Required 300- and 400-level courses for the Advanced Track Program:

MTH 317H - Honors Linear Algebra - Systems of equations, matrix algebra, vector spaces, linear transformations, geometry of Rn, eigenvalues, eigenvectors, diagonalization, inner products. A writing course with emphasis on mathematical reasoning, proofs, and concepts.

MTH 347H - Honors Ordinary Differential Equations - Separable and exact equations, linear equations and variation of parameters, higher order linear equations, Laplace Transforms, first-order linear systems, classification of singularities, nonlinear systems, partial differential equations and Fourier Series, existence and uniqueness theorems. There will be an emphasis on theory.

MTH 327H - Honors Introduction to Analysis - Real and complex numbers, limits of sequences and series, continuity, differentiation, Riemann integration of functions over R, uniform convergence.

MTH 429H – Honors Real Analysis - Continuation of 327H. Convergence of sequences and series of functions, differentiation and integration in higher dimensional settings. Inverse and implicit function theorems.

MTH 428H – Honors Complex Analysis - Analytic functions of a complex variable, line integrals and harmonic functions, Cauchy's theorem and integral formula, power series, Laurent series, isolated singularities, residue calculus, Rouche's theorem, automorphisms of the disk, the Riemann mapping theorem.

MTH 418H - Honors Algebra I - Theory of groups, Sylow theory, the structure of finite Abelian groups, ring theory, ideals, homomorphisms, and polynomial rings.

MTH 419H - Honors Algebra II - Algebraic field extensions, Galois theory. Classification of finite fields. Fundamental Theorem of Algebra.

MTH 496 – Capstone in Mathematics – Topics vary semester to semester. This is a course integrating several areas of mathematics.

#### Many students in the Advanced Track Program choose to take first-year Graduate Courses. These are not required for the BS in Mathematics, Advanced.

MTH 810 – Error-Correcting Codes - Block codes, maximum likelihood decoding, Shannon's theorem. Generalized Reed-Solomon codes, modification of codes, subfield codes. Alterant and Goppa codes, cyclic codes and BCH codes.

MTH 818 – Algebra I - Group theory: Sylow theory, permutation groups, Jordon-Hoelder theory, Abelian groups, free groups. Ring theory: algebra of ideals, unique factorization, polynomial rings, finitely generated modules over PIDs.

MTH 819 – Algebra II - Modules and vector spaces, projectives modules, tensor algebra. Fields and Galois groups, algebraic and transcendental numbers, non-commutative rings. The Jacobson radical, the structure of semisimple rings with the descending chain condition.

MTH 828 – Real Analysis I - Lebesgue measure on real line, general measure theory. Convergence theorems, Lusin's theorem, Egorov's theorem, Lp-spaces, Fubini's theorem. Functions of bounded variation, absolutely continuous functions, Lebesgue differentiation theorem.

MTH 829 – Complex Analysis I - Cauchy theorem, identity principle, Liouville's theorem, maximum modulus theorem. Cauchy formula, residue theorem, Rouche's theorem. Casorati-Weierstrass theorem, Arzela-Ascoli theorem. Conformal mapping, Schwarz lemma, Riemann mapping theorem.

MTH 840 – Chaos and Dynamical Systems - Chaotic or random motions in differential and difference equations.

MTH 841 – Boundary Value Problems I - Methods for solving boundary and initial value problems for ordinary and partial differential equations.

MTH 842 – Boundary Value Problems II - Continuation of MTH 841.

MTH 848 – Ordinary Differential Equations - Existence and uniqueness theorems. Theory of linear differential equations. Floquet theory. Stability theory and Poincare-Bendixson theory. Green's functions and boundary value problems.

MTH 849 – Partial Differential Equations - Cauchy-Kowalewski theorem. Characteristics. Initial-boundary value problems for parabolic and hyperbolic equations. Energy methods, boundary value problems for elliptic equations, potential theory. Green's function, maximum principles, Schauder's method.

MTH 850 – Numerical Analysis I - Convergence and error analysis of numerical methods in applied mathematics.

MTH 851 – Numerical Analysis II - Interpolation theory and approximation of functions. Numerical solutions of nonlinear equations. Numerical integration methods.

MTH 852 – Numerical Methods for Ordinary Differential Equations - Linear multi-step methods and single step nonlinear methods for initial value problems. Consistency, stability and convergence. Finite difference, finite element, shooting methods for boundary value problems.

MTH 864 – Geometric Topology - Topology of surfaces and higher dimensional manifolds, studied from combinatorial, algebraic or differential viewpoints.

MTH 868 – Geometry and Topology I - Fundamental group and covering spaces, van Kampen's theorem. Homology theory, Differentiable manifolds, vector bundles, transversality, calculus on manifolds. Differential forms, tensor bundles, deRham theorem, Frobenius theorem.

MTH 869 – Geometry and Topology II - Continuation of MTH 868.

MTH 880 – Combinatorics - Enumerative combinatorics, recurrence relations, generating functions, asymptotics, applications to graphs, partially ordered sets, generalized Moebius inversions, combinatorial algorithms.

MTH 881 – Graph Theory - Graph theory, connectivity, algebraic and topological methods. Networks, graph algorithms, Hamiltonian and Eulerian graphs, extremal graph theory, random graphs.

#### Course Information

Sample Programs

Course Descriptions

#### Contact information

Director of Advanced Track Program: Russell Schwab