G_i \text{ with } \abs{G_i} = p_i^{k_i} \text{ and } \\ G_i \cong \bigoplus_{j=1}^k \ZZ_{p_i^{\alpha_j}} \text{ where } \sum_{j=1}^k \alpha_j = k_i$$

$G$ decomposes into a direct sum of groups corresponding to its prime factorization. For each component, you take the corresponding prime, write an integer partition of its exponent, and each unique partition yields a unique group.

Ring Theory

  • Definition: $(R, +, \times)$ where $(R, +)$ is abelian and $(R, times)$ is a monoid.

  • Ideals: $(I, +) \leq (R, +)$ and $r\in R, x\in I \implies rx \in I$

  • Noetherian: $I_1 \subseteq I_2 \subseteq \cdots \implies \exists N \suchthat I_N = I_{N+1} = \cdots$

    • (Ascending chain condition)
  • Differences between prime and irreducible elements

    • Prime: $p \mid ab \implies \mid a \or p \mid b$
    • Irreducible: $x \text{ irreducible } \iff \not\exists a,b\in R^\times \suchthat p = ab$.
  • Various types of rings and their relations:

$$ \text{field} \implies \text{Euclidean Domain} \implies \text{PID} \\ \implies \text{UFD} \implies \text{integral domain} $$

Topics: Real Analysis

  • Properties of Metric Spaces
  • The Cauchy-Schwarz Inequality
  • Definitions of Sequences and Series
  • Testing Convergence of sequences and series
  • Cauchy sequences and completeness
  • Commuting limiting operations:
    • $[\frac{\partial}{\partial x}, \int dx]$
  • Uniform and point-wise continuity
  • Lipschitz Continuity

Big Theorems

  • Completeness: Every Cauchy sequence in $\RR^n$ converges.
  • Generalized Mean Value Theorem
    $$f,g\text{ differentiable on } [a,b] \implies \\ \exists c\in[a,b] : \left[f ( b ) - f ( a ) \right] g' ( c ) = \left[g ( b ) - g ( a )\right] f' ( c )$$
    • Take $g(x) = x$ to recover the usual MVT
  • Bolzano-Weierstrass: every bounded sequence in $\RR^n$ has a convergent subsequence.
  • Heine-Borel: in $\RR^n, X$ is compact $\iff X$ is closed and bounded.

Topics: Point-Set Topology

General Concepts

  • Open/closed sets
  • Connected, disconnected, totally disconnected, etc
  • Mostly topics related to metric spaces

Useful Facts

  • Topologies are closed under
    • Arbitrary unions:
      $$U_j \in \mathcal{T} \implies \union_{j\in J} U_i \in \mathcal{T}$$
    • Finite intersections:
      $$U_i \in \mathcal{T} \implies \intersect_{i=1}^n U_i \in \mathcal{T}$$
  • In $\RR^n$, singletons are closed, and thus so are finite sets of points
    • Useful for constructing counterexamples to statements

Topics: Complex Analysis

General Concepts

  • $n\dash$th roots:
    $$e^{\frac{ki}{2\pi n}}, \qquad k = 1, 2, \cdots n-1$$
  • The Residue theorem:
    $$\oint_C f(z)~dz = 2\pi i \sum_k \mathrm{Res}(f, z_k)$$
    • Exams often include one complex integral
    • Need a number of other theorems for actually computing residues

Topics: Discrete Mathematics + Combinatorics

General Concepts

  • Graphs, trees
  • Recurrence relations
  • Counting problems
    • e.g. number of nonisomorphic structures
  • Inclusion-exclusion, etc

$$(x+y)^n = \sum_{k=0}^n {n\choose k}x^ky^{n-k}$$

Example Problems

Example Problem 1

problem1

Example Problem 1

problem1
C, because $\ZZ-\theset{0}$ lacks inverses
(Would need to extend to $\QQ$)

Example Problem 2

problem1

Example Problem 2

problem1

$$L = \lim _ { ( a , b ) \rightarrow 0 } \frac { ( a - b i ) ^ { 2 } } { ( a + b i ) ^ { 2 } } = \lim _ { ( a , b ) \rightarrow 0 } \frac { a ^ { 2 } - b ^ { 2 } - 2 a b i } { a ^ { 2 } - b ^ { 2 } + 2 a b i } \\ a = 0 \implies L = 1 \\ a = b \implies L = -1$$

So E, because the limit needs to be path-independent.

Example Problem 3

problem1

Example Problem 3

problem1

Don’t row-reduce or invert! Just one computation

$$\left( \begin{array}{cccc} 1 & 3 & 2 & 3 \\ 1 & 4 & 1 & 0 \\ 3 & 5 & 10 & 14 \\ 2 & 5 & 5 & 6 \end{array}\right) \left(\begin{array}{c}-5 \\ 1 \\ 1 \\ 0\end{array}\right) = \vector 0$$

Example Problem 3

So D, A are true. C is true because it’s a homogeneous system. B is true because $A\vector x = 0 \implies A (t\vector x) = tA\vector x = 0$ which means $t\vector x$ is a solution for every $t$. By process of elimination, E must be false.

Example Problem 4

problem1

Example Problem 4

problem1

Note $N_{\frac{1}{2}}(x) = \theset{x}$, so every singleton is open. Any subset of $\ZZ$ is a countable union of its singletons, so every subset of $\ZZ$ is open. The complement any set is one such subset, so every subset is clopen. The inverse image of any subset of $\RR$ under any $f:\ZZ \to \RR$ is a subset of $\ZZ$, which is open, so every such $f$ is continuous. So E.