I audited the graduate section of abstract algebra. Groups, homomorphisms, subgroups and quotient groups. Group actions, Sylow theory. Representation theory, regular representations, inner automorphisms. Direct and semidirect products, composition series, the Jordan-Holder theorem. Module theory, tensor products, modules over PID's, canonical forms.
Textbook: Abstract Algebra, 3rd edition, by Dummit and Foote.
First and second order ODE's. Vector spaces, determinants, eigenvalues and eigenvectors, complex roots. Fourier series, boundary value problems, the heat equation.
Textbook: Differential Equations and Their Applications, by Braun.
University of North Texas (TAMS)
Topology (Math 4500/5600), taught by Lior Fishman.
Metric spaces, countability, continuous functions. Topological spaces, open/closed sets, boundary, closure, accumulation points, dense sets. The Cantor ternary set, Baire category theorem, and Banach-Mazur game. Compactness, connectedness, homeomorphisms. Separation axioms, T1, Hausdorff, regular, and normal spaces.
Textbook: Officially Introduction to Topology by Singh, but we didn't really follow one.
Abstract Algebra II (Math 4510/5900), taught by Anne Shepler.
Polynomial rings, factorization of polynomials, homomorphisms and factor rings, prime and maximal ideals. Extension fields, vector spaces, algebraic extensions, algebraic closure(s). Field automorphisms, isomorphic extensions, splitting fields, separable extensions. The fundamental theorem of Galois theory. We never got to see the proof of the unsolvability of the quintic :(.
Textbook: A First Course in Abstract Algebra, 7th edition, by Fraleigh.
Abstract Algebra I (Math 3510), taught by Neal Brand.
Introduction to groups, rings, and fields. Group axioms, isomorphisms, cyclic groups, subgroups, direct products. The symmetric, dihedral, and general linear groups. Lagrange's theorem, normal subgroups, factor groups, classification of finitely generated abelian groups. Basic rings, integral domains, fields. Fermat's and Euler's theorems, applications to RSA encryption.
Textbook: A First Course in Abstract Algebra, 8th edition, by Fraleigh and Brand (!).
Real Analysis II (Math 3610), taught by Mariusz Urbanksi.
Limits, L'hopital's rule, sequences, the monotone convergence theorem. Accumulation points, limit superior and limit inferior, Bolzano-Weierstrauss. Continuity, compactness, uniform continuity, the intermediate value theorem. Differentiation, Rolle's theorem, Taylor's theorem. Riemann integration, Darboux sums.
Textbook: Analysis with an Introduction to Proof, 5th edition, by Lay.
Real Analysis I (Math 3000), taught by reading the textbook in Summer 2019.
Basic set theory, union, intersect, complement. Power sets, orderings, cartesian products. Conditional statements, truth tables, quantifiers, tautologies. Direct proof, contrapositive, contradiction, disproof, induction. Equivalence relations, partitions, functions as relations, cardinality.
Field axioms, real numbers, ordered fields. Completeness, supremum and infimum, density of the rationals. Neighborhoods, accumulation points, boundary sets, interior and closure, open and closed sets. Compactness, Bolzano-Weierstrauss, Heine-Borel theorem. Limits, continuous functions, sequences.
Textbook(s): Book of Proof by Hammack and Analysis with an Introduction to Proof, 5th edition, by Lay.
Linear Algebra (Math 2700), taught by Huguette Tran.
Matrix operations, row reduction, linear independence and dependence. Linear transformations, matrix inversion and factorization, the Leontief model. Vector spaces, subspace, rank, determinants, null and column space. Bases, eigenvalues and eigenvectors, change of basis, and diagonalization. Inner products, orthogonality, least squares, Gram-Schmidt.
Textbook: Linear Algebra and Its Applications, by Lay.
Multivariable Calculus (Math 2730), taught by Arunabha Biswas.
3D coordinates, dot and cross products. Lines and planes, vector-valued functions, arc length. Curvature, normal vectors, multivariate functions. Limits, partial and directional derivatives, gradients. Lagrange multipliers, double and triple integration in polar, cylindrical, and spherical coordinates.
Textbook: Calculus, by Briggs and Cochran.
Calculus II (Math 1710), taught by Neal Brand.
Integration by parts, trigonometric integration, subsitution, partial fractions. Sequences, convergence of series, geometric, harmonic, alternating series. Divergence, p-series, integral, ratio, root, direct comparison, and limit comparison convergence tests. Taylor and Maclaurin polynomials, remainder and error bound, power series, radius of convergence.
Textbook: Calculus, by Briggs and Cochran.
A collection of the blog posts about mathematical topics, written with mathematical language.