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## Fall, 2018 | Mathematics

## Foundations for Calculus

A limited enrollment class designed specifically for students planning to take calculus but who need additional precalculus preparation. The course aims to build both the technical skills and the conceptual understanding needed to succeed in calculus, and gives previews of some selected topics from calculus. The course will emphasize links between the graphical, numeric, and algebraic viewpoints. A variety of approaches will be used to present the material, e.g., technology, group work, writing assignments. Prerequisite: 2 yrs high school algebra and geometry (or the equivalent). MUST BE TAKEN FOR A LETTER GRADE.

## 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. Prerequisite: 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 September 20, October 10, and November 14.

## 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 to 8:30 p.m. on September 17, October 9, and November 13.

## Real Mathematical Applications: Solving Problems with Calculus I

This is a one credit course, that can only be taken concurrently with Math 131, Calculus
I. The purpose of the course is to show how mathematics can solve real world problems, and how calculus dramatically expands the range of problems that can be tackled. Each class will be devoted to the analysis of some problems, which may include: dimensional analysis, the mathematics of convoys, Fibonacci numbers, fractals, linear regression, Euclid's algorithm, Stein's algorithm, network capacities, Braess's paradox, Galton's approach to surnames, how genes spread through populations, SIR model of infectious diseases.
The first few classes will not use differentiation.
Must be taken concurrently with Math 131.
Refere

## Honors Mathematics I

This is the first half of a one-year calculus sequence for first year students with a strong interest in mathematics with an emphasis on rigor and proofs. The course begins at the beginning but assumes the students have already studied the material from a more "mechanical" view. Students who complete both semesters will have completed the material Calc III and other topics that may let them move through the upper level math curriculum more quickly. Sets, functions, real numbers, and methods of proof. The Riemann-Darboux integral, limits and continuity, differentiation, and the fundamental theorems of calculus. Sequences and series of real numbers and of functions. Vector spaces and linear maps. Prerequisite: Score of 5 on the A.P Calculus Exam, BC version, or the equivalent.

## Differential Equations

Introduction to ordinary differential equations: first-order equations, linear equations, systems of equations, series solutions, Laplace transform methods, numerical solutions. Prerequisite: successful completion of, or concurrent enrollment in, Math 233. EXAMINATION SCHEDULE: Tests, at which attendance is required, will be given from 6:30 to 8:30 p.m. on September 17, October 9, and November 13.

## Elementary Probability and Statistics

An elementary introduction to statistical concepts, reasoning and data analysis. Topics include statistical summaries and graphical presentations of data, discrete and continuous random variables, the logic of statistical inference, design of research studies, point and interval estimation, hypothesis testing, and linear regression. Students will learn a critical approach to reading statistical analyses reported in the media, and how to correctly interpret the outputs of common statistical routines for fitting models to data and testing hypotheses. A major objective of the course is to gain familiarity with basic R commands to implement common data analysis procedures. Students intending to pursue a major or minor in mathematics or wishing to take 400 level or above statistics courses should instead take Math 3200.
EXAMINATION SCHEDULE: Tests, at which attendance is required, will be given from 6:30 to 8:30 p.m. on September 20, October 10, and November 14.
Prereqs: Math 131

## Calculus III

Differential and integral calculus of functions of two and three variables. Vectors, curves and surfaces in space, partial derivatives, multiple integrals, line integrals, vector calculus through Green's Theorem. Prerequisite: Math 132, or a score of 4 or 5 on the Advanced Placement Calculus Examination (BC version). EXAMINATION SCHEDULE: Tests, at which attendance is required, will be given from 6:30 to 8:30 p.m. on September 20, October 10, and November 14.

## Elementary Geometry From An Advanced Point of View

A rigorous modern treatment of Euclidean geometry, and an introduction to non-Euclidean geometry. Prereq: Math 310 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. Prerequisite: Math 132. EXAMINATION SCHEDULE: In-semester exams, at which attendance is required, will be given from 6:30 to 8:30 p.m. on October 8, and November 12.

## Foundations for Higher Mathematics

An introduction to the rigorous techniques used in more advanced mathematics. Topics include set theoretic methods of proof, counter-examples, basic logic, foundations of mathematics. Use of these methods in areas such as construction of number systems, counting methods, combinatorial arguments and elementary analysis. Students who want a writing intensive (WI) course should register INSTEAD for Math 310W. Prereq: Math 233.

## Foundations For Higher Mathematics With Writing

An introduction to the rigorous techniques used in advanced work in mathematics. At least 3 papers (of length 4-5 pages) will be required, each with at least one revision. Prerequisite: Math 233.

## Introduction to Calculus of Several Variables

Selected topics for functions of several variables involving some matrix algebra and presented at a level of rigor intermediate between that of Calculus III and higher level analysis courses. Students may not receive credit toward a mathematics major or minor for both Math 308 and 318. Prerequisites: Math 233 and Math 309 (not concurrent).

## Elementary to Intermediate Statistics and Data Analysis

An introduction to probability and statistics. Major topics include elementary probability, special distributions, experimental design, exploratory data analysis, estimation of mean and proportion, hypothesis testing and confidence, regression, and analysis of variance. Emphasis is placed on development of statistical reasoning, basic analytic skills, and critical thinking in empirical research studies. The use of the statistical software R is integrated into lectures and weekly assignments. Required for students pursuing a major or minor in mathematics or wishing to take 400 level or above statistics courses.
EXAMINATION SCHEDULE: Tests, at which attendance is required, will be given from 6:30 to 8:30 p.m. on September 17, October 9, and November 13.
Prereqs: Math 132. Though Math 233 is not essential, it is recommended.

## Undergraduate Independent Study

See the beginning of the mathematics listings and register for the section corresponding to the supervising instructor. Approval of instructor is required.

## Introduction to Analysis

The real number system and the least upper bound property; metric spaces (completeness, compactness, and connectedness); continuous functions (in R^n; on compact spaces; on connected spaces); C(X) (pointwise and uniform convergence; Weierstrass approximation theorem); differentiation (mean value theorem; Taylor's theorem); the contraction mapping theorem; the inverse and implicit function theorems. Prerequisite: Math 310 or permission of instructor.

## Partial Differential Equations

Introduction to the theory of PDE's with applications to selected classical problems in physics and engineering. Linear and quasilinear first order equations, derivation of some of the classical PDE's of physics, and standard solution techniques for boundary and initial value problems. Preliminary topics such as orthogonal functions, Fourier series, and variational methods introduced as needed. Prerequisites: Math 217 and 309, or permission of instructor.

## Topology I

An introduction to the most important ideas of topology, with an emphasis on metric spaces. Course includes any necessary ideas from set theory, topological spaces, subspaces, products and quotients, compactness and connectedness. Prerequisite: Math 4111 or permission of instructor.

## Linear Algebra

Introduction to the linear algebra of finite-dimensional vector spaces. Includes systems of equations, matrices, determinants, inner product spaces, spectral theory. Prerequisite: Math 310 or permission of instructor. Math 309 is not an explicit prerequisite but students should already be familiar with such basic topics from matrix theory as matrix operations, linear systems, row reduction and Gaussian elimination. Material on these topics in early chapters of the text will be covered very quickly.

## Linear Statistical Models

Theory and practice of linear regression, analysis of variance (ANOVA) and their extensions, including testing, estimation, confidence interval procedures, modeling, regression diagnostics and plots, polynomial regression, colinearity and confounding, model selection, geometry of least squares. The theory will be approached mainly from the frequentist perspective and use of the computer (mostly R) to analyze data will be emphasized.
Prerequisite: Math 3200, a course in linear algebra (Math 309 or 429); some acquaintance with fundamentals of computer programming (CSE 131) and Math 493, or permission of instructor.

## Numerical Applied Mathematics

Computer arithmetic, error propagation, condition number and stability; mathematical modeling, approximation and convergence; roots of functions; calculus of finite differences; implicit and explicit methods for initial and boundary value problems; numerical integration; numerical solution of linear systems, matrix equations, and eigensystems; Fourier transforms; optimization. Various software packages may be introduced and used. Prerequisites: Math 217, 309 and CSE 131 or 200 (or other computer background with permission of the instructor); .

## Time Series Analysis

Time series data types; autocorrelation; stationarity and nonstationarity; autoregressive moving average models; model selection methods; bootstrap confidence intervals; trend and seasonality; forecasting; nonlinear time series; filtering and smoothing; autoregressive conditional heteroscedasticity models; multivariate time series; vector autoregression; frequency domain; spectral density;
state-space models; Kalman filter. Emphasis on real-world applications and data analysis using statistical software. Prerequisite: Math 493 and either Math 3200 or 494; or permission of the instructor. Some programming experience may also be helpful (consult with the instructor).

## Analytic Combinatorics

Analytic combinatorics is the study of large structured combinatorial configurations. The course will be broken into two components. First, generating functions will be used to encapsulate counting sequences and their recurrence structures with a formal power series. Second, analytic methods will be used to obtain the precise asymptotic behavior of counting sequences. The informal prerequisites are: familiarity with basic discrete math objects: sets, permutations, combinations, graphs; power series; mathematical maturity (e.g. the ability to write rigorous proofs and to absorb new definitions quickly).
Formal Prerequisites: Math 310.

## Statistical Computation

Introduction to modern computational statistics. Pseudo-random number generators; inverse transform and rejection sampling. Monte Carlo approximation. Nonparametric bootstrap procedures for bias and variance estimation; bootstrap confidence intervals. Markov chain Monte Carlo methods; Gibbs and Metropolis-Hastings sampling; tuning and convergence diagnostics. Cross-validation. Time permitting, optional topics include numerical analysis in R, density estimation, permutation tests, subsampling, and graphical models. Prior knowledge of R at the level used in Math 494 is required.
Prereqs: Math 233, 309, 493, 494 (not concurrently); acquaintance with fundamentals of programming in R.

## Probability

Mathematical theory and application of probability at the advanced undergraduate level; a calculus based introduction to probability theory. Topics include the computational basics of probability theory, combinatorial methods, conditional probability including Bayes Theorem, random variables and distributions, expectations and moments, the classical distributions, and the central limit theorem. Section 2 is specifically for students who enrolled in Math 3200 sections 3 and 4 in Spring 2018. Prereq: Math 233. Math 318 or Math 308 is recommended.

## Linear Representations of Finite Group and Lie Groups

An introduction to the theory of group representations and some of its applications. Topics include: general facts about group and linear algebra; linear representations of finite and compact groups; Lie groups and Lie algebras; applications to partial differential equations and quantum theory. Prerequisites: Math 309 and 318 or permission from the instructor.

## Study for Honors

Prereq: Senior standing, a distinguished performance in upper level mathematics courses, and permission of the Chair of the Undergraduate Committee. Register for the section (listed in department header) corresponding to your honors project supervisor.

## Independent Work

See the beginning of the mathematics listings and register for the section corresponding to supervising instructor. Prerequisite: Graduate standing and permission of the instructor.

## Theoretical Physics

The first part of a two-semester course reviewing the mathematical methods essential for the study of physics. Theory of functions of a complex variable, residue theory; review of ordinary differential equations; introduction to partial differential equations; integral transforms. Prerequisite: undergraduate differential equations (Math 217), or permission of instructor.

## Complex Analysis I

An intensive course in complex analysis at the introductory graduate level. Math 5021-5022 form the basis for the Ph.D. qualifying exam in complex variables. Prerequisite: Math 4111, 4171 and 4181, or permission of the instructor.

## Algebra I

An introductory graduate level course on the basic structures and methods of algebra. Detailed survey of group theory including the Sylow theorems and the structure of finitely generated Abelian groups, followed by a study of basic ring theory and the Galois theory of fields. Math 5031-5032 form the basis for the Ph.D. qualifying exam in algebra. Prerequisite: Math 430 or the equivalent, or permission of the instructor.

## Geometry I

An introductory graduate level course including differential calculus in n-space; differentiable manifolds; vector fields and flows; differential forms and calculus on manifolds; elements of Lie groups and Lie algebras; Frobenius theorem; elements of Riemannian geometry. Math 5041and 5042 or 5043 form the basis for the Ph.D. qualifying exam in geometry / topology. Prerequisites: Math 4121, 429, and 4181, or permission of the instructor.

## Measure Theory and Functional Analysis I

An introductory graduate level course including the theory of integration in abstract and Euclidean spaces, and an introduction to the basic ideas of functional analysis. Math 5051-5052 form the basis for the Ph.D. qualifying exam in analysis. Math 4111, 4171, and 4181, or permission of the instructor.

## Theory of Statistics I

An introductory graduate level course. Probability spaces; derivation and transformation of probability distributions; generating functions and characteristic functions; law of large numbers, central limit theorem; exponential family; sufficiency, uniformly minimum variance unbiased estimators, Rao-Blackwell theorem, information inequality; maximum likelihood estimation; estimating equation; Bayesian estimation; minimax estimation; basics of decision theory. Prerequisite: Math 493 or the equivalent. Some knowledge of basic ideas from analysis (e.g. Math 4111) will be helpful: consult with instructor.

## The Mathematics of Quantum Theory

This is a course primarily aimed at Mathematics students although others are welcome. Topics may include: spectral theory of self-adjoint operators in Hilbert spaces and Schroedinger operators, non-commutative probability theory, elements of unitary representations of groups, assorted physical examples (harmonic oscillator, angular momentum and spin, the hydrogen atom, etc.) Prerequisite: Real Analysis Math 5051-5052 or permission of instructor.

## Information Theory

Discrete source and channel model, definition of information rate and channel capacity, coding theorems for sources and channels, encoding and decoding of data for transmission over noisy channels. Corequisite: ESE 520.

## Geometric Mechanics

A graduate-level introduction to classical mechanics from the modern, differential geometric point of view. Topics include: the Lagrangian and Hamiltonian formalisms, symplectic and Poisson geometry, Lie groups and Lie algebras, symmetries, conservation laws, reduction. Prerequisites: prior exposure to manifolds and to groups or permission of the instructor.

## Research

See the beginning of the mathematics listings and register for the section corresponding to supervising instructor. Prerequisite: Graduate standing and permission of the instructor.

## Seminar

See the beginning of the mathematics listings and register for the section corresponding to supervising instructor. Prerequisite: Graduate standing and permission of the instructor.

## Teaching Seminar

Principles and practice in the teaching of mathematics at the college and university level. Prerequisite: graduate standing, or permission of instructor.