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Basic School Level Courses

Arithmetic

Fundamentals of arithmetic. Fractions, decimals, percents, signed numbers, ratios, proportions and rates, word problems.

Pre-Algebra

Review of fundamentals of elementary Algebra. Focus on mathematical translation of verbal statements.

Algebra I

Operations with polynomials, rational and square root expressions, exponents, solving and understanding linear and quadratic equations, basic applications and graphing.

Algebra II

Absolute value equations,  inequalities, and their graphs, radical expressions, rational exponents, quadratic equations and inequalities, logarithms, introduction to functions, and topics in analytic geometry.

Geometry I

Introduction to Geometry. Lines, angles, triangles, congruent triangles, parallel lines, distances and angle sums. Parallelograms, trapezoids, medians, and midpoints. Methods of proof.

Geometry II

The geometry of the circle, similarity, areas, regular polygons, locus, inequalities and indirect reasoning, transformations, constructions, improvement of reasoning, proof of important theorems.

Geometry III  (Stereometry –  3-Dimensional Geometry)

The geometry of the three-dimensional plane, vectors, solids, functions and their graphs, proofs.

Algebra II

Absolute value equations,  inequalities, and their graphs, radical expressions, rational exponents, quadratic equations and inequalities, logarithms, introduction to functions, and topics in analytic geometry.

Advanced Elementary Geometry

This course covers everything in Geometry I, II, III, from an advanced standpoint. More attention is given to proofs. This course is taken by our Math Club students to prepare for the Mathematics Competitions and Olympiads.

Preparation for the Math Competitions/ Olympiads

This course, which is held at the Math Club prepares students for the Math Competitions or Olympiads. Various topics from advanced elementary mathematics are covered. Special attention is given to problem solving, proofs, and non standard approaches to solving problems.

Trigonometry

Introduction to the fundamental ideas in Trigonometry.  The main trigonometric identity.

New York State Math A Regents

New York State Math B Regents

Pre-Calculus

Functions and their graphs: polynomial, rational, exponential, logarithmic and trigonometric functions; conic sections; topics in trigonometry; graphical and analytical solutions of systems of equations and inequalities, trigonometric forms of a Complex number and DeMovire’s Theorem; sequences and series.

Calculus AB

Limits, continuity, and differentiation of algebraic functions.  Applications of differentiation; exponential, logarithmic, and trigonometric functions and their derivatives; optimization problems, the definite integral and the fundamental theorem of calculus.  Application of the integral. The Riemann sum, Intermediate Value Theorem, Mean Value Theorem, L’Hospital’s rule and more.

Calculus BC

Theoretical treatments and Applications of Definite Integrals. Techniques of Integration. Infinite sequences and series and convergence and divergence of series. Taylor Series and polynomial Approximations. Line integrals. Parametric equations and polar coordinates. Elementary differential equations. Basic proof methods.

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High School Test Prep

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 - Undergraduate Courses

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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Graduate Level Courses

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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Graduate Test Prep

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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