Course Descriptions

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.

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.

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

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.

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.

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 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.

This course will introduce students to writing in mathematics, both technical and otherwise. Writing assignments will include proofs, instructional handouts, resumes, cover letters, presentations, 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, and to tailor that writing for a particular audience.

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.

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

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

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

This course can be used to satisfy the UMass Integrative Experience requirement. The main goal of the class is to learn how to translate real-world situations into mathematical terms and use the model to predict, optimize and generally understand the original situation. The course material will concentrate on topics related to social sciences, such as voting and electoral systems. We will use a variety of mathematical techniques and objects, including networks.

This course 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. Students are required to use primary sources on a weekly basis. Students will study examples of how mathematical advances were made in response to or alongside developments in other branches of science, how these advances were understood at the time, and their implications for modern mathematics. Since many math majors go on to teach mathematics after graduation, in this course the history of math is also studied in the context of the history of education.

Forms of evaluation will include a group presentation, homework assignments, participation in class discussions, and a 15-page final paper.

This is the first part of the introduction to analysis sequence (523 and 524). This course deals with basic concepts of analysis of functions mostly on the real line, and we will try to make many of the concepts one learns in calculus rigorous. Covered topics will include series, and sequences, continuity, differentiability, and integration.

An introduction to PDEs (partial differential equations), covering some of the most basic and ubiquitous linear equations modeling physical problems and arising in a variety of contexts. We shall study the existence and derivation of explicit formulas for their solutions (when feasible) and study their behavior. We will also learn how to read and use specific properties of each individual equation to analyze the behavior of solutions when explicit formulas do not exist. Equations covered include: transport equations and the wave equation, heat/diffusion equations and the Laplace’s equation on domains. Along the way we will discuss topics such as Fourier series, separation of variables, energy methods, maximum principle, harmonic functions and potential theory, etc.

Time-permitting, we will discuss some additional topics (eg.. Schrödinger equations, Fourier transform methods, eigenvalue problems, etc.). The final grade will be determined on the basis of homework, attendance and class participation, a midterm and final projects.

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.

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.

In this course, students complete an applied statistics field project that has been solicited from researchers in biological, physical, or social sciences. The instructor supplies applied as well as statistical methodology readings for the students. The readings serve to extend what students have learned in prior classes, and especially to help students learn to apply statistical methodology to problems from real-world applications. The students work in groups of 2, and they have to write a 10-20 page technical report and prepare a poster to summarize the project. Satisfies the Integrative Experience requirement for BA-Math and BS-Math majors. Prerequisites: Previous coursework in Probability and Statistics, such as Statistc 516 or 501, as well as Statistc 525 and 526 including knowledge of estimation, intervals, and hypothesis testing, or permission of instructor.

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, probability inequalities, the central limit theorem, the Poisson approximation 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, probability inequalities, the central limit theorem, the Poisson approximation 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, probability inequalities, the central limit theorem, the Poisson approximation 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.

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 provides an introduction to fundamental computer science concepts relevant to the statistical analysis of large-scale data sets. Students will collaborate in a team to design and implement analyses of real-world data sets, and communicate their results using mathematical, verbal and visual means. Students will learn how to analyze computational complexity and how to choose an appropriate data structure for an analysis procedure. Students will learn and use the python language to implement and study data structure and statistical algorithms.

This course covers classical methods in applied mathematics and math modeling, including dimensional analysis, asymptotics, regular and singular perturbation theory for ordinary differential equations, random walks and the diffusion limit, and classical solution techniques for PDE. The techniques will be applied to models arising throughout the natural sciences.

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.

The theory of sums of squares (SOS) blends exciting ideas from optimization, real algebraic geometry and convex geometry. Indeed, Hilbert's famous characterization of nonnegative polynomials that are SOS in 1888, and Artin's affirmative answer to Hilbert's 17th problem on whether all nonnegative polynomials are SOS of rational functions are at the origins of this topic. Over the last two decades, interest in the theory and application of SOS polynomials has exploded because of the work of Shor, Nesterov, Lasserre and Parrilo that connects SOS polynomials to modern optimization via semidefinite programming. Since then, there has been many thrilling applications in combinatorics, theoretical computer science, and engineering. This course will cover both the theory and some applications.

This is an introductory course on symplectic topology, along with its connections to differential, complex algebraic, and contact geometry and topology.

Functional analysis deals with the structure of infinite dimensional vector spaces and (mostly) linear on such spaces. Many such spaces are spaces of functions, hence the name functional analysis, but much of the theory will developed for abstract spaces (spaces with a norm or a scale product). We shall assume that the reader has taken Math 624 (or an equivalent course) and is familiar with the basic objects of functional analysis: Banach spaces and Hilbert spaces, linear functionals and duals, bounded linear operators. Our main goal is to develop a series of tools instrumental in the applications of functional analysis to PDE's, probability, ergodic theory, etc... Among the topics covered in this class are: Hahn-Banach theorem, Inverse mapping and closed graph theorems. Compact operators, Fredholm operators and applications; Spectral theory for linear bounded and unbounded operators, Banach algebras, Semigroups. Time permitting we will also cover the theory of distributions and Sobolev spaces.

Hitchin once said that there is no real definition of what an infinite dimensional integrable system should be, but if you encounter one, you know it is one.

This being said, there are a number of descriptive aspects: infinite hierarchies (e.g. KdV, KP, non-linear Schroedinger, etc.) of flows and functionals whose symplectic gradients are those flows; loop Grassmannians; actions of loop groups and loop algebras; zero curvature equations; Lax pairs; spectral curves and linear flows on Jacobians/Prim varieties etc. The objective of the course is to discuss the interrelations among those various aspects by studying some of the classical examples such as KdV or non-linear Schroedinger. A novelty (when compared to the existing literature) is the geometric approach: the (pre)symplectic (or Poisson) manifolds will be spaces of geometric objects, such as manifolds of curves or surfaces in a certain target manifold rather than function spaces.

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.

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.

Distribution and inference for binomial and multinomial variables with contingency tables, generalized linear models, logistic regression for binary responses, logit models for multiple response categories, loglinear models, inference for matched-pairs and correlated clustered data. Prerequisites: Previous course work in probability and mathematical statistics including knowledge of distribution theory, estimation, confidence intervals, hypothesis testing and multiple linear regression; e.g. Stat 516 and Stat 525 (or equivalent). Prior programming experience.