# Course Descriptions

## Lower Division Courses

### MATH 011: Elementary Algebra

See Preregistration guide for instructors and times

Description:

Beginning algebra enhanced with pre-algebra topics such as arithmetic, fractions, and word problems as need indicates. Topics include real numbers, linear equations and inequalities in one variable, polynomials, factoring, algebraic fractions, problem solving, systems of linear equations, rational and irrational numbers, and quadratic equations.

This course is only offered online through Continuing and Professional Education.

### MATH 100: Basic Math Skills for the Modern World

See Preregistration guide for instructors and times

Description:

Topics in mathematics that every educated person needs to know to process, evaluate, and understand the numerical and graphical information in our society. Applications of mathematics in problem solving, finance, probability, statistics, geometry, population growth. Note: This course does not cover the algebra and pre-calculus skills needed for calculus.

### MATH 102: Analytic Geometry and Trigonometry

See Preregistration guide for instructors and times

Prerequisites:

Math 101

Description:

Second semester of the two-semester sequence MATH 101-102. Detailed treatment of analytic geometry, including conic sections and exponential and logarithmic functions. Same trigonometry as in MATH 104.

### MATH 104: Algebra, Analytic Geometry and Trigonometry

See Preregistration guide for instructors and times

Prerequisites:

MATH 011 or Placement Exam Part A score above 15. Students with a weak background should take the two-semester sequence MATH 101-102.

Description:

One-semester review of manipulative algebra, introduction to functions, some topics in analytic geometry, and that portion of trigonometry needed for calculus.

### MATH 113: Math for Elementary Teachers I

See Preregistration guide for instructors and times

Prerequisites:

MATH 011 or satisfaction of R1 requirement.

Description:

Fundamental and relevant mathematics for prospective elementary school teachers, including whole numbers and place value operations with whole numbers, number theory, fractions, ratio and proportion, decimals, and percents. For Pre-Early Childhood and Pre-Elementary Education majors only.

### MATH 114: Math for Elementary Teachers II

See Preregistration guide for instructors and times

Prerequisites:

MATH 113

Description:

Various topics that might enrich an elementary school mathematics program, including probability and statistics, the integers, rational and real numbers, clock arithmetic, diophantine equations, geometry and transformations, the metric system, relations and functions. For Pre-Early Childhood and Pre-Elementary Education majors only.

### MATH 121: Linear Methods and Probability for Business

See Preregistration guide for instructors and times

Prerequisites:

Working knowledge of high school algebra and plane geometry.

Description:

Linear equations and inequalities, matrices, linear programming with applications to business, probability and discrete random variables.

### MATH 127: Calculus for Life and Social Sciences I

See Preregistration guide for instructors and times

Prerequisites:

Proficiency in high school algebra, including word problems.

Description:

Basic calculus with applications to problems in the life and social sciences. Functions and graphs, the derivative, techniques of differentiation, curve sketching, maximum-minimum problems, exponential and logarithmic functions, exponential growth and decay, and introduction to integration.

### MATH 128: Calculus for Life and Social Sciences II

See Preregistration guide for instructors and times

Prerequisites:

Math 127

Description:

Continuation of MATH 127. Elementary techniques of integration, introduction to differential equations, applications to several mathematical models in the life and social sciences, partial derivatives, and some additional topics.

### MATH 128H: Honors Calculus for Life and Social Sciences II

See Preregistration guide for instructors and times.

Prerequisites:

Math 127

Description:

Honors section of Math 128.

### MATH 131: Calculus I

See Preregistration guide for instructors and times

Prerequisites:

High school algebra, plane geometry, trigonometry, and analytic geometry.

Description:

Continuity, limits, and the derivative for algebraic, trigonometric, logarithmic, exponential, and inverse functions. Applications to physics, chemistry, and engineering.

### MATH 132: Calculus II

See Preregistration guide for instructors and times

Prerequisites:

Math 131 or equivalent.

Description:

The definite integral, techniques of integration, and applications to physics, chemistry, and engineering. Sequences, series, and power series. Taylor and MacLaurin series. Students expected to have and use a Texas Instruments 86 graphics, programmable calculator.

### MATH 233: Multivariate Calculus

See Preregistration guide for instructors and times

Prerequisites:

Math 132.

Description:

Techniques of calculus in two and three dimensions. Vectors, partial derivatives, multiple integrals, line integrals.

### MATH 233H: Honors Multivariate Calculus

See Preregistration guide for instructors and times

Prerequisites:

Math 132.

Description:

Honors section of Math 233.

### MATH 235: Introduction to Linear Algebra

See Preregistration guide for instructors and times

Prerequisites:

Math 132 or consent of the instructor.

Description:

Basic concepts of linear algebra. Matrices, determinants, systems of linear equations, vector spaces, linear transformations, and eigenvalues.

### MATH 235H: Honors Introduction to Linear Algebra

See Preregistration guide for instructors and times

Prerequisites:

Math 132 or consent of the instructor.

Description:

Honors section of Math 235.

### MATH 331: Ordinary Differential Equations for Scientists and Engineers

See Preregistration Guide for instructors and times

Prerequisites:

Math 132 or 136; corequisite: Math 233

Text:

TBA

Description:

(Formerly Math 431) Introduction to ordinary differential equations.First and second order linear differential equations, systems of linear differential equations, Laplace transform, numerical methods, applications. (This course is considered upper division with respect to the requirements for the major and minor in mathematics.)

### STAT 111: Elementary Statistics

See Preregistration guide for instructors and times

Prerequisites:

High school algebra.

Description:

Descriptive statistics, elements of probability theory, and basic ideas of statistical inference. Topics include frequency distributions, measures of central tendency and dispersion, commonly occurring distributions (binomial, normal, etc.), estimation, and testing of hypotheses.

### STAT 240: Introduction to Statistics

See Preregistration guide for instructors and times

Description:

Basics of probability, random variables, binomial and normal distributions, central limit theorem, hypothesis testing, and simple linear regression

## Upper Division Courses

### MATH 300.1: Fundamental Concepts of Mathematics

Weimin Chen MWF 11:15-12:05

Prerequisites:

Math 132 with a grade of 'C' or better

Text:

An Introduction to Mathematical Thinking, by William J. Gilbert and Scott A. Vanstone.

Note:

Math 300 students are required to register also for the 1-credit co-seminar Math 391A. Seminar times will be arranged during the first week of classes.

Description:

The goal is that you learn to read, understand, and construct coherent logically correct proofs, so that you may more easily make the transition from calculus to the more theoretical junior-senior courses, especially abstract algebra and modern analysis. Starting with explicit axioms and precisely stated definitions, you will systematically develop basic propositions about induction, equivalence relations, real numbers, infinite sets, group theory, and metric spaces and point set topology. You will be provided with the needed background about logic, sets, and functions. For nearly every class you will create written mathematical proofs. You are expected to participate actively in class, including at the co-seminar.

### MATH 300.2: Fundamental Concepts of Mathematics

Eric Sommers TuTh 10:00-11:15

Prerequisites:

Math 132 with a grade of 'C' or better

Text:

TBD

Note:

Math 300 students are required to register for the 1-credit co-seminar Math 391A. Seminar times will be arranged during the first week of classes.

Description:

The goal of this course is to help students learn the language of rigorous mathematics. Students will learn how to read, understand, devise and communicate proofs of mathematical statements. A number of proof techniques (contrapositive, contradiction, and especially induction) will be emphasized. Topics to be discussed include set theory (Cantor's notion of size for sets and gradations of infinity, maps between sets, equivalence relations, partitions of sets), basic logic (truth tables, negation, quantifiers), and number theory (divisibility, Euclidean algorithm, congruences). Other topics will be included as time allows. Math 300 is designed to help students make the transition from calculus courses to the more theoretical junior-senior level mathematics courses.

### MATH 300.3: Fundamental Concepts of Mathematics

Paul Hacking TuTh 1:00-2:15

Prerequisites:

Math 132 with a grade of C or better

Text:

Description:

The goal of this course is to help students learn the language of rigorous mathematics. Students will learn how to read, understand, devise and communicate proofs of mathematical statements. A number of proof techniques (contrapositive, contradiction, and especially induction) will be emphasized. Topics to be discussed include set theory (Cantor's notion of size for sets and gradations of infinity, maps between sets, equivalence relations, partitions of sets), basic logic (truth tables, negation, quantifiers), and number theory (divisibility, Euclidean algorithm, congruences). Other topics will be included as time allows. Math 300 is designed to help students make the transition from calculus courses to the more theoretical junior-senior level mathematics courses.

### MATH 370.1: Writing in Mathematics

Robin Koytcheff MWF 11:15-12:05

Prerequisites:

MATH 300 or CS 250 and completion of the College Writing (CW) requirement.

Text:

None.

Recommended Text:

None.

Note:

This course satisfies the Junior Year Writing requirement.

Description:

Students will develop skills in writing, oral presentation, and teamwork. The first part of the course will focus on pre-professional skills, such as writing a resume, cover letter, and graduate school essay and preparing for interviews. Subsequent topics will include presenting mathematics to a general audience, the role of mathematics in society, mathematics education, and clear communication of mathematical content. The end of the term will be dedicated to a group research paper, expository in nature.

### MATH 370.2: Writing in Mathematics

Robin Koytcheff MWF 12:20-1:10

Prerequisites:

MATH 300 or CS 250 and completion of College Writing (CW) requirement.

Text:

None.

Recommended Text:

None.

Note:

This course satisfies the Junior Year Writing requirement.

Description:

Students will develop skills in writing, oral presentation, and teamwork. The first part of the course will focus on pre-professional skills, such as writing a resume, cover letter, and graduate school essay and preparing for interviews. Subsequent topics will include presenting mathematics to a general audience, the role of mathematics in society, mathematics education, and clear communication of mathematical content. The end of the term will be dedicated to a group research paper, expository in nature.

### MATH 411: Introduction to Abstract Algebra I

Gufang Zhao MWF 10:10-11:00

Prerequisites:

Math 235, Math 300

Text:

Dan Saracino, "Abstract Algebra: A First Course", second edition, Waveland Press.

Recommended Text:

Michael Artin, "Algebra", second edition, Pearson.

Description:

Introduction to abstract algebra, focusing primarily on the theory of groups. There will be regular problem assignments and quizzes, as well as midterm and final examinations.

### MATH 412: Introduction to Abstract Algebra II

Yaping Yang TuTh 8:30-9:45

Prerequisites:

Math 411

Text:

Dan Saracino, "Abstract Algebra: A First Course", Second Edition, Waveland Press.

Description:

This course is a continuation of Math 411. We will study properties of rings and fields. A ring is an algebraic system with two operations (addition and multiplication) satisfying various axioms. Main examples are the ring of integers and the ring of polynomials in one variable. Later in the course we will apply some of the results of ring theory to construct and study fields. At the end we will outline the main results of Galois theory which relates properties of algebraic equations to properties of certain finite groups called Galois groups. For example, we will see that a general equation of degree 5 can not be solved in radicals.

### MATH 421: Complex Variables

Weimin Chen MWF 1:25-2:15

Prerequisites:

Math 233

Text:

Complex Variables and Applications, 8th Edition, James W. Brown and Ruel V. Churchill, McGraw-Hill, 2009.

Description:

An introduction to functions of a complex variable. Topics include: Complex numbers, functions of a complex variable and their derivatives (Cauchy-Riemann equations). Harmonic functions. Contour integration and Cauchy's integral formula. Liouville's theorem, Maximum modulus theorem, and the Fundamental Theorem of Algebra. Taylor and Laurant series. Classification of isolated singularities. Evaluation of Improper integrals via residues. Conformal mappings.

### MATH 425.1: Advanced Multivariate Calculus

Franz Pedit MWF 1:25-2:15

Prerequisites:

Multivariable calculus (Math 233) and Linear algebra (Math 235)

Text:

H. M. Schey, div, grad, curl and all that [ISBN-13: 978-0393925166; ISBN-10: 0393925161]
M. Spivak, Calculus on Manifolds [ISBN-13: 978-0805390216; ISBN-10: 0805390219]

Recommended Text:

J. Marsden and A. Tromba, Vector Calculus (any edition is fine)

Note:

Description:

This course covers the basics of differential and integral calculus in many variables: differentiability, directional and partial derivatives and gradient of functions; inverse and implicit function theorems; critical points, also with constraints (Lagrange-multipliers/tangential-gradient), and the Hessian; vector fields and differential forms; divergence, curl and exterior derivative; line- and surface-integrals; the fundamental theorem of calculus (Gauss/Green/Stokes).

### MATH 425.2: Advanced Multivariate Calculus

Ava Mauro MWF 9:05-9:55

Prerequisites:

Multivariable calculus (Math 233) and Linear algebra (Math 235)

Recommended Text:

Vector Calculus by Marsden and Tromba, 5th Ed., W. H. Freeman

Description:

This is a course in differential and integral multivariate calculus from a more advanced perspective than Math 233. We will begin by studying limits, continuity, and differentiation of functions of several variables and vector-valued functions. We will then study integration over regions, the change of variables formula, and integrals over paths and surfaces. The relationship between differentiation and integration will be explored through the theorems of Green, Gauss, and Stokes. Various physical applications, such as fluid flows, force fields, and heat flow, will be covered.

### MATH 455: Introduction to Discrete Structures

Prerequisites:

Calculus (MATH 131, 132, 233), Linear Algebra (MATH 235), and Math 300 or CS 250. For students who have not taken Math 300 or CS 250, the instructor may permit students with sufficient experience in reading and writing mathematical arguments to enroll.

Text:

Harris, Hirst, and Mossinghoff, Combinatorics and Graph Theory, 2nd edition

Note:

### MATH 691Y: Applied Math Project Sem.

Nathaniel Whitaker F 2:30-3:45

Prerequisites:

Graduate Student in Applied Math MS Program

Text:

None

Description:

Continuation of Project

### MATH 697MS: ST - Multiscale Methods

Matthew Dobson MWF 10:10-11:00

Prerequisites:

Math 523 or 623, Math 331.

Text:

Pavliotis and Stuart, Multiscale Methods: Averaging and Homogenization

Description:

This course will study problems with multiple scales in time and/or space and derive simplified equations that feel the fine-scale effects in an averaged way. Multiscale structure is widely found in mathematics applied to biological, chemical, and engineering applications, and the study of such structures is a rapidly developing
field. We will study multiscale discrete and continuum models starting with ordinary differential equations and progressing to include Markov processes, stochastic differential equations, and partial differential equations.

In order to keep prerequisites minimal, an introduction to stochastic differential equations, partial differential equations and Markov processes will be given before their treatment, and the focus will be on the unified technique of formal expansions.

### MATH 697NA: ST - Numerical Algorithms

Eric Polizzi TuTh 2:30-3:45

Description:

Objectives: Provide a practical understanding of matrix computations for science, engineering and industrial applications;
Provide solid foundations in computational linear algebra; Introduction to parallel computing and programming practices

Contents: Introduction to Scientific Computing- Basic numerical techniques of linear algebra and their applications- Data formats and Practices - Matrix computations with an emphasis on solving sparse linear systems of equations and eigenvalue problems - Parallel architectures and parallel programming with OpenMP, MPI and hybrid -. Numerical parallel algorithms

### MATH 704: Topics in Geometry II

Paul Gunnells TuTh 8:30-9:45

Prerequisites:

Topics In Geometry 703 or knowledge of abstract manifolds, vector bundles, ODEs on manifolds and basics of Lie theory.

Text:

Introduction fo Smooth Manifold, John M. Lee, second edition. Riemannian Manifolds - An Introduction to Curvature, by John M. Lee

Description:

This course is the continuation of Math 703. We will discuss differential forms, integration on manifolds and deRham cohomology, as well as basic Riemannian geometry (connections and curvature).

### MATH 797FR: ST - Financial Math and Risk Management

HongKun Zhang TuTh 1:00-2:15

Prerequisites:

Stochastic Calculus

Description:

This is the second course of the series. We will introduce the concepts of arbitrage, state price densities and equivalent martingale measures. European and American claims. Optimal stopping. We will study admissible strategies, pricing and hedging in Markovian models. Admissible strategies. Pricing and hedging in Markovian models. We will consider different ways of of modeling markets with uncertain volatility. We will cover stochastic volatility models, auto-regressive models as well as recently de- rived nonlinear pricing models. We will emphasize, in particular, worst-case scenario risk-management techniques, and managing volatility risk with options. The main probability tool we will use is the Ex- treme Value Theory. Finally, we will cover more practical risk-management techniques and guidelines such as the “Value at Risk” outlined in the J.P. Morgan technical document on measuring financial risk.

### MATH 797MC: ST - Mapping Class Groups

R. Inanc Baykur TuTh 11:30-12:45

Prerequisites:

* Basic algebra and complex analysis (undergraduate level)
* Point-set topology (671)
* Algebraic topology (cellular decomposition, homotopy, covering spaces, homology)
* Differentiable manifolds (703-704)

Text:

We will draw the material from several books and research articles.

Recommended Text:

* Benson Farb and Dan Margalit, "A primer on mapping class groups"
* Joan Birman, “Braids, links and mapping class groups”
* William Thurston, "Three dimensional geometry and topology"
* Andrew Casson, “Automorphisms of surfaces after Nielsen and Thurston”
* Wilhelm Magnus, Abraham Karrass and Donald Solitar, “Combinatorial group theory”
* Yukio Matsumoto, “An introduction to Morse theory”
* Burak Ozbagci and Andras Stipsicz, “Surgery on contact 3-manifolds and Stein surfaces”
* Benson Farb, “Problems on mapping class groups and related topics"

Note:

The course grade will be based on homework (70%) and final presentations (30%). Homework will be typically due every one or two weeks. Final presentations will consist of a 30-40 min in-class presentation (15%), accompanied by an expository paper prepared with TeX (15%).

Description:

This is an introductory course on the theory of mapping class groups of surfaces, which connects to several fields of mathematics, such as geometry, topology, algebra and combinatorics. After presenting the rich algebraic structure and the intrinsic geometric nature of mapping class groups, the course will discuss various facets and applications of this newly emerging beautiful subject.

### MATH 797MD: ST - Moduli Space and Invariant Theory

Jenia Tevelev MWF 9:05-9:55

Prerequisites:

Math 612, Math 621, and at least one semester of Algebraic Geometry, for example MATH 797W: ST-Algebraic Geometry (or consent of the instructor).

Text:

I have notes on my website from the previous time I taught this course. I probably won't follow them too closely but they will give you some idea what to expect, see http://people.math.umass.edu/~tevelev/moduli797.pdf

Description:

A moduli space parametrizes or classifies all geometric objects of some sort. For example, elliptic curves are classified by the J-invariant, so the moduli space of elliptic curves is an affine line (with coordinate J). The Jacobian of a Riemann surface is also a moduli space: it parametrizes line bundles on the Riemann surface. The study of moduli spaces is an old subject and many methods were developed to understand them including GIT (geometric invariant theory), variation of Hodge structures, stacks, etc. Reading modern literature on moduli theory can be quite hard because it requires deep understanding of these advanced topics. Instead of following this ontogeny recapitulates phylogeny'' philosophy, we will introduce interesting examples of moduli spaces and then develop whatever machinery is necessary to work with them. Along the way we will learn a lot of Algebraic Geometry.

### MATH 797PM: ST - Predictive Modeling and Uncertainty Quantification

Markos Katsoulakis TuTh 2:30-3:45

Prerequisites:

Instructor consent and graduate courses in probability theory (STAT 605), numerical analysis for ODE/PDE (MATH 652) and familiarity with the basic concepts of statistical inference (STAT608).

Description:

In this course we will discuss an array of topics that straddle probabilistic modeling and simulation, uncertainty quantification and statistical learning methods. We focus on developing systematic mathematical and computational tools for building data-driven, predictive models for complex systems and dynamics. In particular, we will discuss material that includes: uncertainty quantification, local and global sensitivity analysis methods for dynamical systems, data assimilation and approximate inference methods, stochastic optimization algorithms and model selection. A significant component of the course focuses on related software such as DAKOTA and others.

### STAT 605: Probability Theory I

Yao Li TuTh 8:30-9:45

Prerequisites:

Math 623 (Real Analysis) or permission of instructor. In order to understand the material in this course, knowledge of measure theory is required (Math 623 or equivalent).

Text:

A Course in Probability Theory, Third Edition by Kai-Lai Chung.

Recommended Text:

Probability (GTM 95) by A.N. Shiryaev.

Note:

Most homework problems will be from Chung's book. Some materials may come from GTM 95.

Description:

This course aims to develop the probability basis that is required in modern statistical theories and stochastic processes. Knowledge of measure theory is required to understand the materials in this course. Topics of this course include measure and integration foundations of probability, distribution functions, convergence of random variables, laws of large numbers and central limit theory. If time permits, I will cover some basic martingale theory and random walks.

### STAT 608: Mathematical Statistics II

Krista J Gile MW 2:30-3:45

Prerequisites:

STAT 607 or permission of the instructor.

Text:

Statistical Inference, 2nd edition,
by Casella and Berger

Description:

This course is the second half of the STAT 607-608 sequence, which together provide the foundational theory of mathematical statistics. STAT 607 emphasizes concepts in probability, while 608 builds on those concepts to build statistical theory. STAT 608 topics include point and interval estimation, hypothesis testing, large sample results in estimation and testing, decision theory, Bayesian methods, and analysis of discrete data. Other areas may include nonparametric methods, sequential methods, regression, and analysis of variance.

### STAT 691P: Project Seminar

John Staudenmayer MWF 12:20-1:10

Prerequisites:

Permission of instructor.

Text:

None.

Description:

This course is designed for students to complete the master's project requirement in statistics, with guidance from faculty. The course will begin with determining student topics and groups. Each student will complete a group project. Each group will work together for one semester and be responsible for its own schedule, work plan, and final report. Regular class meetings will involve student presentations on progress of projects, with input from the instructor. Students will learn about the statistical methods employed by each group. Students in the course will learn new statistical methods, how to work collaboratively, how to use R and other software packages, and how to present oral and written reports.

### STAT 697D: ST - Appl Stat and Data Analysis

Krista J Gile MWF 11:15-12:05

Prerequisites:

co-requisites: Regression (525 or 597R), Mathematical Statistics II (516 or 608)

Description:

This course gives students a brief overview of several topics of practical importance to statisticians doing data analysis. It focuses on topics not typically covered in the required curriculum, but of use to students earning advanced degrees in statistics.

Each topic provides a foundation from which students may extend their understanding, should they need to use the method in practice, or wish to consider more detailed study or research.

The course begins with a very fast introduction to R (students unfamiliar with R are encouraged to contact the professor to begin this process before the semester). Each topic covered includes both a technical overview and an application, implemented in software. Specific topics may include: Simulation studies, randomization methods, Markov chain Monte Carlo, missing data, casual modeling, survival analysis, network analysis, exploratory data analysis, and cross-validation. Other topics may arise from student interest or consulting problems.

One hour each week will meet concurrently with the Statistical Consulting Practicum.

Students will complete projects of their choosing.

### STAT 697TS: ST - Time Series Analysis and Appl

Daeyoung Kim TuTh 11:30-12:45

Prerequisites:

Two prerequisites are required - 1) Probability and Statistics at a calculus based graduate level such as Stat 607 and Stat 608 (concurrent), 2) a previous course on regression analysis covering multiple linear regression (e.g., Stat 505, BioEpi 744, RESEC 702) with some exposure to regression models in matrix form. Prior computing experience with R is desirable.

Text:

Time Series Analysis and Its Applications, 3rd ed., by Shumway, Robert H. and Stoffer, David S.

Description:

Time series analysis is an effective statistical methodology for modelling time series data (a series of observations collected over time) and forecasting future observations in many areas, economics, the social sciences, the physical and environmental sciences, medicine, and signal processing. For example, monthly unemployment rates in economics, yearly birth rates in social science, global warming trends in environmental studies, and magnetic resonance imaging of brain waves in medicine. This course presents the fundamental principles of time series analysis including mathematical modeling of time series data and methods for statistical inference. Topics covered will include modeling and inference in the following models : autoregressive (AR) and autoregressive moving average (ARMA) models, (nonseasonal/seasonal) autoregressive integrated moving average (ARIMA) models, unit root and differencing, spectral analysis, (generalized) autoregressive conditionally heteroscedastic models and state-space models.