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.

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.

Honors section of Math 128.

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.

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

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

(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.

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.

The course is an in depth version of a vector calculus course. We will cover intergration and differentiation in multidimensional setting. The Stokes and Green theorem and their multidimensional versions will
be discussed. Some application to engineering and science are covered if time permits.

In this course we will study multivariable differential and integral calculus from a more advanced point of view. We will study limits, continuity, and differentiation of functions of several variables and vector-valued functions. Then we will continue with multivariable integrals (double-triple), line integrals and surface integrals. 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.

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, and matchings; the pigeonhole principle, induction and recursion, generating functions, and discrete probability proofs (time permitting). The course integrates learning mathematical theories with applications to concrete problems from other disciplines using discrete modeling techniques. Student groups will be formed to investigate a concept or an application related to discrete mathematics, and each group will report its findings to the class in a final presentation. This course satisfies the university's Integrative Experience (IE) requirement for math majors.

This course is an introduction to mathematical modeling. 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. Topics to be covered include systems of nonlinear differential equations and Markov chains, with emphasis on ideas such as equilibria, stability, and long time behavior, to name a few. Most of the theory in the course will deal with Markov chains. Although Stats 515 is not a pre-req for this class, it would be an advantage to have seen its material beforehand.

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.

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.

Forms of evaluation will include a group presentation, homework assignments, class discussions, and a final paper.
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. Students will be required to write an interdisciplinary 15-page final paper. Essay topics are developed by the student with assistance from the professor. 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.

This course is the second part of the Introduction to Analysis sequence (Math 523 and Math 524). We will first extend the notions of convergence developed for real numbers in 523 to more general settings-metric spaces. This will reveal the need for a deeper theory of integration -- measure theory -- which we will then explore. Finally, we will apply our understanding to develop Fourier Analysis.

Math 536 is three credit hours course, which 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 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, we will often use MATLAB.

The course will introduce basic numerical methods used for solving problems that arise in
a variety of 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. We will cover the following topics (not necessarily in the order listed): Finite Precision Arithmetic
and Error Propagation, Linear Systems of Equations (direct and iterative methods), Root Finding
(fixed point methods such as Newton's method), interpolation, least squares and Numerical Quadrature.

The course will introduce basic numerical methods used for solving problems that arise in a variety 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. The use of MATLAB for homework assignments will be mandatory. We will also discuss some very important practical considerations of implementing numerical methods using such languages 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. We will develop their local invariants (curvature and torsion for curves, fundamental forms and curvature for surfaces.) We will also explore the interplay between local and global properties.

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.

Introduction to some statistical regression and classification techniques including logistic regression, nearest neighbor methods, discriminant analysis, kernel smoothing, smoothing spline, local regression, generalized additive models, decision trees, random forests, and support vector machines. Clustering methods such as K-means and hierarchical clustering will be introduced. Finally, there will also be topics on resampling based model evaluation methods and regularization based model selection methods.

This fast-paced course is an introduction to the basic tools of algebraic topology, which studies topological spaces and continuous maps by producing associated algebraic structures (groups, rings, vector spaces, and homomorphism between them). Emphasis will be placed on being able to compute these invariants. Topics include: Cell complexes, homotopy, fundamental group, Van-Kampen's theorem, covering spaces (all reviewed from Math 671), simplicial complexes, singular and cellular homology, exact sequences, Mayer-Vietoris, cohomology, universal coefficients theorem, Künneth formulas, Poincaré and Lefschetz dualities.
Grade will be based on regularly assigned homework, as well as exams.

Continuation of Project

In this course, we will consider maximization and minimization problems in graphs and networks. We will cover a broad range of topics such as matchings in bipartite graphs and in general graphs, assignment problem, polyhedral combinatorics, total unimodularity, matroids, matroid intersection, min arborescence, max flow/min cut, max cut, traveling salesman problem, stable sets and perfect graphs. One of our main tools will be integer programming, and we will also sometimes rely on semidefinite programming. Many of these problems come from real-world applications, so we will also sometimes discuss the algorithms necessary to solve them. This is a rigorous mathematical introduction to combinatorial optimization with proofs.

This course aims to give an introduction to the fundamental topics in modern differential geometry, as organized in the following four units.

1. Theory of fiber bundles and connections.
2. Riemannian geometry.
3. Characteristics classes via Chern-Weil theory.
4. Hermitian and Kahler geometry.

We will present the basic concepts and theorems in each unit listed above, illustrated with interesting examples and detailed proofs of some selected results to demonstrate
the various basic techniques in these subjects. We also hope to enhance the learning experience with homework assignments/projects, which form the basis of the course grade.

This course is a second course in algebraic geometry. The prerequisites are a basic knowledge of algebraic geometry and commutative algebra. Some prior experience with sheaf cohomology would be useful but not essential. We will study complex algebraic surfaces, introducing the main computational tools of algebraic geometry along the way. Complex surfaces play a central role in algebraic, differential, and symplectic geometry, and number theory.

Tentative syllabus:

1. Sheaves and cohomology. Hodge theory.
2. Line bundles, divisors, morphisms to projective space.
3. Birational geometry of surfaces.
4. Characterization of the projective plane. del Pezzo surfaces.
5. Ruled surfaces.
6. K3 surfaces.
7. Elliptic fibrations.
8. Surfaces of general type.

We first review some basic probability and useful tools, including random walk, Law of large numbers and central limit
theorem. Conditional expectation and martingales. The topics of the course include the theory of stochastic differential equations oriented towards topics useful in applications, such as Brownian motion, stochastic integrals, and diffusion as solutions of stochastic differential equations. Then we study about diffusion in general: forward and backward Kolmogorov equations, stochastic differential equations and the Ito calculus, as well as Girsanov's theorem, Feynman-Kac formula, Martingale representation theorem. We will also include some applications to mathematical finance or networks as time permits.

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.