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Course Criteria
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3.00 Credits
Theory and applications of ordinary differential equations. Existence and uniqueness of solutions, methods for solving equations, linear differential equations, singularities, qualitative analysis of solutions, systems of equations. Prerequisite for undergraduates and enrollment requirement for graduate students: MATH 374 and 430.
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3.00 Credits
Theory and applications of partial differential equations. Cauchy problems, boundary problems, the Cauchy-Kovalevsky Theorem, Fourier Series, harmonic functions, elliptic equations, hyperbolic equations. Enrollment Requirement: MATH 260. Prerequisite for undergraduates and enrollment requirement for graduate students: MATH 374 and 430.
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3.00 Credits
Analysis in several variables including multivariable derivatives and integrals, inverse function theorem, implicit function theorem, generalizations of the fundamental theorem of calculus (e.g., Stokes' Theorem). Some of these topics may be presented from the point of view of differential forms. Enrollment Requirement: MATH 260. Prerequisite for undergraduates and enrollment requirement for graduate students: MATH 374 and 430.
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3.00 Credits
Study of functions of a complex variable, including analytic functions, contour integrals, Cauchy's Theorem, poles and residues, Liouville's Theorem, Laurent Series, the Residue Theorem, analytic continuation, conformal mappings. Prerequisite for undergraduates and enrollment requirement for graduate students: MATH 430.
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3.00 Credits
Foundations of functional analysis; linear and metric spaces; different modes of convergence; Hilbert Space; applications. May include topics such as calculus of variations, fixed point theorems, and operator theory. Prerequisites for undergraduates and enrollment requirement for graduate students: MATH 362, 374 and 430.
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3.00 Credits
Blend of continuous and discrete topics including sums, recurrences, elementary number theory, binomial coefficients, generating functions, discrete probability, and asymtotic methods. Prerequisite for undergraduates and enrollment requirement for graduate students: MATH 350 or 370 or 470 or 472 or 474.
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3.00 Credits
Graphs and digraphs; traversability; factorization; planarity and embedding; coloring; graph Ramsey theory; probabilistic methods; extremal graph theory; algebraic graph theory. Prerequisite for undergraduates and enrollment requirement for graduate students: MATH 350 or 370 or 470 or 472 or 474.
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3.00 Credits
Enumeration; combinatorial set systems; combinatorial designs; Ramsey theory; combinatorial optimization; matroids; axiomatic social choice. Prerequisite for undergraduates and enrollment requirement for graduate students: MATH 350 or 370 or 470 or 472 or 474.
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3.00 Credits
Geometric ideas selected from the following fields: euclidean geometry, hyperbolic geometry, projective geometry, introductory algebraic geometry, and computational geometry. Combines theoretical ideas with hands-on laboratory experience. Prerequisite for undergraduates and enrollment requirement for graduate students: MATH 374 and 470.
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3.00 Credits
Introduction to curves, surfaces, and possibly higher dimensional manifolds from the point of view of differential topology and/or differential geometry. Includes some of the following: Curves (e.g., Frenet-Serret Theorem and its consequences, isoperimetric inequality, four-vertex theorem, line integrals, Fenchel's Theorem), the topological classification of surfaces, vector fields, curvature on surfaces (leading up to some of the following: geodesics, minimal surfaces, Gauss's Theorema Egregium, and the Gauss-Bonnet Theorem), introduction to higher dimensional manifolds, differential forms and integration (possibly including Stokes' Theorem and global invariants such as the Euler Characteristic and De Rham Cohomology). Enrollment Requirement: MATH 260. Prerequisite for undergraduates and enrollment requirement for graduate students: MATH 374 and 430.
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