Course Descriptions

First semester of the two-semester sequence MATH 101-102. Detailed, in-depth review of manipulative algebra; introduction to functions and graphs, including linear, quadratic, and rational functions.

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

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.

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.

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

Techniques of calculus in two and three dimensions. Vectors, partial derivatives, multiple integrals, line integrals. Theorems of Green, Stokes and Gauss. Honors section available. (Gen.Ed. R2)

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

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.)

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.

Satisfies Junior Year Writing requirement. Develops research and writing skills in mathematics through peer review and revision. Students write on mathematical subject areas, prominent mathematicians, and famous mathematical problems. Some goals of the course are to learn how to write documents effectively in LaTeX, learn how to prepare and give good presentations, learn about possible career paths in mathematics, and to attend and listen actively during math talks. Prerequisites: MATH 300 and completion of College Writing (CW) requirement.

This course will introduce students to technical writing in mathematics. Writing assignments will include proofs, instructional handouts, resumes, cover letters, and a final paper. All assignments will be completed using LaTeX. By the end of the semester, students should be able to clearly convey mathematical ideas through their writing, as geared to a particular audience.

This course provides future secondary math teachers an introduction to a range of topics related to the teaching of mathematics in the public schools. The focus will be on increasing the participants' mathematical content knowledge for teaching by exploring the mathematical content and practices of secondary math. Through these explorations, students will have opportunities to gain some familiarity with the Massachusetts Frameworks, Common Core, standardized and local assessments, curriculum resources, and other topics related to secondary math teaching. As part of the course, students will also explore connections between secondary math and the higher-level mathematics courses they have been taking.

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.

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.

This course covers the basics of differential and integral calculus in many variables: differentiability, directional and partial derivatives and gradient of functions; critical points without or 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/Thomson). If time and taste permit, topics from physics (fluids and electromagnetism) and differential geometry (curves and surfaces in space) may also be explored.

In this course we will study multivariate differential and integral calculus. We will dive deeper into the material from Math 233 and also cover new material on vector calculus.

The first part of the course will cover with differential calculus of functions of several variables. Then we will study multivariate integration and integration over paths and surfaces. We will conclude with the capstone of this course: vector calculus. Our goal will be to understand Green's theorem, Gauss' theorem, and Stokes' theorems. Some motivation for vector calculus will come from physics, for example Maxwell's equations that describe electromagnetic waves.

Mostly we will focus on functions of three variables but will also sometimes discuss how the theory works with any number of variables.

This is a rigorous introduction to some topics in mathematics that underlie areas in computer science and computer engineering, including graphs and trees, spanning trees, colorings and matchings; the pigeonhole principle, induction and recursion, enumeration, and generating functions. As part of the course, student groups will be assigned and a final project will be presented. This course satisfies the university's Integrative Experience (IE) requirement for math majors.

We learn how to build, use, and critique mathematical models. In modeling we translate scientific questions into mathematical language, and thereby we aim to explain the scientific phenomena under investigation. Models can be simple or very complex, easy to understand or extremely difficult to analyze. We introduce some classic models from different branches of science that serve as prototypes for all models. Student groups will be formed to investigate a modeling problem themselves and each group will report its findings to the class in a final presentation. The choice of modeling topics will be largely determined by the interests and background of the enrolled students. Satisfies the Integrative Experience requirement for BA-Math and BS-Math majors.

This is an introduction to the history of mathematics from ancient civilizations to present day. Students will study major mathematical discoveries in their cultural, historical, and scientific contexts. This course explores how the study of mathematics evolved through time, and the ways of thinking of mathematicians of different eras - their breakthroughs and failures. Students will have an opportunity to integrate their knowledge of mathematical theories with material covered in General Education courses. Forms of evaluation will include a group presentation, class discussions, and a final paper.
The math major is made up of technical courses on the theory of mathematics from Calculus to more complex concepts. This course is unique in providing a humanities-based approach to understanding math. For example, students are required to use primary sources on a weekly basis. Students study examples of how mathematical advances were made in response to or alongside developments in other branches of science - such as Ptolemy’s work in trigonometry being motivated by applications in astronomy, and Newton as the father of both calculus and modern physics. Students also learn to understand mathematicians as people of their times - for example, how Babylonian mathematicians were motivated by the needs of the empire, or how Evariste Galois was both a brilliant mathematician and a passionate French revolutionary. Additionally, many math majors go on to teach mathematics after graduation, and in this course the history of math is is studied in the context of the history of education.
The homework assignments contain a component where students are required to write short compositions. Many of these assignments will ask students to engage in self-reflection on how their study of the history of mathematics in the current course is influenced by the General Education courses they took. This is not limited to courses carrying the Historical Studies designation, but Economics, Sociology, Science and other modes of human thought all present lenses through which one can study the mathematics and mathematicians that are the focus of the course; the homework assignments will invite students to apply all of those lenses to the topics at hand. Students will also be required to write an interdisciplinary 20-page final essay. Essay topics are developed by the student with assistance from the professor. Students draw not only on their mathematics coursework, but also on the knowledge they have accrued from other GenEd curriculum courses they have experienced in their first two years of college. After seeing how mathematical tools have developed in conversation with history, culture, and science, students can better appreciate the uses and possibilities of advanced mathematics.
A substantial fraction of the course grade is based on a class presentation; this is a major change from standard upper division math/stat requirements. Each group, which typically consists of three students, collaboratively researches and presents an interdisciplinary mathematical topic, chosen by the group from the instructor’s list of suggested topics. This is highly unusual for mathematics classes, where problems are typically presented abstracted from their scientific and cultural roots. These presentations are spaced out throughout the semester, and the students’ work is referenced and discussed in a class discussion later in class.
Additionally, highly unusually for a mathematics course, a large part of the course grade is based on participation in class discussions. Each week, the students are assigned readings. A large amount of weekly class time is devoted to a roundtable discussion of the reading - its implications for modern mathematics, how it was understood at the time, mathematical concepts in the reading, and the close-reading of assigned passages.
See http://people.math.umass.edu/~tevelev/475_2017/ for more information about the course and examples of essays, class presentations, and reading assignments from previous years.

Topology of the euclidean space and functions of several variables (implicit function theorem), introduction to Fourier analysis, metric spaces and normed spaces. Applications to differential equations, calculus of variations, and others.

Math 536 serves as a preparation for the first SOA/CAS actuarial exam on the fundamental probability tools for quantitatively assessing risk, known as Exam P (SOA) or Exam 1 (CAS). The course covers general probability, random variables with univariate probability distributions (including binomial, negative binomial, geometric, hypergeometric, Poisson, uniform, exponential, gamma, and normal), random variables with multivariate probability distributions (including the bivariate normal), basic knowledge of insurance and risk management, and other topics specified by the SOA/CAS exam syllabus.

This is a course in Advanced Linear Algebra and Applications. We will cover LU decom-
position, Vector and Inner Product Spaces, Orthogonality and Least Squares, Determinants and
Eigenvalues, Jordan form, Spectral theorem, symmetric positive definite matrices. Other decom-
positions such as SVD and QR, will be covered as well. If time permitting. basic numerical linear
algebra will be included. There will be elements of proof and computation in the course. No coding
will be taught in the class, but the students will have the option to do a final project instead of the
exam. Homework will be assigned every one or two weeks. Late homework will NOT be accepted.
The final grade will be based on both exams and homework assignments.

The course will introduce basic numerical methods used for solving problems that arise in different scientific fields. Properties such as accuracy of methods, their stability and efficiency will be studied. Students will gain practical programming experience in implementing the methods. We will cover the following topics (not necessarily in the order listed): Finite Precision Arithmetic and Error Propagation, Linear Systems of Equations, Root Finding, Interpolation, Least squares, Numerical Integration, Numerical Solution of Ordinary Differential Equations.

This course will introduce basic numerical methods used for solving problems that arise in different scientific fields. The following topics (not necessarily in the order listed) will be covered: finite precision arithmetic and error propagation, linear systems of equations, root finding, interpolation, least squares and numerical integration. Students will gain practical programming experience in implementing such numerical methods using MATLAB. The use of MATLAB for homework assignments will be mandatory, although any other scientific language for solving the homework problems will be accepted. We will also discuss some very important practical considerations of implementing numerical methods using other languages such as fortran, C or C++.

The course covers standard materials in the theory of curves in the plane and the space, and the theory of surfaces in the space.
Emphasis is given on the interplay between the local numerical quantities such as curvature and the global properties of the geometric objects,
which is a central theme in modern differential geometry.

A non-calculus-based applied statistics course for graduate students and upper level undergraduates with no previous background in statistics who will need statistics in their future studies and their work. The focus is on understanding and using statistical methods in research and applications. Topics include: descriptive statistics, probability theory, random variables, random sampling, estimation and hypothesis testing, basic concepts in the design of experiments and analysis of variance, linear regression, and contingency tables. The course has a large data-analytic component using a statistical computing package.

This course provides a calculus based introduction to probability (an emphasis on probabilistic concepts used in statistical modeling) and the beginning of statistical inference (continued in Stat516). Coverage includes basic axioms of probability, sample spaces, counting rules, conditional probability, independence, random variables (and various associated discrete and continuous distributions), expectation, variance, covariance and correlation, the central limit theorem, and sampling distributions. Introduction to basic concepts of estimation (bias, standard error, etc.) and confidence intervals.

This course provides a calculus based introduction to probability (an emphasis on probabilistic concepts used in statistical modeling) and the beginning of statistical inference (continued in Stat516). Coverage includes basic axioms of probability, sample spaces, counting rules, conditional probability, independence, random variables (and various associated discrete and continuous distributions), expectation, variance, covariance and correlation, the central limit theorem, and sampling distributions. Introduction to basic concepts of estimation (bias, standard error, etc.) and confidence intervals.

Continuation of Stat 515. Overall objective of the course is the development of basic theory and methods for statistical inference. Topics include: Sampling distributions; General techniques for statistical inference (point estimation, confidence intervals, tests of hypotheses); Development of methods for inferences on one or more means (one-sample, two-sample, many samples - one-way analysis of variance), inference on proportions (including contingency tables), simple linear regression and non-parametric methods (time permitting).

Continuation of Stat 515. Overall objective of the course is the development of basic theory and methods for statistical inference. Topics include: Sampling distributions; General techniques for statistical inference (point estimation, confidence intervals, tests of hypotheses); Development of methods for inferences on one or more means (one-sample, two-sample, many samples - one-way analysis of variance), inference on proportions (including contingency tables), simple linear regression and non-parametric methods (time permitting).

Regression is the most widely used statistical technique. In addition to learning about regression methods, this course will also reinforce basic statistical concepts and introduce students to "statistical thinking" in a broader context. This is primarily an applied statistics course. While models and methods are written out carefully with some basic derivations, the primary focus of the course is on the understanding and presentation of regression models and associated methods, data analysis, interpretation of results, statistical computation and model building. Topics covered include simple and multiple linear regression; correlation; the use of dummy variables; residuals and diagnostics; model building/variable selection, regression models and methods in matrix form. With time permitting, further topics include an introduction to weighted least squares, regression with correlated errors and nonlinear (including binary) regression.

The course will introduce computing tools needed for statistical analysis including data acquisition from database, data exploration and analysis, numerical analysis and result presentation. Advanced topics include parallel computing, simulation and optimization, and package creation. The class will be taught in a modern statistical computing language.

We will cover the basic theory of functions of one complex variable, at a pace that will allow for the inclusion of some non-elementary topics at the end. Basic Theory: Holomorphic functions, conformal mappings, Cauchy's Theorem and consequences, Taylor and Laurent series, singularities, residues, elliptic functions, other topics as time permits.

This course gives an introduction to the basic tools of algebraic topology, which studies topological spaces and continuous maps by producing associated algebraic structures (groups, vector spaces, rings, and homomorphism between them). Emphasis will be placed on being able to compute these invariants, not just on their definitions and associated theorems.

Topics include: Homotopy, fundamental group and covering spaces (reviewed from Math 671), simplical and cell complexes, singular and simplicial homology, long exact sequences and excision, cohomology, Künneth formulas, Poincaré duality.
If time permits, more advanced topics will be discussed at the end, such as higher homotopy groups, sheaf cohomology, the de Rham theorem, and equivariant cohomology.

Continuation of Project

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

The aim of this course will be to give an overview of the mathematical background, physical applications and numerical computations associated with a number of prototypical wave systems both at the continuum and at the lattice level. We will start from finite dimensional Hamiltonian systems, discuss their symmetries and Lagrangian/Hamiltonian structure, and then extend considerations to infinite dimensional systems of partial differential and differential-difference equations. Prototypical case examples will be the continuum and the discrete nonlinear Schrodinger equation, the Korteweg-de Vries equation, the sine-Gordon equation, and the Fermi-Pasta-Ulam lattice, among others. We will examine the symmetries, conservation laws, solitary wave solutions, linearization spectral properties and dynamics of such equations and attempt to connect them with physical applications from nonlinear optics, fluid mechanics, materials science and atomic physics, as well as develop computational tools (such as bifurcation analysis and time-stepping algorithms) about how to address them. Time permitting, we will also make short excursions to systems of multiple components, higher dimensions or of dissipative character (e.g. reaction-diffusion type) to discuss some similarities and differences with these.

An introductory course to complex algebraic
geometry. The basic techniques of Kahler geometry, Hodge theory, line and
vector bundles, needed for the study of the geometry and topology of complex projective
algebraic varieties, will be introduced and illustrated in basic examples such as Riemann surfaces, algebraic surfaces, abelian
varieties, and Grassmannians.

This is the second part of a two semester sequence on probability and mathematical statistics. Stat 607 covered probability, discrete/continuous distributions, basic convergence theory and basic statistical modelling. In Stat 608 we cover an introduction to the basic methods of statistical inference, pick up some additional probability topics as needed and examine further issues in methods of inference including more on likelihood based methods, optimal methods of inference, more large sample methods, Bayesian inference and Resampling methods. The theory is utilized in addressing problems in parametric/nonparametric methods, two and multi-sample problems, and regression. As with Stat 607, this is primarily a theory course emphasizing fundamental concepts and techniques.