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## Spring, 2018 | Mathematics

## Introduction to Statistics

Basic concepts of statistics. Data collection (sampling and designing experiments), data organization (tables, graphs, frequency distributions, numerical summarization of data), statistical inference (elementary probability and hypothesis testing). Prereq: high school algebra.

## Mathematics and Music

An elementary introduction to the connections between mathematics and musical sound. Review of integers, ratios, prime numbers, functions, rationality, exponents, logarithms, trigonometry. Review of scales, clefs, key signatures, intervals, time signatures. Frequency and pitch. The connection between intervals and logarithms. Tuning and temperament, just intonation. Scales and modular arithmetic. The mathematics of harmony; the sound of the low prime numbers and their roles in harmony. Harmonics, partials and overtones. Numerical integration and basic Fourier analysis. The nature of complex tones. Analysis of instrument sounds. Human vowels and formants. Prerequisites: 2 years of high school algebra, and trigonometry.

## Calculus I

Derivatives of algebraic, trigonometric, and transcendental functions, techniques of differentiation and applications of the derivative. The definite integral and Fundamental Theorem of Calculus. Areas. Simpler integration techniques. Prereq: high school algebra and precalculus (including trigonometry). EXAMINATION SCHEDULE: Tests, at which attendance is required, will be given from 6:30 to 8:30 p.m. on: Wednesday January 31, Wednesday March 7, and Wednesday April 11.

## Calculus II

Continuation of Math 131. A brief review of the definite integral and Fundamental Theorem of Calculus. Techniques of integration, applications of the integral, sequences and series, and some material on differential equations. Prereq: Math 131 or a grade of B or better in a one year high school calculus course. EXAMINATION SCHEDULE: Tests, at which attendance is required, will be given from 6:30-8:30 p.m. on Tuesday January 30, Tuesday March 6 and Tuesday April 10.

## Honors Mathematics II

Matrices, linear systems, and determinants. Eigenvalues and eigenvectors, diagonalization, and the spectral theorem. Scalar and vector fields, differential and integral calculus of several variables, and the fundamental theorems of Green, Gauss, and Stokes. Restricted to first year students who have completed Math 203 in the fall semester. Math 204 can replace Math 233 in major/minor requirements.

## Differential Equations

Introduction to ordinary differential equations: first order equations, linear equations, systems of equations, series solutions, Laplace transform methods. Computer aided study of numerical solutions and graphic phase planes. Prereq: Math 233 or 233 concurrently. Examination Schedule: Tests, at which attendance is required, will be given from 6:30-8:30 p.m. on the following dates: Tuesday January 30, Tuesday March 6 and Tuesday April 10.

## Elementary Probability and Statistics

An elementary introduction to probability and statistics. Discrete and continuous random variables, mean and variance, hypothesis testing and confidence limits, nonparametric methods, Student's t, analysis of variance (ANOVA), (multiple) regression, contingency tables. Graphing calculator with statistical distribution functions (such as the TI-83) is required. Students considering a major or minor in mathematics should take Math 3200, NOT Math 2200. Examination Schedule: Tests, at which attendance is required, will be given from 6:30 to 8:30 p.m. on the following dates: Wednesday January 31, Wednesday March 7 and Wednesday April 11. Prerequisite: Math 131.

## Calculus III

Multivariable calculus. Topics include differential and integral calculus of functions of two or three variables: vectors and curves in space, partial derivatives, multiple integrals, line integrals, vector calculus at least through Green's Theorem. Prereq: Math 132, or a score of 4-5 on the Advanced Placement Calculus Exam (BC version). EXAMINATION SCHEDULE: Tests, at which attendance is required, will be given from 6:30-8:30 p.m. on: Wednesday January 31, Wednesday March 7 and Wednesday April 11.

## Mathematics for the Physical Sciences

Continuation of Math 233 that emphasizes topics of interest to the physical sciences. Topics in multivariable and vector calculus include: vector fields, div, grad, curl; line and surface integrals and connections to electromagnetism; Fourier series and integrals, boundary value problems (diffusion and wave equations); topics from calculus of variations. STUDENTS MAY NOT RECEIVE CREDIT FOR BOTH MATH 308 AND MATH 318. Prerequisite: Math 217 and Math 233, or permission of instructor.

## Matrix Algebra

An introductory course in linear algebra that focuses on Euclidean n-space, matrices and related computations. Topics include: systems of linear equations, row reduction, matrix operations, determinants, linear independence, dimension, rank, change of basis, diagonalization, eigenvalues, eigenvectors, orthogonality, symmetric matrices, least square approximation, quadratic forms. Introduction to abstract vector spaces. Tests, at which attendance is required, will be given from 6:30-8:30 p.m. on Wednesday February 21, and Wednesday April 4.
Prerequisite: Math 132.

## Foundations for Higher Mathematics

An introduction to the rigorous techniques used in advanced mathematics. Topics include basic logic, set theory, methods of proof and counterexamples, foundations of mathematics, construction of number systems, counting methods, combinatorial arguments and elementary analysis. Preq: Math 233 or Math 233 concurrently with permission of instructor.

## Differential Equations and Dynamical Systems

Qualitative theory of ordinary differential equations. Picard's existence and uniqueness theorem, the phase plane, Poincare-Bendixon theory, stationary points, attractors and repellors, graphical methods. Physical applications, including chaos, are indicated. Prerequisite: Math 217.

## Introduction to Calculus of Several Variables

Differential and integral calculus of functions of n-variables making some use of matrix algebra, and at a level of rigor intermediate between that of Calculus III and upper level analysis courses. Students may not receive credit for both Math 308 and 318. Prereq: Math 233 and 309 (not concurrent)

## Elementary to Intermediate Statistics and Data Analysis

An introduction to probability and statistics. Discrete and continuous random variables, mean and variance, hypothesis testing and confidence limits, Bayesian inference, nonparametric methods, Student's t-test, contingency table analysis, multifactor analysis of variance, random effects models, mixed models, multiple regression, maximum likelihood and logistic regression. Graphing calculator with Z, t, chi-square and F distribution functions (such as the TI-83 series) may be required. Calculus and the SAS software package are both used in an essential way. EXAMINATION SCHEDULE: Tests, at which attendance is required, will be given from 6:30-8:30 p.m. on Tuesday January 30, Tuesday March 6, and Tuesday April 10. Prerequisite: Math 233 or permission of the instructor.

## Biostatistics

A second course in elementary statistics with applications to life sciences and medicine. Review of basic statistics using biological and medical examples. New topics include incidence and prevalence, medical diagnosis, sensitivity and specificity, Bayes' rule, decision making, maximum likelihood, logistic regression, ROC curves and survival analysis. Each student will be required to perform and write a report on a data analysis project. Prereq: Math 3200, or (Math 2200 AND permission of instructor).

## Undergraduate Independent Study

Register for the section corresponding to the supervising instructor. Approval of instructor required.

## Introduction to Lebesgue Integration

Riemann integration; measurable functions; measures; the Lebesgue
integral; integrable functions; L^p spaces; modes of convergence;
decomposition of measures; product measures; Lebesgue measure. Prerequisite: Math 4111 or permission of the instructor.

## Topology II

A continuation of Math 4171 featuring more advanced topics in topology. The content may with each offering. Prerequisite: Math 4171, or permission of instructor.

## Experimental Design

A first course in the design and analysis of experiments, from the point of view of regression. Factorial, randomized block, split-plot, Latin square, and similar design. Prerequisite: CSE 131 or 200, Math 3200, or permission of instructor.

## Modern Algebra

An introduction to groups, rings, and fields. Includes permutation groups, group and ring homomorphisms, field extensions, connections with linear algebra. Prereq: Math 429.

## Algebraic Geometry

Introduction to affine and projective algebraic varieties, the Zariski topology, regular and rational mappings, simple and singular points, divisors and differential forms, genus, the Riemann-Roch theorem. Prerequisites: Math 318, 429, and 430, or permission of the instructor.

## Advanced Linear Statistical Models

Review of basic linear models relevant for the course; generalized linear models including logistic and Poisson regression (heterogeneous variance structure, quasilikelihood); linear mixed-effects models (estimation of variance components, maximum likelihood estimation, restricted maximum likelihood, generalized estimating equations), generalized linear mixed-effects models for discrete data, models for longitudinal data, optional multivariate models as time permits. The computer software R will be used for examples and homework problems, and some of the scheduled classroom time will be spent in a computer lab. Implementation in SAS will be mentioned for several specialized models. Prerequisites: Math 439 and a course in linear algebra (such as Math 309 or 429), or consent of instructor.

## Topics in Applied Mathematics

Application and analysis of numerical methods for ordinary and partial differential equations. Specific topics may include: Runge-Kutta methods, geometric numerical integrators, finite difference methods, finite element methods, spectral methods, etc. Prerequisites: Math 449 or permission of instructor.

## Bayesian Statistics

Introduces the Bayesian approach to statistical inference for data analysis in a variety of applications. Topics include: comparison of Bayesian and frequentist methods, Bayesian model specification, choice of priors, computational methods such as rejection sampling, and stochastic simulation (Markov chain Monte Carlo), empirical Bayes method, hands-on Bayesian data analysis using appropriate software. Prerequisite: Math 493 and either Math 3200 or 494; and some acquaintance with fundamentals of computer programming (such as CSE 131 or CSE 200), or permission of instructor.

## Multivariate Statistical Analysis

A modern course in multivariate statistics. Elements of classical multivariate analysis as needed, including multivariate normal and Wishart distributions. Clustering; principal component analysis. Model selection and evaluation; prediction error; variable selection; stepwise regression; regularized regression. Cross-validation. Classification; linear discriminant analysis. Tree-based methods. Time permitting, optional topics may include nonparametric density estimation, multivariate regression, support vector machines, and random forests.
Prerequisite: Multivariable calculus (Math 233), linear or matrix algebra (Math 429 or Math 309), multivariable-calculus-based probability and mathematical statistics (Math 493, Math 494) and linear models (Math 439). Prior knowledge of R at the level introduced in Math 439 is assumed.

## MATHEMATICAL FOUNDATIONS OF BIG DATA

Mathematical foundations of data science. Core topics include: Probability in high dimensions; curses and blessings of dimensionality; concentration of measure; matrix concentration inequalities. Essentials of random matrix theory. Randomized numerical linear algebra. Data clustering. Depending on time and interests, additional topics will be chosen from: Compressive sensing; efficient acquisition of data; sparsity; low-rank matrix recovery. Divide, conquer and combine methods. Elements of topological data analysis; point cloud; Cech complex; persistent homology. Selected aspects of high-
dimensional computational geometry and dimension reduction; embeddings; Johnson-Lindenstrauss; sketching; random projections. Diffusion maps; manifold learning; intrinsic geometry of massive data sets. Optimization and stochastic gradient descent. Random graphs and complex networks. Combinatorial group testing.
Prerequisite: Multivariable calculus (Math 233), linear or matrix algebra (Math 429 or 309), and multivariable-calculus-based probability and mathematical statistics (Math 493-494). Prior familiarity with analysis, topology, and geometry is strongly recommended. A willingness to learn new mathematics as needed is essential.

## Mathematical Statistics

Theory of estimation, minimum variance and unbiased estimators, maximum likelihood theory, Bayesian estimation, prior and posterior distributions, confidence intervals for general estimators, standard estimators and distributions such as the Student-t and F-distribution from a more advanced viewpoint, hypothesis testing, the Neymann-Pearson Lemma (about best possible tests), linear models, and other topics as time permits. Prereq: Math 3200 and 493, or permission of the instructor.

## Stochastic Processes

Content varies with each offering of the course. Past offerings have included such topics as random walks, Markov chains, Gaussian processes, empirical processes, Markov jump processes, and a short introduction to martingales, Brownian motion and stochastic integrals. Prerequisites: Math 318 and 493, or permission of instructor.

## Study for Honors

Prerequisites: junior or senior standing, eligibility for honors work in mathematics and permission of the Department's Director of Undergraduate Studies.

## Complex Analysis II

Continuation of Math 5021. Prerequisite: Math 5021 or permission of intstructor.

## Methods of Theoretical Physics II

Continuation of Phys 501. Introduction to function spaces; self-adjoint and unitary operators; eigenvalue problems, partial differential equations, special functions; integral equations; introduction to group theory. Prerequisite: Phys 501, or permission of instructor.

## Algebra II

Continuation of Math 5031. Prereq: Math 5031 or permission of instructor.

## Geometry II

Continuation of Math 5041. Math 5042 and Math 5043 are offered in alternate spring semesters as a sequel to Math 5041. Prerequisite: Math 5041 or permission of instructor.

## Measure Theory and Functional Analysis II

Continuation of Math 5051. Prereq: Math 5051 or permission of instructor.

## Theory of Statistics II

Continuation of Math 5061. Prerequisite: Math 5061 or permission of instructor.

## Teaching Seminar

Techniques of mathematical instruction for prospective teaching assistants. Prerequisite: Permission of instructor.