# Course Descriptions

## Lower Division Courses

### MATH 100: Basic Math Skills for the Modern World

See Preregistration guide for instructors and times

Description:

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.

### MATH 101: Precalculus, Algebra, Functions and Graphs

See Preregistration guide for instructors and times

Prerequisites:

Prereq: MATH 011 or Placement Exam Part A score above 10. Students needing a less extensive review should register for MATH 104.

Note:

Students cannot receive credit for MATH 101 if they have already received credit for any MATH or STATISTC course numbered 127 or higher.

Description:

First semester of the two-semester sequence MATH 101-102. Detailed, in-depth review of manipulative algebra; introduction to functions and graphs, including linear, quadratic, and rational functions.

### MATH 102: Analytic Geometry and Trigonometry

See Preregistration guide for instructors and times

Prerequisites:

Math 101

Description:

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.

### MATH 103: Precalculus and Trigonometry

See Preregistration guide for instructors and times

Prerequisites:

The equivalent of the algebra and geometry portions of MATH 104. (See also MATH 101, 102, 104.)

Description:

The trigonometry topics of MATH 104.

### MATH 104: Algebra, Analytic Geometry and Trigonometry

See Preregistration guide for instructors and times

Prerequisites:

MATH 011 or Placement Exam Part A score above 15. Students with a weak background should take the two-semester sequence MATH 101-102.

Description:

One-semester review of manipulative algebra, introduction to functions, some topics in analytic geometry, and that portion of trigonometry needed for calculus.

### MATH 113: Math for Elementary Teachers I

See Preregistration guide for instructors and times

Prerequisites:

MATH 011 or satisfaction of R1 requirement.

Description:

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.

### MATH 114: Math for Elementary Teachers II

See Preregistration guide for instructors and times

Prerequisites:

MATH 113

Description:

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.

### MATH 121: Linear Methods and Probability for Business

See Preregistration guide for instructors and times

Prerequisites:

Working knowledge of high school algebra and plane geometry.

Description:

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

### MATH 127: Calculus for Life and Social Sciences I

See Preregistration guide for instructors and times

Prerequisites:

Proficiency in high school algebra, including word problems.

Description:

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.

### MATH 127H: Honors Calculus for Life and Social Sciences I

See Preregistration guide for instructors and times

Prerequisites:

Proficiency in high school algebra, including word problems.

Description:

Honors section of Math 127.

### MATH 128: Calculus for Life and Social Sciences II

See Preregistration guide for instructors and times

Prerequisites:

Math 127

Description:

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.

### MATH 131: Calculus I

See Preregistration guide for instructors and times

Prerequisites:

High school algebra, plane geometry, trigonometry, and analytic geometry.

Description:

Continuity, limits, and the derivative for algebraic, trigonometric, logarithmic, exponential, and inverse functions. Applications to physics, chemistry, and engineering.

### MATH 131H: Honors Calculus I

See Preregistration guide for instructors and times

Prerequisites:

High school algebra, plane geometry, trigonometry, and analytic geometry.

Description:

Honors section of Math 131.

### MATH 132: Calculus II

See Preregistration guide for instructors and times

Prerequisites:

Math 131 or equivalent.

Description:

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.

### MATH 132H: Honors Calculus II

See Preregistration guide for instructors and times

Prerequisites:

Math 131 or equivalent.

Description:

Honors section of Math 132.

### MATH 233: Multivariate Calculus

See Preregistration guide for instructors and times

Prerequisites:

Math 132.

Description:

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)

### MATH 233H: Honors Multivariate Calculus

See Preregistration guide for instructors and times

Prerequisites:

Math 132.

Description:

Honors section of Math 233.

### MATH 235: Introduction to Linear Algebra

See Preregistration guide for instructors and times

Prerequisites:

Math 132 or consent of the instructor.

Description:

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

### MATH 235H: Honors Introduction to Linear Algebra

See Preregistration guide for instructors and times

Prerequisites:

Math 132 or consent of the instructor.

Description:

Honors section of Math 235.

### MATH 331: Ordinary Differential Equations for Scientists and Engineers

See Preregistration Guide for instructors and times

Prerequisites:

Math 132; corequisite: Math 233

Text:

TBA

Description:

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.)

### STAT 111: Elementary Statistics

See Preregistration guide for instructors and times

Prerequisites:

High school algebra.

Description:

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.

### STAT 240: Introduction to Statistics

See Preregistration guide for instructors and times

Description:

Basics of probability, random variables, binomial and normal distributions, central limit theorem, hypothesis testing, and simple linear regression

## Upper Division Courses

### MATH 300.1: Fundamental Concepts of Mathematics

Prerequisites:

Math 132 with a grade of 'C' or better

Description:

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). 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.

### MATH 300.2: Fundamental Concepts of Mathematics

Prerequisites:

Math 132 with a grade of 'C' or better

Description:

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). 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.

### MATH 300.3: Fundamental Concepts of Mathematics

Prerequisites:

Math 132 with a grade of 'C' or better

Text:

An Introduction to Mathematical Thinking (Algebra and Number Systems), by Gilbert and Vanstone, Prentice Hall, 2005.

Description:

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). 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.

### MATH 370.1: Writing in Mathematics

Prerequisites:

Math 300 or Comp Sci 250 and completion of the College Writing (CW) requirement.

Description:

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.

### MATH 370.2: Writing in Mathematics

Prerequisites:

MATH 300 or CS 250 and completion of the College Writing (CW) requirement.

Description:

Essentially all serious mathematics is written using some variant of the LaTeX software, and developing proficiency with this tool is an important part of the course. We will cover a variety of types of writing related to mathematics, such as: critique of seminar presentations and written articles; (auto)biography of mathematicians; expository writing about mathematical topics; aspects of writing a research article; opinion pieces; precise short communications (e.g. abstract, memo, tweet, blog post, newspaper headline); essays about aspects of mathematical culture (e.g. jokes, songs, anecdotes, sociological aspects).

Satisfies Junior Year Writing requirement.

### MATH 370.3: Writing in Mathematics

Prerequisites:

MATH 300 or CS 250 and completion of the College Writing (CW) requirement.

Description:

Essentially all serious mathematics is written using some variant of the LaTeX software, and developing proficiency with this tool is an important part of the course. We will cover a variety of types of writing related to mathematics, such as: critique of seminar presentations and written articles; (auto)biography of mathematicians; expository writing about mathematical topics; aspects of writing a research article; opinion pieces; precise short communications (e.g. abstract, memo, tweet, blog post, newspaper headline); essays about aspects of mathematical culture (e.g. jokes, songs, anecdotes, sociological aspects).

Satisfies Junior Year Writing requirement.

### MATH 370.4: Writing in Mathematics

Prerequisites:

MATH 300 or CS 250 and completion of the College Writing (CW) requirement

Description:

Satisfies Junior Year Writing requirement. Develops research, presentation, and writing skills in mathematics, including LaTex through team work, peer review, and revision. Students write on mathematical subject areas, prominent mathematicians, and famous mathematical problems.

### MATH 397C: ST - Mathematical Computing

Prerequisites:

COMPSCI 121, MATH 235, and MATH 300

Text:

There is no required text. Notes will be provided by the instructor and online resources will be used.

Description:

This course is about how to write and use computer code to explore and solve problems in pure and applied mathematics. The first part of the course will be an introduction to programming in Python. The remainder of the course (and its goal) is to help students develop the skills to translate mathematical problems and solution techniques into algorithms and code. Students will work on projects, both individually and in groups, with a variety applications throughout the curriculum.

### MATH 411.1: Introduction to Abstract Algebra I

Prerequisites:

Math 235; Math 300 or CS 250.

Description:

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

### MATH 411.2: Introduction to Abstract Algebra I

Prerequisites:

MATH 235; MATH 300 or CS 250

Text:

Abstract Algebra with Applications by Audrey Terras. The course will start in Chapter 2 and cover Chapters 2, 3 and 4. It is recommended that students also read Chapter 1, which covers the material from the introduction to proofs course such as Math 300. Some of this material will also be reviewed as the course progresses. Some assignments in this class will require access to Wolfram Mathematica software. A free account with https://www.wolframcloud.com/ will be sufficient.

Description:

The focus of the course will be on learning group theory. A group is a central concept of mathematics which is used to describe algebraic operations and symmetries of every possible kind, from modular arithmetic to symmetries of geometric objects.

Learning objectives: The emphasis will be on development of careful mathematical reasoning. Theoretical constructions and applications will be tested on many examples, both by hand and using computer algebra systems, specifically Wolfram Mathematica.

Approximate Schedule:

• Group axioms. Examples of groups. Numbers and matrices. Transformation groups.
• Multiplication tables and Cayley graphs. Properties of groups.
• Subgroups. Order of an element. Cyclic groups.
• Symmetric group. Properties of permutations. Cayley theorem. Isomorphisms.
• Cosets. Lagrange theorem. Normal subgroups.
• Quotient group. Isomorphism theorems. Homomorphisms.
• Direct product of groups. Classification of finite Abelian groups.
• Group actions. Orbit/stabilizer theorem. Burnside lemma. Conjugacy classes. Cauchy theorem. Sylow theorems (without proof).
• Classification of crystallographic groups.
• Group-theoretic aspects of public-key cryptography.
• Subgroups of the symmetric group. Solving puzzles using groups.
• Classification of groups of small order.

### MATH 411.3: Introduction to Abstract Algebra I

Prerequisites:

MATH 235; MATH 300 or CS 250.

Description:

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

### MATH 421: Complex Variables

Prerequisites:

Math 233

Text:

Complex Variables and Applications, 8-th Edition, by James Ward Brown and Ruel V. Churchill, McGraw-Hill.

Description:

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. The Argument Principle and Rouche's Theorem. Evaluation of Improper integrals via residues. Conformal mappings.

### MATH 437: Actuarial Financial Math

Prerequisites:

Math 131 and 132 or equivalent courses with C or better

Recommended Text:

ASM Study Manual for Exam FM 15th Edition by Cherry & Shaban, you can buy it at https://www.studymanuals.com/Product/Show/453142747.
Texas Instruments BA II plus calculator is recommended

Description:

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 455: Introduction to Discrete Structures

Prerequisites:

Calculus (MATH 131, 132, 233), Linear Algebra (MATH 235), and Math 300 or CS 250.

Text:

Combinatorics and Graph Theory, John M. Harris, Jeffry L. Hirst, Michael J. Mossinghoff (second edition)

Note:

TTh 1:00-2:15pm

Description:

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; big-Oh notation, 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.

### MATH 456.1: Mathematical Modeling

Prerequisites:

Math 235 and Math 300/CS 250. Some familiarity with a programming language is very desirable (Python, Java, Matlab, etc.).

Description:

This course is an introduction to mathematical modeling. The main goal of the class is to learn how to translate real-world problems into quantitative terms for interpretation, suggestions of improvement and future predictions. Since this is too broad of a topic for one semester, this class will focus on linear and integer programming to study real world problems that affect real people. The course will culminate in a final modeling research project that will involve optimizing different aspects of a community partner. This course satisfies the Integrative Experience requirement for BA-Math and BS-Math majors.

### MATH 456.2: Mathematical Modeling

Prerequisites:

Math 233, Math 235, Math 331. Some familiarity with a programming language is desirable (Mathematica, Matlab, Java, C++, Python, etc.). Some familiarity with statistics and probability is desirable.

Description:

Mathematics is usually termed as “the language of nature”. Complex physical phenomenon can be described by mathematical models with sufficient accuracy. In this course, students learn how to formulate and analyze some real-world problems by utilizing concepts, methods and theories from mathematics, thus coming to understand the interplay between mathematical theory and practice. Since mathematical models can be very broad and can appear in every discipline, this course will mainly focus on problems rising from data science. The goal is to discuss how to build and learn mathematical models in data-driven applications. Students will form several groups to investigate a modeling problem and each group will report their findings in a final presentation. Prerequisites of the course include Calculus (Math 131, 132, 233), Linear Algebra (Math 235) and Differential Equations (Math 331). Some familiarity with a programming language is desirable (Matlab, Python, etc.)

### MATH 461: Affine and Projective Geometry

Prerequisites:

Math 235 and Math 300

Description:

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.

### MATH 471: Theory of Numbers

Prerequisites:

Math 233 and Math 235 and either Math 300 or CS250.

Text:

TBA

Description:

The goal of this course is to give a rigorous introduction to elementary number theory. While no prior background in number theory will be assumed, the ability to read and write proofs is essential for this course. The list of topics include (but is not limited to): Euclidean Algorithm, Linear Diophantine Equations, the Fundamental Theorem of Arithmetic, congruence arithmetic, continued fractions, the theory of prime numbers, primitive roots, and quadratic reciprocity, with an emphasis on applications to and connections with cryptography. Homework will consist of both rewritten assignments and computer projects.

### MATH 490A: Intro to Abstract Algebra for Future Teachers

Prerequisites:

Math 235 and either Math 300 or CMPSCI 250, or permission of the instructor

Text:

Ronald S. Irving. Integers, Polynomials, and Rings. Springer-Verlag. You can read this online through the UMass library, and buy ebook, softcover, or hardcover editions for fairly low prices from the publisher.

Description:

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 some of the connections between these topics and high school mathematics; this is definitely a course in abstract algebra, not a course on how to understand or teach high school mathematics, but I hope that the things you learn in this course will deepen your understanding of, and change the way you think about, some important parts of high school algebra.

### MATH 491A: Seminar - Putnam Exam Prep (1 credit)

Instructor and Time TBA

Prerequisites:

One variable Calculus, Linear Algebra

Description:

The William Lowell Putnam Mathematics Competition is the most prestigious annual contest for college students. While the problems employ topics from a standard undergraduate curriculum, the ability to solve them requires a great deal of ingenuity, which can be developed through systematic and specific training. This class aims to assist the interested students in their preparation for the Putnam exam, and also, more generally, to treat some topics in undergraduate mathematics through the use of competition problems.

### MATH 491P: GRE Prep Seminar

Instructor and Time TBA

Prerequisites:

MATH 233 & 235 and either MATH 300 or COMPSCI 250.
Students should have already completed, or be currently taking Math 331.
Students should have already completed, or be currently taking Math 411 or Math 523H.

Description:

This class 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.

### MATH 497K: Knot Theory

Prerequisites:

Math 235; Math 300 or CS 250. Math 411 is strongly recommended as a co-requisite.

Description:

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.

### MATH 513: Combinatorics

Description:

Cross-listed with CompSci 575. A basic introduction to combinatorics and graph theory for advanced students in computer science, mathematics, and related fields. Topics include elements of graph theory, Euler and Hamiltonian circuits, graph coloring, matching, basic counting methods, generating functions, recurrences, inclusion-exclusion, Polya's theory of counting. Prerequisites: mathematical maturity, calculus, linear algebra, discrete mathematics course such as CompSci 250 or Math 455. Math 411 recommended but not required.

### MATH 523H: Introduction to Modern Analysis

Prerequisites:

Math 300 or CS 250

Description:

This course is an introduction to mathematical analysis. A rigorous treatment of the topics covered in calculus will be presented with a particular emphasis on proofs. Topics include: properties of real numbers, sequences and series, continuity, Riemann integral, differentiability, sequences of functions and uniform convergence.

### MATH 532H: Nonlinear Dynamics

Prerequisites:

Math 235 (Linear Algebra), Math 331 (Differential Equations) and the calculus sequence (Math 131, 132, 233), or equivalent background in elementary differential equations, linear algebra, and calculus

Description:

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 Physics, Chemistry, and Biology. 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 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.

### MATH 537.1: Intro to Mathematics of Finance

Prerequisites:

Math 233 and either Stat 515 or MIE 273

Description:

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.

### MATH 537.2: Intro to Mathematics of Finance

Prerequisites:

Math 233 and either Stat 515 or MIE 273

Description:

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.

### MATH 545.1: Linear Algebra for Applied Mathematics

Prerequisites:

Math 233 and 235 with a grade of C or better, and either Math 300 or CS 250

Description:

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. Homeworks include programming projects.

### MATH 545.2: Linear Algebra for Applied Mathematics

Prerequisites:

Math 233 and 235 with a grade of C or better, and either Math 300 or CS 250

Text:

Strang - Linear Algebra with Applications - 3rd or 4th edition

Description:

Vector spaces, subspaces, norms and inner products,
spanning sets and independence, basis and dimension.

Matrices, multiplication, LU decomposition.

Orthorgonality, Gram-Schmidt and QR decomposition,
Least Squares.

FFT, determinants.
Eigenvalues, Jordan decomposition.
Singular Values, Computational Methods.

### MATH 545.3: Linear Algebra for Applied Mathematics

Prerequisites:

Math 233 and 235 with a grade of C or better, and either Math 300 or CS 250

Description:

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.

### MATH 551.1: Intr. Scientific Computing

Prerequisites:

MATH 233 & 235 and either COMPSCI 121, E&C-ENG 122, PHYSICS 281, or E&C-ENG 242. Knowledge of a scientific programming language, e.g. MATLAB, Fortran, C, C++, Python, Java.

Recommended Text:

A First Course in Numerical Methods, Authors: Uri M. Ascher and Chen Greif, Publisher: Society for Industrial and Applied Mathematics (SIAM), 2011.

Note: An electronic version of the textbook is free from SIAM for students.

Description:

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.

### MATH 551.2: Intr. Scientific Computing

Prerequisites:

MATH 233 & 235 and either COMPSCI 121, E&C-ENG 122, PHYSICS 281, or E&C-ENG 242. Knowledge of a scientific programming language, e.g. MATLAB, Fortran, C, C++, Python, Java.

Description:

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.

### MATH 551.3: Intr. Scientific Computing

Prerequisites:

MATH 233 & 235 and either COMPSCI 121, E&C-ENG 122, PHYSICS 281, or E&C-ENG 242. Knowledge of a scientific programming language, e.g. MATLAB, Fortran, C, C++, Python, Java.

Description:

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.

### MATH 597U: Stochastic Processes and Simulations

Prerequisites:

Stat 515

Recommended Text:

1. Essentials of Stochastic Processes, Rick Durrett

2. Simulation, Sheldon M. Ross

Description:

This is a second course in Probability, studying stochastic/random process, intended for majors in Applied Math, Statistics and related fields. The prerequisite is STAT 515 or similar upper-division course. If you did not get at least a B in that course then you will find this course very tough. Moreover, students are required to have some knowledge of Python, as we will cover simulation topics, including sampling of probability distributions, Monte Carlo algorithms, etc. A major focus of the course is on solving problems extending the scope of the lectures, developing analytical skills and probabilistic intuition. The course will cover the following topics in the core of the theory of Random and Stochastic Processes.

1. Review of probability theory : probability space, random variables, expectations, independence, conditional expectations.
2. Random walks and finite state Markov chains: Transition matrix, transience and recurrence, limiting distributions, convergence of Markov Chains.
3. Poisson processes: definition, inter-arrival and waiting time.
4. Continuous Markov chains: Strong Markov properties, Chapman Kolmogorov equations, irreducible and recurrence. Long time behaviour.
5. Brownian Motions: Definitions, scaled random walk, Brownian motion.

In addition, we also add a parallel part to the theoretical lectures - the stochastic simulations.

1.Sampling of basic probability distributions, generation of pseudorandom numbers,
2. Monte Carlo integration Simulation of random samples from discrete distributions and continuous distributions
3. Discrete event simulation for stochastic models of queueing systems
4. MCMC

### STAT 310.1: Fundamental Concepts/Stats

Prerequisites:

MATH 132

Description:

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.

### STAT 310.2: Fundamental Concepts/Stats

Prerequisites:

MATH 132

Description:

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.

### STAT 501: Methods of Applied Statistics

Prerequisites:

Knowledge of high school algebra, junior standing or higher.

Description:

For graduate and upper-level undergraduate students, with focus on practical aspects of statistical methods.Topics include: data description and display, probability, random variables, random sampling, estimation and hypothesis testing, one and two sample problems, analysis of variance, simple and multiple linear regression, contingency tables. Includes data analysis using a computer package (R).

### STAT 515.1: Introduction to Statistics I

Prerequisites:

Two semesters of single variable calculus (Math 131-132) or the equivalent, with a grade of "C" or better in Math 132. Math 233 is recommended for this course. However, some necessary concepts for multiple integration or partial derivatives will be re-introduced in the course as needed.

Description:

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.

### STAT 515.2: Introduction to Statistics I

Prerequisites:

Two semesters of single variable calculus (Math 131-132) or the equivalent, with a grade of "C" or better in Math 132. Math 233 is recommended for this course. However, some necessary concepts for multiple integration or partial derivatives will be re-introduced in the course as needed.

Description:

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.

### STAT 515.3: Introduction to Statistics I

Prerequisites:

Two semesters of single variable calculus (Math 131-132) or the equivalent, with a grade of "C" or better in Math 132. Math 233 is recommended for this course. However, some necessary concepts for multiple integration or partial derivatives will be re-introduced in the course as needed.

Description:

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.

### STAT 515.4: Introduction to Statistics I

Prerequisites:

Two semesters of single variable calculus (Math 131-132) or the equivalent, with a grade of "C" or better in Math 132. Math 233 is recommended for this course. However, some necessary concepts for multiple integration or partial derivatives will be re-introduced in the course as needed.

Description:

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.

### STAT 515.5: Introduction to Statistics I

Prerequisites:

Two semesters of single variable calculus (Math 131-132) or the equivalent, with a grade of "C" or better in Math 132. Math 233 is recommended for this course. However, some necessary concepts for multiple integration or partial derivatives will be re-introduced in the course as needed.

Description:

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.

### STAT 515.6: Introduction to Statistics I

Prerequisites:

Two semesters of single variable calculus (Math 131-132) or the equivalent, with a grade of "C" or better in Math 132. Math 233 is recommended for this course. However, some necessary concepts for multiple integration or partial derivatives will be re-introduced in the course as needed.

Description:

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.

### STAT 516.1: Statistics II

Prerequisites:

Stat 515 or 515H with a grade of “C" or better

Text:

Wackerly, Mendenhall, Schaeffer, Mathematical Statistics with Applications, 7th Ed. (ISBN-13: 978-0495110811).

Description:

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).

### STAT 516.3: Statistics II

Prerequisites:

Stat 515 or 515H with a grade of “C" or better

Description:

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).

### STAT 516.4: Statistics II

Prerequisites:

Stat 515 or 515H with a grade of “C" or better

Text:

Wackerly, Mendenhall, Schaeffer, Mathematical Statistics with Applications, 7th Ed. (ISBN-13: 978-0495110811).

Description:

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).

### STAT 525.1: Regression Analysis

Prerequisites:

Stat 516 or equivalent : Previous coursework in Probability and Statistics, including knowledge of estimation, confidence intervals, and hypothesis testing and its use in at least one and two sample problems. You must be familiar with these statistical concepts beforehand. Stat 515 by itself is NOT a sufficient background for this course! Familiarity with basic matrix notation and operations is helpful.

Description:

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.

### STAT 525.2: Regression Analysis

Prerequisites:

Stat 516 or equivalent : Previous coursework in Probability and Statistics, including knowledge of estimation, confidence intervals, and hypothesis testing and its use in at least one and two sample problems. You must be familiar with these statistical concepts beforehand. Stat 515 by itself is NOT a sufficient background for this course! Familiarity with basic matrix notation and operations is helpful.

Description:

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.

### STAT 535.1: Statistical Computing

Prerequisites:

Stat 516 and CS 121

Description:

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.

### STAT 535.2: Statistical Computing

Prerequisites:

Stat 516 and CS 121

Note:

This course may be taken remotely. Please enroll and contact the instructor if you would like to take the course remotely.

This class meets on the Newton Mount Ida Campus of UMass-Amherst.

Description:

This course will introduce computing tools needed for statistical analysis including data acquisition from database, data exploration and analysis, numerical analysis and result presentation. Advanced topics include parallel computing, simulation and optimization, and package creation. The class will be taught in a modern statistical computing language.

### STAT 598C: Statistical Consulting Practicum (1 cr)

Prerequisites:

Graduate standing, STAT 515, 516, 525 or equivalent, and consent of instructor.

Description:

This course provides a forum for training in statistical consulting. Application of statistical methods to real problems, as well as interpersonal and communication aspects of consulting are explored in the consulting environment. Students enrolled in this class will become eligible to conduct consulting projects as consultants in the Statistical Consulting and Collaboration Services group in the Department of Mathematics and Statistics. Consulting projects arising during the semester will be matched to students enrolled in the course according to student background, interests, and availability. Taking on consulting projects is not required, although enrolled students are expected to have interest in consulting at some point. The class will include some presented classroom material; most of the class will be devoted to discussing the status of and issues encountered in students' ongoing consulting projects.

### MATH 605: Probability Th I

Prerequisites:

Stat 515 or equivalent, Math 523 or equivalent is extremely useful. A good working knowledge of undergraduate probability and analysis, contact the instructor if in doubt.

Description:

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.

### MATH 611: Algebra I

Prerequisites:

Undergraduate algebra (equivalent of our Math 411-412).

Description:

This fast-paced course (and its continuation - Math 612) will introduce modern algebra concepts with an emphasis on topics required for the qualifying exam in algebra. Syllabus of Math 611 - Math 612:

I. Group Theory and Representation Theory.

Group actions. Counting with groups. P-groups and Sylow theorems. Composition series. Jordan-Holder theorem. Solvable groups. Automorphisms. Semi-direct products. Finitely generated Abelian groups. Complex representations of finite groups. Schur's Lemma. Maschke's Theorem. Representations of Abelian groups. Characters. Schur's orthogonality relations. The number of irreducible representations is equal to the number of conjugacy classes. The sum of squares of dimensions of irreducible representation is equal to the size of the group. The dimension of any irreducible representation divides the size of the group.

II. Linear Algebra and Commutative Algebra.

Euclidean domain is a PID. PID is a UFD. Gauss Lemma. Eisenstein's Criterion. Exact sequences. First and second isomorphism theorems for R-modules. Free R-modules. Hom. Tensor product of vector spaces, Abelian groups, and R-modules. Right-exactness of the tensor product. Restriction and extension of scalars. Complexification. Bilinear forms. Symmetric and alternating forms. Symmetric and exterior algebras. Structure Theorem for finitely generated modules of a PID. Rational canonical form. Jordan canonical form. Chain conditions. Noetherian rings. Hilbert's Basis Theorem. Prime and maximal ideals. Field of fractions. Localization of rings and modules. Exactness of localization. Local rings. Nakayama's Lemma. Integral extensions. Noether's Normalization Lemma. Integral closure. Nullstellensatz. Closed affine algebraic sets.

III. Field Theory and Galois Theory

Algebraic extensions. Finite extensions. Degree. Minimal polynomial. Adjoining roots of polynomials. Splitting field. Algebraic closure. Separable extensions. Theorem of the primitive element. Galois extensions. Fundamental Theorem of Galois Theory. Finite fields and their Galois groups. Frobenius endomorphism. Cyclotomic polynomial. Cyclotomic fields and their Galois groups. Cyclic extensions. Lagrange resolvents. Solvable extensions. Solving polynomial equations in radicals. Norm and trace. Transcendence degree.

### MATH 623: Real Analysis I

Prerequisites:

Working knowledge of undergraduate Analysis (with rigorous proofs) as well as basics of metric spaces and linear algebra as for example taught in classes like M523H and M524 at UMass Amherst.

Text:

Real Analysis - Measure Theory, Integration and Hilbert Spaces by Elias M. Stein and Rami Shakarchi

Princeton Lectures in Analysis, Vol. III (2005)    Princeton University Press (required)

Note:  Some early editions of the book have an Erratum for Theorem 4.2 Chapter 1: which can be found here as well as other errata that can be found here.

For some of the material and homework problems (optional). Depending on the printing, you can find errata here (first 5 printings) and here (printings 6th and later).

Description:

This is the first part of a 2-semester introduction to Real Analysis: Math 623 in the Fall, and in the Spring Math 624 which covers part of Vol. IV of Stein & Shakarchi also.  In the Fall semester we will cover the following material from Stein-Shakarchi's Vol III:

1) Measure theory: Lebesgue measure and integrable functions on Euclidean spaces  (Chapter 1)

2) Integration theory: Lebesgue integral, convergence theorems and Fubini theorem (Chapter 2)

3) Differentiation and Integration. Functions of bounded variation (Chapter 3)

4) Abstract measure theory over more general spaces (first part of Chapter 6)

The topics covered in Math 623 lay at the foundation not just Analysis but also of many other areas of mathematics and are essential to all mathematicians.

### MATH 645: ODE and Dynamical Systems

Prerequisites:

Advanced Calculus, Linear Algebra, Elementary Differential Equations (one semester at the undergraduate level)

Text:

The course discusses stability in nonlinear systems of differential equations, bifurcation theory, chaos, strange attractors, iteration of nonlinear mappings and fractals,
along with their applications in science and engineering. Some topics from, Clark Robinson - "An Introduction to Dynamical Systems: Continuous and Discrete" - are also covered.
I will provide all the slides for these advanced topics.

textbook: 'Dynamical Systems with Applications Using Python' by Stephen Lynch

Description:

Classical theory of ordinary differential equations and some of its modern developments in dynamical systems theory. Linear systems and exponential matrix solutions. Well-posedness for nonlinear systems. Qualitative theory: limit sets, invariant set and manifolds. Stability theory: linearization about an equilibrium, Lyapunov functions. Autonomous two-dimensional systems and other special systems.

### MATH 651: Numerical Analysis I

Prerequisites:

Knowledge of Math 523 and 235 (or 545) or permission of the instructor

Description:

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.

### MATH 671: Topology I

Prerequisites:

Strong performance in Math 300, 411, and 523, or equivalent classes.

Text:

John M. Lee, "Introduction to Topological Manifolds", second edition.

Description:

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.

### MATH 691T: S-Teachng In Univ C

Instructor TBA M 4:00-5:15

Prerequisites:

Open to Graduate Teaching Assistants in Math and Statistics

Description:

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.

### MATH 691Y: Applied Math Project Seminar

Prerequisites:

Graduate Student in Applied Math MS Program

Description:

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.

### MATH 697MA: Mathematical Theory of Machine Learning

Prerequisites:

There is no prerequisite, although some background on probability and statistics (STAT 515), optimization, and approximation theory will be helpful.

Description:

Data science and machine learning (deep learning in particular) have become a burgeoning domain with a great number of successes in science and technology. Most of the recently developed deep learning techniques are still at the “engineering” level based on trial and error. A complete theory of deep learning is still under development.

The purpose of this course is to introduce the theoretical foundation of data science with an emphasis on the mathematical understanding of machine learning. The course is divided into two semesters:

• In the first semester, we will introduce the basic set up of statistical learning, optimization, classical learning methods such as support vector machines, kernel methods, dimensionality reduction, as well as advanced learning theories, providing a mathematical foundation for the study of neural networks in the following semester.

• The second semester will contain two parts. The first half of the course introduces some useful fundamental tools from probability and statistics, and more extensively the theory of neural networks including their approximation power and generalization properties. The second half is a seminar part of the course, covering selected advanced topics on optimization and generative modeling.

The expected outcome of this course is to prepare students with solid mathematical background of modern machine learning, and to get students engaged with new research topics in this area. This course complements some earlier courses on machine learning and data sciences, such as MATH 697PA: ST-Math Foundtns/ProbabilistAI and STAT 697ML: ST- Stat Machine Learning.

### MATH 703: Topics in Geometry I

Prerequisites:

Solid understanding of abstract linear algebra, topology (e.g., as in Math 671) and calculus in n dimensions.

Description:

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.

### MATH 797EC: Elliptic Curves

Prerequisites:

Math 611 and 612. Basic knowledge of complex analysis will be helpful but not required.

Text:

Joseph Silverman, The Arithmetic of Elliptic Curves, 2nd Edition. Springer-Verlag, Berlin, 2009.

Description:

The subject of arithmetic geometry is about studying the solutions of diophantine equations by exploiting the geometry of the solution sets of these equations. This is a very active field of study and makes use of techniques from algebra, analysis, geometry, and topology. In this course we introduce the audience to this exciting field by focusing on the case of elliptic curves, a central object in mathematics from the time of Euler and Gauss and continues to play a key role in contemporary research. We will review and develop tools from the theory of algebraic curves, algebraic number theory, representation and cohomology theory, cumulating in the celebrated Mordell-Weil theorem that describes the structure of rational solutions of a smooth curve equation.

### MATH 797P: Stochastic Calculus

Prerequisites:

STATISTC 605

Description:

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 as time permits.

### MATH 797RT: Intro to Representation Theory

Prerequisites:

MATH 235, MATH 411, MATH 611

Text:

Introduction to Representation Theory,
by Pavel Etingof, Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex Schwendner

Description:

We are planning to cover basics of representation theory as well as key methods.
The key examples that are covered in class are finite groups, $GL_n(\mathbb{C})$, $S_n$, $GL_2$ over a finite field, and quiver algebras.
If time permits, we cover the theory of Soergel bimodules and applications to knot homology.

### MATH 797TT: Information Theory and Optimal Transport

Prerequisites:

A good working knowledge of Analysis and Probability Theory and an adventurous spirit.

Description:

This is a class on the mathematical foundations and the applications of distances and divergences between probability measures. Among the topics to be treated: Entropies, Kullback-Leibler and general f-divergences; Optimal transport and Wasserstein metrics; Maximum mean discrepancy and reproducing kernel Hilbert spaces, Stein discrepancies, and integral probability metrics. We will provide the analytical and probabilistic foundations for these objects with an emphasis on variational representations, We will discuss inequalities in information theory, and also present algorithms for the computations or statistical estimation of divergences or metrics, including generative adversarial networks and gradient flows. We illustrate the theory with examples from applied mathematics and data science, for example mutual information, generative adversarial networks, and so on.

### STAT 607.1: Mathematical Statistics I

Prerequisites:

Prerequisite: advanced calculus and linear algebra, or consent of instructor.

Description:

Probability theory, including random variables, independence, laws of large numbers, central limit theorem; statistical models; introduction to point estimation, confidence intervals, and hypothesis testing.

### STAT 607.2: Mathematical Statistics I

Prerequisites:

For graduates students: Multivariable calculus and linear algebra; For undergraduate students: permission of instructor

Text:

Statistical Inference (second edition), by George Casella and Roger L. Berger

Recommended Text:

All of Statistics: A Concise Course in Statistical Inference, by Larry Wasserman

Note:

This course may be taken remotely. Please enroll and contact the instructor if you would like to take the course remotely.

This class meets on the Newton Mount Ida Campus of UMass-Amherst.

Description:

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.

### STAT 625.1: Regression Modeling

Prerequisites:

Previous coursework in Probability and Statistics, including knowledge of estimation, confidence intervals, and hypothesis testing and its use in at least one and two sample problems; e.g., ST516 or equivalent. You must be familiar with these statistical concepts beforehand. ST515 by itself is NOT a sufficient background for this course! Familiarity with basic matrix notation and operations is helpful.

Description:

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.

### STAT 625.2: Regression Modeling

Prerequisites:

Previous coursework in Probability and Statistics, including knowledge of estimation, confidence intervals, and hypothesis testing and its use in at least one and two sample problems; e.g., ST516 or equivalent. You must be familiar with these statistical concepts beforehand. ST515 by itself is NOT a sufficient background for this course! Familiarity with basic matrix notation and operations is helpful.

Note:

This course may be taken remotely. Please enroll and contact the instructor if you would like to take the course remotely.

This class meets on the Newton Mount Ida Campus of UMass-Amherst.

Description:

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.

### STAT 691P: S - Project Seminar

Prerequisites:

Permission of instructor.

Note:

This course may be taken remotely. Please enroll and contact the instructor if you would like to take the course remotely.

This class meets on the Newton Mount Ida Campus of UMass-Amherst.

Description:

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.

### STAT 697BD: Biomed and Health Data Analysis

Prerequisites:

Knowledge of calculus and linear algebra are required.
Knowledge of statistics (Stat 516, Stat 608, or equivalent) preferred, but not required.
Knowledge of regression (Stat 525, Stat 625, Stat 697R, or equivalent) preferred, but not required.

Advanced undergraduate students may request permission of instructor to enroll.

Description:

In this course, we will apply several novel machine learning algorithms, including normalization methods, classification and regression analysis on cancer patient data sets to arrive at personalized cancer treatments. We will develop several algorithms for analyzing cancer data sets, including gene expression data sets. We will review, develop, and evaluate some computational biology methods. We will implement most of these methods in Python. Although programming skills, machine learning, or computational biology background are preferred, they are not required for this course. Importantly, this is a research based course; it is an introduction on how to do research in computational biology. We all work as a team to learn cutting-edge methods in computational biology and hopefully find ways to improve them. We will read some recently published papers and implement methods that have been introduced in these papers. Except the first few lectures, a team of students will present the papers and their implementation of methods. Students should be interested in Python programming, computational biology, and doing research as a team member. There is no exam. Students will be evaluated based on their participation, presentations, and works, including their codes and HWs.

### STAT 697L: Categorical Data Analysis

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.

Note:

This course may be taken remotely. Please enroll and contact the instructor if you would like to take the course remotely.

This class meets on the Newton Mount Ida Campus of UMass-Amherst.

Description:

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.

### STAT 697SC: Statistical Consulting: Bringing theory to practice

Prerequisites:

STAT 525, or STAT 625 or similar regression class or permission of instructor

Description:

This class focuses on skills statisticians need to bring their classroom knowledge to practical problems. Half of the class will meet concurrently with the Statistical Consulting Practicum, and be focused around real consulting problems brought in by clients. The other half will explore in greater depths methods, techniques, and approaches useful in practical problems but not usually covered in the standard curriculum. Specific topics covered may vary slightly based on the mix of clients this semester, but may include methods like power analysis, missing data, practical mixed models, and practical model selection and interpretation, as well as soft skills like maintaining professional relationships, asking questions, listening, and communicating with non-statisticians. Wherever possible, content will be motivated by specific current or past consulting projects.

### STAT 725: Estmtn Th and Hypo Tst I

Prerequisites:

Stat 607-608

Recommended Text:

Elements of Large-Sample Theory (by Erich Lehmann),
Theory of Point Estimation (by Erich Lehmann and George Casella),
A Course in Large Sample Theory (by Thomas S. Ferguson),
Asymptotic Statistics (by A. W. van der Vaart)

Description:

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