Courses Offered at Math Academy

High School Admissions Test

SAT I

Test prep and test taking strategies; algebra and functions; geometry; numbers and operations; data analysis

SAT II Subject Tests

Mathematics Levels 1 and 2; Physics

ACT

Test prep and test taking strategies; pre-algebra; elementary and intermediate algebra; coordinate and plane geometry; trigonometry

SHSAT

Reading Comprehension; Logical Reasoning; Scrambled Paragraphs; Mathematics; Multiple Choice Questions; Various mathematical topics; Basic math; Algebra; Factoring; Substitution; Geometry; Basic Coordinate Graphing; Logic; Word Problems

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College Algebra

Solving quadratic equations; solving linear equations; factoring polynomials; completing the square; exponents; logarithms; radical expressions; rational expressions; graphing in the x-y plane; midpoint and distance formulas; parallel and perpendicular lines; solving linear equations; systems of equations; inequalities; functions and function notation; function transformation and graphing; matrix properties; basic trigonometric functions; conic sections

Calculus 1

Representation of functions; rate of change; limits at a point; infinite limits; two-sided limits; continuity of functions; criteria for differentiability; first and higher derivatives; derivatives of trigonometric functions; product and quotient rule; chain rule; implicit differentiation; optimization; Liebniz notation; Extreme Value Theorem; Mean Value Theorem; inflection points; first and second derivative tests for finding inflection points

Calculus 2

L’Hospital’s Rule and indeterminate forms; Reimann sums; average value theorem; definite and indefinite integrals; antiderivative techniques; u-substitution; trig substitution; integration by parts; partial fraction decomposition; solids of revolution; differential equations; Fundamental Theorem of Calculus; modeling physical and biological systems using calculus; differentiation under the integral

Calculus 3

Parametric functions; calculus in polar, spherical, and cylindrical coordinates; multi-variable calculus; partial derivatives; Green’s Theorem; Stokes’ Theorem; vector calculus; dot and cross products; sequences and series; convergence of series; harmonic, geometric series, and power series; Taylor and Maclaurin series; Fourier series; radius of convergence; ratio test; comparison test; alternating series test; root test

Ordinary Differential Equations

Slope fields; separation of variables; Fourier Transforms; Laplace Transforms; Taylor and Maclaurin series; Bessel Functions; general and particular solution to differential equations; damped harmonic motion; oscillation problems; bifurcations; non-homogeneous differential equations; strategies for non-linear differential equations; boundary value problems; dynamical systems; existence and uniqueness; reduction of order method; Cauchy-Euler Equations

Partial Differential Equations

Distribution theory; Fourier transforms; Laplace transforms; Schrodinger equations; Green’s theorem; Euler-Lagrange equations; boundary condition; boundary value problems; Dirichlet problems; heat and wave equations; Maxwell’s equations; Navier-Stokes equations; Poisson’s equation; Green’s function; applications of partial differential equations; numerical methods for finding solutions

Linear Algebra

Matrix addition and subtraction; matrix multiplication; matrix row operations; matrix inversion; transpose matrices; skew and zsymmetric matrices; determinants; solving systems of equations; Gaussian elimination; reduced row echelon form; rank of a matrix; Cramer’s Rule; minors and cofactors; eigenvalues and eigenvectors; vector spaces; orthogonality; transformation matrices; vector spaces; subspaces; linear span; projection; matrix decompositions

Statistics and Probability

Advanced Statistics: Fundamentals and axioms; combinatorial probability; conditional probability and independence; binomial, poisson and normal distributions; law of large numbers and the central limit theorem and random variables and generating functions.

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Abstract Algebra

Group theory; abelian groups; cyclic groups and generators; subgroups; simple groups; dihedral groups; finite groups; homomorphisms and isomorphisms; group actions; kernel of a function; Lagrange’s Theorem; ring theory; integral domains; field properties; algebra over a field; field extensions; norms; Gaussian integers; ideals; quaternions; algebraic numbers; polynomial fields; lie algebra; Galois theory

Number Theory

Composite numbers; prime numbers; divisors; Euclid’s algorithm and divisibility; square-free objects; modular arithmetic and congruence; Fermat’s Little Theorem; Euler’s totient function; Chinese Remainder Theorem; quadratic residue; Diophantine equations; prime numbers; Lagrange’s theorem; combinatorical theorem; primality tests; algebraic number theory

Real Analysis

Limits of sequences and functions; arithmetic, geometric, and harmonic series; convergence and divergence of series; Cauchy sequence; power series; telescoping series; alternating series; Fourier series; pointwise, uniform, and absolute convergence; tests for convergence; integral test; ratio test; direct comparison test; limit comparison test; root test; alternating series test; Dirichlet’s test; continuity of functions; uniform and absolute continuity; differentiability of functions; derivatives; partial derivatives; integrals and antiderivatives; Fundamental Theorem of Calculus; multiple integrals; differentiating under the integral; Reimann sums; measure theory; compactness; metric spaces; solution of differential equations; change of variables

Complex Variables

Analytic functions; complex differentiation and integration; Cauchy integral formula; Laplace transforms; Power series, Taylor series, and Laurent series; residue theory; complex analysis; isometries in the complex plane; conformal mapping; power series; uniform convergence; radius of convergence; poles; zeroes; singularities; branch points; countour integrals; gamma and beta functions; Reimann zeta functions; Reimann surfaces; infinite products; complex dynamics

Topology

Topological spaces; uniform and metric spaces; open and closed sets; continuity; bases and subbases; open cover; limit point; Blaire category theorem; compact spaces; connectedness; separation axioms; Kolmogorov, Frechet, and Haudsorff spaces; Banach spaces; Lp spaces; normal space; tensor products; finite products; order theory; homotopy equivalence; algebraic topology; differential topology; topological and differentiable manifolds; polytopes; covering spaces; vector fields; Frobenius Theorem; Lie groups; Stokes’ Theorem; geometric topology; knot theory

Differential Geometry

Curvature; differential geometry of surfaces; first and second fundamental forms; calculus on manifolds; Reimannian manifolds; Finsler manifolds; connections; Levi-Civita connection; tensor analysis; lie groups; bundles and connections; Symplectic geometry; differential topology; critical values; diffeomorphism; character classes; non-Euclidean geometry; geodesics; symmetric spaces; complex manifolds; index theory; homogeneous spaces

Advanced Topics in Geometry

Complex manifolds; Reimannian manifolds; geodesics; symmetric spaces; Lie groups; sympletic geometry; conformal geometry; Hodge theory and Fredholm theory; representation theory; projective spaces; non-Euclidean geometry; finite geometry; computational geometry; discrete geometry; algebraic geometry; fractals

Functional Analysis

Normed vector spaces; Hilbert spaces; Euclidean space; Banach spaces; Lp spaces; duality and separation theory; function spaces; locally convex spaces; Banach-Steinaus theorems; orthonormal basis; uniform boundedness; spectral theory; open mapping theorem; Hausdorff space; operator theory; real and complex algebras; topological vector spaces; wavelets

Numerical Analysis

Iterative methods; rate of convergence; series acceleration; error analysis; interval arithmetic; significant figures; least squares; numerical algorithms; numerical linear algebra; eigenvalues; orthogonalization; approximation formulas; interpolation methods; spline, polynomial, and trigonometric interpolation; linear systems; root-finding algorithms for nonlinear equations; numeric integration and differentiation; numerics for ordinary and partial differential equations; optimization; finite element methods; grids and meshes; Monte Carlo method

Cryptography

Encryption and decryption; cryptanalysis; ciphers; zero-knowledge proof; concurrency and protocol security; symmetric-key algorithms; asymmetric-key algorithms; authentication; transport and exchange theory; weak keys; robustness; symmetric algorithms; hash functions; attack models; database security; cryptographic game theory; multivariate cryptography; quantum cryptography; stenography

Operations Research

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GRE

Quantitative reasoning; arithmetic; algebra; geometry; data analysis; numerical reasoning

GMAT

Quantitative reasoning; data sufficiency; arithmetic; algebra; geometry; word problems; combinatorics; data analysis

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