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.

Honors section of Math 235.

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.

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 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.

Introduction to groups, rings, fields, vector spaces, and related concepts. Emphasis on development of careful mathematical reasoning.

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 Laurent series. Classification of isolated singularities. Evaluation of Improper integrals via residues. Conformal mappings.

Calculus of several variables, Jacobians, implicit functions, inverse functions; multiple integrals, line and surface integrals, divergence theorem, Stokes' theorem.

This 3 credit hours 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.

Math 456 is an introduction to mathematical modeling, and is one of the Integrated Experience courses approved by the General Education Council. The main goal of the class is to learn how to translate problems from "real-life" into a mathematical model and how to use mathematics to solve the problem. We will learn how to build, use and critique mathematical models.

In the beginning, we'll focus on differential equation based models. For the second half, we will study a number of topics from games and gambling, economics, social sciences,for which we will use elementary tools from probability, game theory, information theory, and optimization.

In addition, each student will join a group of 3 students, and the group will investigate a modeling problem. Each group will give a final presentation at the end of the semester. Each student will write an individual report on the group project at the end of the course. After discussion/consultation with the instructor, the choice of modeling topics will be determined by the interests and background of the enrolled students, and the mathematical methods applied will draw upon whatever the students have already learned.

There are three types of surfaces which look the same at every point and in every direction: the plane, the sphere, and the hyperbolic plane. The hyperbolic plane is a remarkable surface in which the circumference of a circle grows exponentially as the radius increases; it was only discovered in the 18th century. We will begin by studying the geometry of the plane and the sphere and their symmetries. Then we will describe and study the hyperbolic plane. The emphasis will be on developing our geometric intuition in each case.

Abstract algebra forms a key part of the ideas behind high school mathematics and is the basis for several parts of the Massachusetts Test for Educator Licensure for secondary school math teachers. This course will cover the parts of abstract algebra most important for building a deep understanding of the ideas of high school mathematics and their interconnections. It will focus on the properties of rings (especially the integers and polynomial rings over fields), and fields. During the course, we will be making connections between these topics and high school mathematics.

This one-credit course is designed to help students review and prepare for the GRE Mathematics subject exam, which is a required exam for entrance into many PhD programs in mathematics. Students should have completed the three courses in calculus, a course in linear algebra, and have some familiarity with differential equations. The focus will be on solving problems based on the core material covered in the exam. Students are expected to do practice problems before each meeting and discuss the solutions in class.

This course provides an introduction to systems of differential equations and dynamical systems, and will 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. In addition to the theoretical component, a self-contained computational component towards addressing these systems will be developed with the assistance of Matlab.

This course is an introduction to the mathematical models used in finance and economics with particular emphasis on models for pricing financial instruments, or "derivatives." The central topic will be options, culminating in the Black-Scholes formula. The goal is to understand how the models derive from basic principles of economics, and to provide the necessary mathematical tools for their analysis.

Basic concepts (over real or complex numbers): vector spaces, basis, dimension, linear transformations and matrices, change of basis, similarity. Study of a single linear operator: minimal and characteristic polynomial, eigenvalues, invariant subspaces, triangular form, Cayley-Hamilton theorem. Inner product spaces and special types of linear operators (over real or complex fields): orthogonal, unitary, self-adjoint, hermitian. Diagonalization of symmetric matrices, applications.

The course will introduce foundational numerical methods used for problems that arise in many scientific fields. Properties such as accuracy of methods, their stability and efficiency will be considered. Students will gain practical programming experience in implementing the methods using MATLAB, which will be taught through increasingly complex codes over the term, with examples in class and students homework assignments. From time to time we will also discuss practical considerations of implementing these methods on modern computer architectures using C, C++ or Fortran.

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).

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.

This is an introductory class to R programming language designed for senior undergrads and junior graduate students. It lays out all the fundamentals of R, such as syntax, data structure, function design, package design, etc. Some important advanced features and packages will also be introduced, including dplyr, ggplot2, Rmarkdown and shiny. The objective is to give a comprehensive view about R, to provide as many guidance for future studying and practicing, and at the same time to distribute as many raw materials as possible for a jump start.

Complex number field, elementary functions, holomorphic functions, integration, power and Laurent series, harmonic functions, conformal mappings, applications.

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.

Topological spaces. Metric spaces. Compactness, local compactness. Product and quotient topology. Separation axioms. Connectedness. Function spaces. Fundamental group and covering spaces.

This course is the group project that is required for the MS program in Applied Mathematics. Each academic year we undertake an in-depth study of select applied science problems, combining modeling, theory, and computation to understand it. The main goal of the course is to emulate the process of teamwork in problem solving, such as is the norm in industrial applied mathematics.

Topics to be covered: smooth manifolds, smooth maps, tangent vectors, vector fields, vector bundles (in particular, tangent and cotangent bundles), submersions,immersions and embeddings, sub-manifolds, Lie groups and Lie group actions, Whitney's theorems and transversality, tensors and tensor fields, differential forms, orientations and integration on manifolds, The De Rham Cohomology, integral curves and flows, Lie derivatives, The Frobenius Theorem.

The goal of the course is the classification of semisimple Lie algebras over the complex numbers and an introduction to their representation theory. The classification is essentially the same as that for simply-connected compact Lie groups, which are central objects in mathematics and physics. Topics covered include representations of sl(2), Jordan decomposition, structure theorems, Weyl groups, roots systems, Dynkin diagrams, complete reducibility, finite-dimensional representations and their characters formulas. Students with a good background in both linear algebra and group theory may contact the instructor to enroll without the Math 611 prerequisite.

This class introduces the fundamental concepts in probability. Prerequisite are a solid working knowledge of undergraduate probability and analysis. Measure theory is not a prerequisite.
Among the topics covered are

1)Axioms of probability and the construction of probability spaces.

2) Random variables, integration, convergence of sequences of random variables, and the law of large numbers.

3) Gaussian random variables, characteristic and moment generating functions, and the central limit theorem.

4) Conditional expectation, the Radon--Nikodym theorem, and martingales.

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 is intended to provide a foundation in statistical machine learning with emphasis on statistical methodology as it applies to large-scale data applications using the graphical model framework. At the end of this course, students will be able to build and test a latent variable statistical model with companion inference algorithm to solve real problems in a domain of their interest including genomics, recommendation systems, topic modeling, and other high-dimensional data problems. Course topics include: exponential families, sufficiency and conjugacy, graphical model framework and approximate inference methods such as expectation-maximization, variational inference, and sampling-based methods. Additional topics may include: nonparametric Bayesian methods, cross-validation, bootstrap, empirical Bayes, and deep learning networks.