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## Spring, 2017 | 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.

## 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: Tuesday February 7, Tuesday March 7, and Tuesday April 4.

## 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 Wednesday February 8, Wednesday March 8, and Wednesday April 5,.

## 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: Wednesday February 8, Wednesday March 8, and Wednesday April 5,.

## 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: Tuesday February 7, Tuesday March 7, and Tuesday April 4. 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: Tuesday February 7, Tuesday March 7, and Tuesday April 4.

## 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 22, and Monday April 3.
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.

## 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 February 7, Tuesday March 7, and Tuesday April 4. 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).

## Introduction to Combinatorics

Basics of enumeration (combinations, permutations and enumeration of functions between finite sets), generating functions; the inclusion-exclusion principle, partition
theory and introductory graph theory. As time permits, additional topics may include Ramsey's Theorem, probabilistic methods in combinatorics and algebraic methods in combinatorics. Prerequisites: Math 132, 309 and 310, or permission of the instructor.

## UNDERGRADUATE INDEPENDENT STUDY

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

## An Introduction to Differential Geometry

A study of properties of curves and surfaces in 3-dimensional Euclidean space. The course is essentially a modern recounting of a seminal paper of Gauss. Prerequisites: Math 233 and Math 309.

## 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.

## 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.

## 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: Mathematics for Multimedia

Survey of the mathematics behind the algorithms and computer software used to present large amounts of digitally coded information (text, images, sound). The subject matter divides into the "continuous" (the mathematical properties of physical signals and the algorithms used to process them, treating them as an approximation to reality whose quality must be kept up) and the "discrete" (algorithms that manipulate exact data, where the main problem is reducing the computer's workload). Concepts will be explored both through mathematics and hands-on computer experiments. Prereqs: 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.

## 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.

## Algebraic Topology

Algebraic topology, including homology, cohomology, the fundamental group, and covering spaces. Math 5043 and Math 5042 are offered in alternate spring semesters as a sequel to Math 5041. Prerequisites: Math 5041 and a course in abstract algebra.

## MEASURE THEORY & 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.

## Statistics for Medical and Public Health Researchers

This course is an introduction to basic statistical analysis for graduate students in medicine, biology, and public health. Students will be introduced to core statistical tools used to study human health outcomes. Topics include: measurement, descriptive analysis, correlation, graphical analysis, hypothesis testing, confidence intervals, analysis of variance, and regression analysis. Major components of the course include learning how to collect, manage, and analyze data using computer software, and how to effectively communicate to others results from statistical analyses. The second aspect of the course is focused on the statistical package R, which is is the most powerful, extensively featured, and capable statistical computing tool available. Course may not be used for credit in undergraduate math major/minor programs, nor in any Mathematics or Statistics graduate programs. Prerequisite: Current graduate enrollment in a program in DBBS, medicine or public health, or permission of instructor.

## Harmonic Analysis in the Complex Domain

An introduction to harmonic analysis in the complex variable context. We will treat both one-variable and several-variable questions. These will include the Bergman and Szego kernels, the Heisenberg group, the inhomogeneous Cauchy-Riemann equations, and other topics.
Prerequisites: 5051-5052, 5021-5022, 5041.
It is not necesssary to know anything about several complex variables.

## Topics in Topology

An introduction to the geometry of the hyperbolic plane and hyperbolic 3-space, with an emphasis on the impact of this geometry on the study of surfaces and 3-manifolds. Topics will include the Nielsen-Thurston classification of homeomorphisms of surfaces, and the background necessary to understand the statement and import of the Thurston-Perelman Geometrization Theorem.
Prerequisites: Mathematics 4171 or its equivalent

## Asymptotic Statistics and Differential Geometry

An exploration of the differential geometric aspects of asymptotic statistics. Topics include reviews of relevant statistical and geometric concepts, such as exponential families, Fisher information, series expansions, manifolds, tensors and connections. Measures of statistical curvature will be introduced and studied in terms of their importance in the large-sample theory of statistical inference. The course is intended for graduate students in mathematics and statistics, but other students are welcome.
Instructor approval required for registration. The course format is 6 weeks of lectures.

## TEACHING SEMINAR

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