COURSE REQUIREMENTS

Course Requirements (30 UNITS) The standard load is two courses per semester.

 MATH 510A/B Algebra and Number Theory MATH 511A/B Linear Algebra and Geometry MATH 512 A/B Concepts of Analysis MATH 513 A/B Discrete Mathematics MATH 514 A/B Probability and Statistics

All courses include active learning and presentations on selected topics. In addition, each B section of the courses will culminate with the student working on a project and writing an article on it.

MATH 510A/B. Algebra and Number Theory: (3) Numbers: integers and prime numbers, rational and complex algebraic numbers. Symmetry and group theory. Rings of polynomials and algebraic integers. Basic algebraic geometry and algebraic extensions. Elementary Galois theory and the theory of equations.

MATH 511A/B. Linear Algebra and Geometry: (3) Modern applications of mathematics that involve matrices. Basic properties of vectors of R2 and R3. Linear transformations of Euclidean 2- and 3-Space and the classification of its rigid motions. Symmetric bilinear forms, conics and quadrics. Basic topology of Rn . Spherical geometry and isometries of the sphere. Poincaré’s models of the hyperbolic plane, and the study of isometries of the hyperbolic plane through Möbius transformations.

MATH 512A/B. Concepts of Analysis: (3) The real number system, countable and uncountable sets, cardinal numbers, and Cantor diagonal argument. Well-ordered sets. Ordinal numbers. Numerical sequences and numerical series of real numbers. Continuity, differentiability, and integration of functions of one variable. Sequences and series of functions, uniform convergence. Ordinary differential equations.

MATH 513A/B. Discrete Mathematics: (3) Permutations, Combinations, Multinomial Coefficients and Pascal Triangles, Pigeon Hole Principle, Inclusion-Exclusion Principle, Ramsey Numbers, Characteristic Functions and algorithms, Generating Functions, Finite Probabilities, Recurrence Relations. Graph Theory: Connectedness, Graph Colorings, Planar Graphs, Trees, Adjacency Matrices, Eulerian Paths, Hamiltonian Paths, Tournaments, Matching and Covering, Optimization problems for networks. Coding Theory: Information transmission, coding and decoding, error correcting codes. Applications: Power in Simple Games and the UN security council, Cost Allocation, Chemical bonds and the Number of Trees, Secondary Structure in RNA, Organic Compounds Built from Benzene Rings, One-Way street assignments, Finding Unknown RNA/DNA chains, De Bruijn Sequences and Telecommunications, Scheduling Problems.

Math 514A/B. Probability and Statistics: (3) Probability rules, continuous and random variables and their distributions, central limit theorem. Examination of elementary topics in statistics from the advanced point of view: exploratory analysis, graphical display, random phenomena, probability distributions, simulation, correlation and regression, survey sampling and experimental design, sampling distributions, confidence intervals and significance tests for proportions and means, chi-square tests.

Culminating Experience - At the end of their course work students will submit a portfolio that summarizes their individual contributions to the courses through their projects and articles. A committee chosen by the candidate will assess this portfolio.