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.

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.

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.

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.

Honors section of Math 131.

Honors section of Math 132.

Honors section of Math 233.

(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.)

The field of Data Science encompasses methods, processes, and systems that enable the extraction of useful knowledge from data. Foundations of Data Science introduces core data science concepts including computational and inferential thinking, along with core data science skills including computer programming and statistical methods. The course presents these topics in the context of hands-on analysis of real-world data sets, including economic data, document collections, geographical data, and social networks. The course also explores social issues surrounding data analysis such as privacy and design.

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 basic logic (truth tables, negation, quantifiers), set theory (Cantor's notion of size for sets and gradations of infinity, maps between sets, equivalence relations, partitions of sets), and elementary 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.

The overall paradigm of the course is to provide the students with a non-compartmentalized view of mathematics. The course is based on material covered in the general audience book ``The Poincare Conjecture: In search of the shape of the universe" by Donald O'Shea. The content of this book is tantamount to what humans know about the physical world and its mathematical pendant, and hence can be considered a bedrock for anyone who aspires to work as an educator and/or researcher. All students are required to read the book. There will be additional reading assignments, and short lectures by the instructor, on background material as the semester progresses. (Groups of) students will choose topics from the book and will have to write a self contained, educated audience article on the material. All papers have to be written using LaTex word processing (open source). The papers will go through a number of editing cycles and will finally be presented to the class. Some of the class time will be reserved for video taped general mathematical audience lectures by Fields medallists and other prize winning mathematicians on related topics. In addition, the course will reserve one week during which advisors from the career office of the University will provide information about the job application process, resume/cover letter writing, and job interviews. Each student will be requested to hand in a putative job application (cover letter, resume) of their choice.

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. Prerequisites: MATH 300 and completion of College Writing (CW) requirement.

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 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. While the text covers functions of up to 3 variables we will sketch how everything works for any number of variables.

This 3 credit course serves as a preparation for SOA's second actuarial exam in financial mathematics, known as Exam FM or Exam 2. The course provides an understanding of the fundamental concepts of financial mathematics, and how those concepts are applied in calculating present and accumulated values for various streams of cash flows as a basis for future use in: reserving, valuation, pricing, asset/liability management, investment income, capital budgeting, and valuing contingent cash flows. The main topics include time value of money, annuities, loans, bonds, general cash flows and portfolios, immunization, interest rate swaps and determinants of interest rates etc. Many questions from old exam FM will be practiced in the course.

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. Prerequisites: Calculus (Math 131, 132, 233), required; Linear Algebra (Math 235) and Differential Equations (Math 331), or permission of instructor required

Basic properties of the positive integers including congruence arithmetic, the theory of prime numbers, quadratic reciprocity, and continued fractions. Theory applied to develop algorithms and computational techniques of computer science and to cryptography. To help learn these materials, students will be assigned computational projects using computer algebra software. Prerequisite: MATH 233 and 235. Math 300 or COMPSCI 250 as a co-requisite is not absolutely necessary but highly recommended.

This course is a basic introduction to combinatorics and graph theory for advanced undergraduates in computer science,
mathematics, engineering and science. Topics covered include: elements of graph theory; Euler and Hamiltonian circuits; graph
coloring; matching; basic counting methods; generating functions; recurrences; inclusion-exclusion; and Polya's theory of counting.

This course is intended to provide an introduction to systems of differential equations and dynamical systems, as well as to touch upon chaotic dynamics, while providing a significant set of connections with phenomena modeled through these approaches in Physics, Chemistry and Biology. From the mathematical perspective, geometric and analytic methods of describing the behavior of solutions will be developed and illustrated in the context of low-dimensional systems, including behavior near fixed points and periodic orbits, phase portraits, Lyapunov stability, Hamiltonian systems, bifurcation phenomena, and concluding with chaotic dynamics. From the applied perspective, numerous specific applications will be touched upon ranging from the laser to the synchronization of fireflies, and from the outbreaks of insects to chemical reactions or even prototypical models of love affairs. In addition to the theoretical component, a self-contained computational component towards addressing these systems will be developed with the assistance of Matlab (and wherever relevant Mathematica). However, no prior knowledge of these packages will be assumed.

This is a second course in Linear Algebra building upon the concepts and techniques introduced in Math 235. We will study the decomposition of matrices, particularly the LU, QR, and singular value decompositions. We also study vector spaces and linear transformations, inner product spaces, orthogonality, and spectral theory. We will emphasize applications of these techniques to various problems including, as time permits: solutions of linear systems, least-square fitting, search engine algorithms, error-correcting codes, fast Fourier transform, dynamical systems.

The coursework will be a mix of proof and computation. For the latter students may choose Matlab or one of the open source programs: Scilab or Octave (both similar to Matlab).

Simulations and numerical computation drive large portions of technological and scientific developments in the modern world. The course will introduce the foundations for formulating, numerically solving, and assessing the accuracy of solutions to 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 using MATLAB or Scilab. 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, and numerical integration.

This course is a one-credit seminar on security research across departments at UMass. Each presentation will cover an active research topic at UMass in a way that assumes only a basic background in security. External speakers may also be invited. Note that this course is not intended to be an introduction to cybersecurity, and will not teach the fundamentals of security in a way that would be useful as a foundation for future security coursework. The intended audience is graduate and advanced undergraduate students, as well as faculty. Meets with COMPSCI 591CF and E&C-ENG 591CF. May be taken repeatedly for credit up to 2 times. This course does not count toward any requirements for the Math major or minor.

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.

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).

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.

Classical theory of ordinary differential equations and some of its modern developments in dynamical systems theory. Topics to be chosen from: Linear systems and exponential matrix solutions; Well-posedness for nonlinear systems; Floquet theory for linear periodic systems. Qualitative theory: limit sets, invariant sets and manifolds. Stability theory: linearization about equilibria and periodic orbits, Lyapunov functions. Numerical simulations will be used to illustrate the behavior of solutions and to motivate the theoretical discussion.

This fast-paced course (and its sequel, Math 672) is an introduction to topology, from point-set to geometric and algebraic topology.
Part I: Basic point-set topology, constructions of topological spaces, connectedness, compactness, countability and separation axioms, topological manifolds.
Part II: Introduction to algebraic topology, cell complexes, homotopy, fundamental group, covering spaces.
Grade will be based on regularly assigned homework, as well as exams.

This course is an introduction to the theory of convex polytopes and their applications to algebraic combinatorics. The course will cover basic facts and properties of polytopes including faces of polytopes, valuation theory, Ehrhart theory, triangulations, and zonotopes. We will also tentatively cover the following families of polytopes with interest in other fields including root polytopes, flow polytopes, polytopes from partially ordered sets, and generalized permutahedra.

This is a course in linear partial differential equations. We will discuss the Laplace, heat, and wave equations on Euclidean space, using purely analytical methods. If time allows we will also consider other elliptic, parabolic, and hyperbolic equations, possibly with a view toward nonlinear problems. Along the way may also discuss distributions and Sobolev spaces which are important tools in the modern study of partial differential equations. A good understanding of the material in the graduate real analysis class (623 and possibly 624) as well as fluency in multivariable calculus (in particular the divergence and Stokes theorems and integration by parts) are required.

Elliptic curves, as the only smooth projective algebraic curves equipped with a group law, play a central role in modern arithmetic geometry. The goal of this course is to learn the tools and techniques required to study these groups over the rational numbers by first studying them over finite fields, p-adic fields and archimedean fields.

Math 605: In this course we will build up the tools and foundations of probability theory needed for understanding and using statistical theories, stochastic simulation, and stochastic processes. Topics covered in this class are measure and integration (construction of probability spaces), distribution functions, random variables and their simulation, convergence of random variables, laws of large numbers and Monte-Carlo methods, concentration inequalities, central limit theorem, information theory, random walks.
A class in real analysis (integration theory, Math 623) is not required for this class as we will review, as needed, the basic constructions of probability spaces. It can be beneficial to take these two classes concurrently for the student who wants and in-depth understanding of the mathematical foundations of probability.
The primary goal of the class is to understand and master the basic probability tools and concepts needed in modern applications such as in stochastic simulation, data science and machine learning, engineering, and economics. Motivated students with various backgrounds are welcome.
The class is the first part of the sequence Math605/606 and Math 606 will cover the theory of stochastic processes and stochastic simulation: Martingale theory, Poisson processes, Markov chains in discrete and continuous time, Markov chain Monte-carlo, Brownian motion.

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.

This course treats the advanced theory of statistics, going into a more advanced treatment of some topics first seen in Stat 607-608, from the viewpoint of large-sample (asymptotic) theory. Topics include Mathematical and Statistical Preliminaries; (Weak/Strong) Convergence; Central Limit Theorems (including Lindeberg-Feller Central Limit Theorem and Stationary m-Dependent Sequences); Delta Method and Applications; Order Statistics and Quantiles; Maximum Likelihood Estimation; Set estimation and Hypothesis Testing; U-statistics; Bootstrap and Applications