A Virtual Mathematics Library

Undergraduate Resources

Functions, limits, continuity, derivative with applications, integrals, techniques of integration, improper integrals, infinite series, analytic geometry, and polar coordinates.

Graphics (some animated!) displaying some fundamental concepts of calculus.

Euclidean 3-space, vectors and their algebra and geometry, vector-valued functions, matrices, derivatives, continuity, directional derivatives, Taylor polynomials, sequences and series, iterated integrals. Path, line, surface, and flux integrals. Theorems of Green, Gauss, and Stokes. Applications to physics.

Solving linear systems; vector spaces, linear independence, and bases; linear maps, homomorphisms and isomorphisms, change of basis, projection; determinants; Similarity, eigenvalues and eigenvectors, Jordan canonical form.

Set theory, relations, the real number system, axioms of an ordered field, mathematical induction, Euclidean n-space and its geometry, vectors and vector spaces.

Discrete and continuous probability distributions, combinatorics, expected value and variance, random variables, the law of large numbers, the central limit theorem, generating functions, Markov chains, and random walks.

Analysis

Axioms for the Real Numbers, sequences, point-set topology, connectedness and compactness, limits, continuity and uniform continuity, differentiation, applications of the Mean Value Theorem.

Sequences, monotone convergence, limits and continuity, differentiability, infinite series, power series. Functions of several variables: partial derivatives, tangent planes, integration, Fubini's theorem, change of variables.

Sets and relations, induction, sequences, convergence, point-set topology, limits and differentiation, Riemann and Lebesgue intergration.

Complex numbers, complex functions, elementary functions, integration, Cauchy's Theorem, Cauchy's integral formula, harmonic series, Taylor and Laurent series, residues and poles, Rouche's Theorem.

Riemann and Lebesgue integration, measures, outer measures and Caratheodory's Theorem, Lebesgue measure, measurable functions, monotone converge theorem, Fatou's lemma, dominated convergence theorem, L^p spaces, Holder and Minkowski inequalities.

Measures, outer measures and Caratheodory's Theorem, Lebesgue-Stieltjes measure, integration, monotone convergence theorem, Fatou's lemma, dominated convergence theorem, product measures and Fubini's theorem, Radon-Nikodym theorem, differentiation, functions of bounded variation, L^p spaces, Holder and Minkowski inequalities.

Normed linear spaces, Banach spaces, linear operators, the Hahn-Banach theorem, Lebesgue measure, Fubini's Theorem, Hilbert spaces, Fourier Analysis.

Brief introduction to the Fourier transform and transform properties. Definitions of Discrete and Fast Fourier Transformations.

Bounded operators on Hilbert spaces, algebras of operators, compact and Hilbert-Schmidt operators, abstract C*-algebras and functional calculus, AF-algebras and reduced C*-group-algebras, states and the GNS construction, the algebra of canonical anticommutation relations, quasi-free states on the car algebra, unitary projective representations of groups.

Abstract Algebra

Groups, Rings, Matrices and Matrix Rings, Linear Algebra. (Emphasis on integrating linear algebra with abstract algebra.)

Groups, Rings, Fields, Modules, More Group Theory (e.g. Sylow's theorems), Galois theory, Algebraic Number Theory, Algebraic Geometry, Noncommutative Algebra, Homological Algebra. Includes written solutions to exercises.

Module Theory, Divisibility and UFDs, Noetherian and Artinian Modules, Localization, Projective/Injective/Flat Modules, Tensor Products, Basic Field Theory.

Algebraic sets, Algebaic varieties, Local study: tangent spaces/tangent cones/singularities, Projective varieties, Finite maps, Dimension theory, Regular maps and their fibres, Algebraic geometry over an arbitrary field, Divisors and intersection theory, Sheaves, Differentials, Algebraic varieties over the complex numbers.

Lattices, basic theory of Universal Algebra, Latin Squares, Boolean Algebras, connections with Model Theory, recent developments and open problems.

Combinatorics

Generating functions: examples, series, the exponential formula, applications, analytic and asymptotic methods.

Definitions in graph theory, matching, connectivity, planar graphs, coloring, flows, substructures in sparse/dense graphs, Ramsey theory, Hamilton cycles, random graphs, minors, trees, well-quasi-ordering.

Number Theory

Elementary number theory, rational and irrational numbers, the division algorithm, the Euclidean algorithm, continued fractions, modular arithmetic, Fermat's Little Theorem, public-key encryptions, the Chinese Remainder Theorem, primitive roots, quadratic reciprocity, asymptotics, the distribution of primes, sieve techniques.

Factorization, divisibility, g.c.d.s and the Euclidean algorithm, Euclidean domains, linear congruences, Chinese Remainder Theorem, integers and units modulo n, Euler's phi function and applications, quadratic congruences, law of quadratic reciprocity, diophantine equations, Gaussian Integers and applications, p-adic numbers.

Algebraic numbers, number fields and number rings, norm, trace, discriminant, factorization in number rings, ideal classes, geometric methods

Elementary number theory, Euler's summation formula, Dirichlet convolution, Euler products, analytic continuation of the Riemann zeta function, the gamma function, the theta function, Fourier analysis, The functional equation, primes in an arithmetic progression, characters of finite Abelian groups, Dirichlet characters and L-functions, analytic continuation of L-functions.

Arithmetic functions, elementary prime number theory, Dirichlet series, primes in arithmetic progressions, the prime number theorem, the Riemann zeta function.

Topology

Topological spaces, closed sets and closure, continuity and homeomorphism, compactness, connectedness, separability, nets, first countable spaces, product spaces, seperation axioms.

Fundamental group, path homotopy, Van Kampen's theorem, covering spaces, simplicial and singular homology, cohomology groups, cup product, Poincare duality, homotopy groups and connections with cohomology, and additional advanced topics.

Point-set topology: bases, metric spaces, continuous maps, homeomorphisms, connectedness, compactness, seperation and countability axioms, product and quotient spaces, projective spaces, topological groups. Algebraic topology: homotopy, fundamental group, covering spaces, retractions and fixed points, one-dimensional homology and cohomology. Manifolds: definitions, low-dimensional manifolds, smooth manifolds, immersions, embeddings, tangent vectors, vector bundles, orientation, transversality and cobordisms.

Geometry

Fundamentals of geometry, geometric algebra, theory of circles, constructions for inscribed and circumscribed figures, theory of abstract proportions, similiar figures and proportions in geometry, fundamentals of number theory, continued proportions in number theory, number theory, classification of incommensurables, solid geometry, measurement of figures, regular solids.

Introduction to manifolds, tangent vectors and tangent bundles, algebra of vector fields, Lie derivatives, algebraic language in geometry, differential forms on manifolds.

Differentiable manifolds, tangent space and tangent bundles, Riemannian manifolds, the Levi-Civita connection, geodesics, the curvature tensor, curvature and local geometry.

Additional Topics

Assorted topics in recreational mathematics.

Biographies of mathematicians (including posters of mathematicians and maps of birthplaces), histories of various mathematical topics, mathematics in various cultures, quotation index, lists of societies and honors, and famous curves index.

Volume I: propositional and first-order logic through the Soundness, Completeness, and Compactness Theorems; applications of the Compactness Theorem. Volume II: The basics of computability using Turing machines and recursive functions, plus the Gödel Incompleteness Theorems.

Categories, Functors, Naturality, Adjunctions, Duality, Constructions (Iso, Epic, Monic, Initial, Final, Products, Sums, Coequalizers, Colimits)

Invariant theory, physics, modular forms, algebraic geometry, string theory.

Projective Modules and Vector Bundles, the Grothendieck group, K₁ and K₂ of a ring, definitions of higher K-theory.