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.

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 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.

We will study the concept of dimension in geometry and physics beginning from dimension 0 to dimension 3 (and perhaps 4) leading to the Poincare Conjecture, which provides the possible shapes of the spacial 3-dimensional universe. The various ideas and view points humanity had over the past 2000 or so years about those topics will be part of our exploration.There will be lectures on this topic by the instructor and video taped lectures/demonstrations by eminent mathematicians who worked on these problems. We will also explore (in the above context) how mathematics and physics interact, why (whether) mathematics describes the "physical" universe so accurately, how (whether) aesthetics, art, philosophy has an impact on mathematics, and how mathematical ideas could be conveyed to a non-expert audience. The course is structured around writing assignments which will be peer reviewed and/or graded by the instructor and the course TA. During the last third of the semester there will be group project presentations.
All writing has to be done in the word processing system LaTex, which is the only word processing system capable of producing a professional layout. At the beginning of the semester there will be a presentation by the UMass career center director. We will not spend time on resume and job application writing, since there is ample opportunity to receive expert help from the career center.

We will study the concept of dimension in geometry and physics beginning from dimension 0 to dimension 3 (and perhaps 4) leading to the Poincare Conjecture, which provides the possible shapes of the spacial 3-dimensional universe. The various ideas and view points humanity had over the past 2000 or so years about those topics will be part of our exploration.There will be lectures on this topic by the instructor and video taped lectures/demonstrations by eminent mathematicians who worked on these problems. We will also explore (in the above context) how mathematics and physics interact, why (whether) mathematics describes the "physical" universe so accurately, how (whether) aesthetics, art, philosophy has an impact on mathematics, and how mathematical ideas could be conveyed to a non-expert audience. The course is structured around writing assignments which will be peer reviewed and/or graded by the instructor and the course TA. During the last third of the semester there will be group project presentations.
All writing has to be done in the word processing system LaTex, which is the only word processing system capable of producing a professional layout. At the beginning of the semester there will be a presentation by the UMass career center director. We will not spend time on resume and job application writing, since there is ample opportunity to receive expert help from the career center.

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

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

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.

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

We will explore several approaches to geometry: constructions with straight-edge and compass, axiomatic approach of Euclid and Hilbert, analytic geometry via linear algebra, and Klein's approach using symmetries and transformations. This will open the doors to many non-Euclidean flavors of geometry. Projective and spherical geometry will be studied in some detail.

Introduction to the fascinating theory of knots, links, and surfaces in 3- and 4-dimensional spaces. This course will combine geometric, algebraic, and combinatorial methods, where the students will learn how to utilize visualization and make rigorous arguments.

This course provides an introduction to systems of differential equations and dynamical systems, as well as chaotic dynamics, while providing a significant set of connections with phenomena modeled through these approaches in engineering, chemistry, biology, and social sciences. From the mathematical perspective, geometric and analytical 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 chaotic dynamics. From an applications perspective, numerous specific applications will be touched upon ranging from epidemiological models to chemical reactions and from lasers to the synchronization of fireflies. In addition to the theoretical and modeling aspects of the class, a self-contained Machine Learning component will be introduced and developed that will allow for both forward simulation and statistical learning of dynamical systems from data. However, no prior knowledge of Machine Learning tools will be assumed.

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.

Introduction to computational techniques used in science and industry. Topics selected from root-finding, interpolation, data fitting, linear systems, numerical integration, numerical solution of differential equations, and error analysis.

Introduction to computational techniques used in science and industry. Topics selected from root-finding, interpolation, data fitting, linear systems, numerical integration, numerical solution of differential equations, and error analysis.

This course is an introduction to the fundamental principles of statistical science.  It does not rely on detailed derivations of mathematical concepts, but does require mathematical sophistication and reasoning.  It is an introduction to statistical thinking/reasoning, data management, statistical analysis, and statistical computation.  Concepts in this course will be developed in greater mathematical rigor later in the statistical curriculum, including in STAT 515, 516, 525, and 535. It is intended to be the first course in statistics taken by math majors interested in statistics.  Concepts covered include point estimation, interval estimation, prediction, testing, and regression, with focus on sampling distributions and the properties of statistical procedures.  The course will be taught in a hands-on manner, introducing powerful statistical software used in practical settings and including methods for descriptive statistics, visualization, and data management.

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.

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. The overall objective of the course is the development of basic theory and methods for statistical inference from a mathematical and probabilistic perspective. Topics include: sampling distributions; point estimators and their properties; method of moments; maximum likelihood estimation; Rao-Blackwell Theorem; confidence intervals, hypothesis testing; contingency tables; and non-parametric methods (time permitting).

Continuation of Stat 515. The overall objective of the course is the development of basic theory and methods for statistical inference from a mathematical and probabilistic perspective. Topics include: sampling distributions; point estimators and their properties; method of moments; maximum likelihood estimation; Rao-Blackwell Theorem; confidence intervals, hypothesis testing; contingency tables; and non-parametric methods (time permitting).

Regression analysis is the most popularly used statistical technique with application in almost every imaginable field. The focus of this course is on a careful understanding and of regression models and associated methods of statistical inference, 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; expressing regression models and methods in matrix form; an introduction to weighted least squares, regression with correlated errors and nonlinear regression. Extensive data analysis using R or SAS (no previous computer experience assumed). Requires prior coursework in Statistics, preferably ST516, and basic matrix algebra. Satisfies the Integrative Experience requirement for BA-Math and BS-Math majors.

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.

General theory of measure and integration and its specialization to Euclidean spaces and Lebesgue measure; modes of convergence, Lp spaces, product spaces, differentiation of measures and functions, signed measures, Radon-Nikodym theorem.

This course covers a broad range of fundamental numerical methods, including: machine zeros nonlinear equations and systems, interpolation and least square method, numerical integration, and the methods solving initial value problems of ODEs, linear algebra, direct and iterative methods for solving large linear systems. There will be regular homework assignments and programming assignments (in MATLAB). The grade will be based on homework assignments, class participation, and exams.

The purpose of the teaching seminar is to support graduate students as they teach their first discussion section at UMass. The seminar will focus on four components of teaching: Who the students are, teaching calculus concepts, instruction techniques, and assessment.

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.

Lie algebras are linear algebra devices of great usefulness in mathematics and physics as an efficient tool for the study of symmetries of objects. This course will cover the fundamentals of the subject, including nilpotent and solvable Lie algebras, as well as semisimple Lie algebras and their representations.

Homological algebra can be seen as a generalization and extension of linear algebra, and so it plays an essential role in many areas of contemporary mathematics. Like linear algebra, it is important to understand both algorithms but also powerful structural results, and we will give due attention to both aspects. This course will lay foundations in a way guided by modern views (e.g., we will introduce model categories) but will also explore classic applications, like group cohomology and Lie algebra cohomology, so that computational facility is developed. By the end of the course, you will know what a derived category is and how to run a spectral sequence.

The first part of a two-semester graduate level sequence in probability and statistics, this course develops probability theory at an intermediate level (i.e., non measure-theoretic - Stat 605 is a course in measure-theoretic probability) and introduces the basic concepts of statistics.
Topics include: general probability concepts; discrete probability; random variables (including special discrete and continuous distributions) and random vectors; independence; laws of large numbers; central limit theorem; statistical models and sampling distributions; and a brief introduction to statistical inference. Statistical inference will be developed more fully in Stat 608.
This course is also suitable for graduate students in a wide variety of disciplines and will give strong preparation for further courses in statistics, econometrics, and stochastic processes, time series, decision theory, operations research, etc.
You will be expected to read sections of the text book in parallel with topics covered in lectures, since important part of graduate study is to learn how to study independently.

This course will introduce students to Bayesian data analysis, including modeling and computation. We will begin with a description of the components of a Bayesian model and analysis (including the likelihood, prior, posterior, conjugacy and credible intervals). We will then develop Bayesian approaches to models such as regression models, hierarchical models and ANOVA. Computing topics include Markov chain Monte Carlo methods. The course will have students carry out analyses using statistical programming languages and software packages.

Regression is the most widely used statistical technique. In addition to learning about regression methods this course will also reinforce basic statistical concepts and expose students (for many for the first time) 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; an introduction to weighted least squares, regression with correlated errors and nonlinear including binary) regression.

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.

This course will cover several workhorse models for analysis of time series data. The course will begin with a thorough and careful review of linear and general linear regression models, with a focus on model selection and uncertainty quantification. Basic time series concepts will then be introduced. Having built a strong foundation to work from, we will delve into several foundational time series models: autoregressive and vector autoregressive models. We will then introduce the state-space modeling framework, which generalizes the foundational time series models and offers greater flexibility. Time series models are especially computationally challenging to work with - throughout the course we will explore and implement the specialized algorithms that make computation feasible in R and/or STAN. Weekly problem sets, two-to-three short exams, and a final project will be required.