# Math 420: Advanced Topics in Mathematics

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. The course may be repeated on a different topic for credit. Prerequisites vary with topic. Offered at least once a year.

This course number serves as an umbrella for individual mathematics classes that are offered less frequently than every other year. We anticipate offering three Math 420 courses in any consecutive two years.

The department is planning to offer:

• Fall 2015:  Numerical Analysis
• Spring 2016:  No offering
• Fall 2016: Combinatorics
• Spring 2017: TBD
• Fall 2017: Number Theory
• Spring 2018: Topology

### Description of Courses

• Advanced Linear Algebra This course begins as a review and continuation of Math 232, Linear Algebra.  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. Applications include topics such as least-squares fit, error-correcting codes, linear programming, physical problems employing eigenvalues, Markov chains and secret sharing.

Satisfies the List B requirement in major contracts and the standard major. Prerequisite: Math 232.

• Combinatorics:
The 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.

Satisfies the proof-based and List B requirement in major contracts and the standard major. Prerequisite: MATH 290.
• Numerical Analysis:
Students learn about numerical solutions to linear systems; numerical linear algebra; polynomial approximations (interpolation and quadrature); numerical differentiation and integration. Students also learn about error analysis and how to select appropriate algorithms for specific problems. A portion of the course will be devoted to learning how to implement these ideas on a computer. No formal programming experience is required, but as this a course that explores the interface between mathematics and computer science, students should be prepared both to prove theorems and program computers.

Prerequisites: MATH 280, 290, and CSCI 161 or equivalent. Cross-listed as CSCI 310. Satisfies the List B requirement in major contracts and the standard major.
• Number Theory:
This course is 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.

Prerequisite: Math 290. Satisfies the proof-based and List B requirements in major contracts and the standard major.
• TBD:
Topic to be chosen by the instructor and to be announced several months before registration. The topic usually reflects an area of particular interest of the instructor.
• Topology:
This course covers the basics of Pointset Topology. The course focuses on what type of structures need to be placed on a set in order to make it a ‘topological space’ and what properties are deduced from these structures. Topics include open and closed sets, continuity, compactness, connectedness, and a selection of topics from metric spaces, manifolds, functions spaces or quotient spaces.

Satisfies the proof-based and List A requirements in major contracts and the standard major. Prerequisite: MATH 290 or permission of instructor.