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.

**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.

**Note:** More information about Michigan State University may be found at: http://www.msu.edu/ More information about the Mathematics Department may be found at: http://www.math.msu.edu/ More information about the course scheduling may be found at: http://www.schedule.msu.edu/ Course descriptions (including semesters offered) may be found at: http://www.reg.msu.edu/Courses/Request.aspx?SubjectCode=mth&Term=current

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Director of Advanced Track Program: **Bruce Sagan**

Advisors for the Advanced Track Program:

**Julie Cioni**: cionijul@math.msu.edu**Tsveta Sendova**: tsendova@math.msu.edu

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