MATH

This course allows students to explore mathematical topics beyond those covered in the standard mathematics curriculum. Some semester-long topics include combinatorics, number theory, numerical analysis, and topology. See the department website for further information on topics to be offered during the next two years, including the prerequisites for each topic. The course may be repeated on a different topic for credit. Prerequisites vary with topic.

This course begins as a review and continuation of MATH 290. Topics covered include invariant subspaces, Jordan canonical form, and rational canonical forms of linear transformations. The remainder of the course is split between advanced topics and applications. Advanced topics include decompositions (such as the LU decomposition), principal axis theorem, alternate definitions of the determinant, singular values, and quadratic forms.

The calculus of functions with complex numbers as inputs and outputs has surprising depth and richness. The basic theory of these functions is developed in this course. The standard topics of calculus (function, limit, continuity, derivative, integral, series) are explored in this new context of complex numbers leading to some powerful and beautiful results. Applications include using conformal mappings to solve boundary-value problems for Laplace's equation.

This course introduces the theory of linear regression and uses it as a vehicle to investigate the mathematics behind applied statistics. The theory combines probability theory and linear algebra to arrive at commonly used results in statistics. The theory helps students understand the assumptions on which these results are based and decide what to do when these assumptions are not met, as is usually the case in applied statistics.

This course provides an introduction to the standard topics of probability theory, including probability spaces, random variables and expectations, discrete and continuous distributions, generating functions, independence, sampling distributions, laws of large numbers, and the central limit theorem. The course emphasizes modeling real-world phenomena throughout.

This course covers advanced methods in applied statistics, beyond those of MATH 260. The analyses will be conducted using R, so students entering the course should already have a working knowledge of R. Topics may include generalized linear models, Bayesian statistics, time series analysis, categorical data analysis, and/or statistical graphics.

This course is an introduction to the application of calculus and linear algebra to the geometry of curves and surfaces. Topics include the geometry of curves, Frenet formulas, tangent planes, normal vectors and orientation, curvature, geodesics, metrics, and isometries. Additional topics may include the Gauss-Bonnet Theorem, minimal surfaces, calculus of variations, and hyperbolic geometry.

Building on the foundation of point-set topology, this course introduces more advanced topics in topology such as metric spaces, quotient spaces, covering spaces, homotopy, the fundamental group, mathematical knots, and manifolds.

This course entails the study of the properties of numbers, with emphasis on the positive integers. Topics include divisibility, factorization, congruences, prime numbers, arithmetic functions, quadratic residues, and Diophantine equations. Additional topics may include primitive roots, continued fractions, cryptography, Dirichlet series, binomial coefficients, and Fibonacci numbers.

This course entails study of the basic principles of combinatorial analysis. Topics include combinations, permutations, inclusion-exclusion, recurrence relations, generating functions, and graph theory. Additional material may be chosen from among the following topics: Latin squares, Hadamard matrices, designs, coding theory, and combinatorial optimization.