# Course Descriptions

## Lower Division Courses

### MATH 011: Elementary Algebra

See Preregistration guide for instructors and times

Description:

Beginning algebra enhanced with pre-algebra topics such as arithmetic, fractions, and word problems as need indicates. Topics include real numbers, linear equations and inequalities in one variable, polynomials, factoring, algebraic fractions, problem solving, systems of linear equations, rational and irrational numbers, and quadratic equations.

This course is only offered online through Continuing and Professional Education.

### 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 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 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 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 128H: Honors Calculus for Life and Social Sciences II

See Preregistration guide for instructors and times.

Prerequisites:

Math 127

Description:

Honors section of Math 128.

### 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 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.

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

See Preregistration Guide for instructors and times

Prerequisites:

Math 132 or 136; corequisite: Math 233

Text:

TBA

Description:

(Formerly Math 431) Introduction to ordinary differential equations.First and second order linear differential equations, systems of linear differential equations, Laplace transform, numerical methods, applications. (This course is considered upper division with respect to the requirements for the major and minor in mathematics.)

### 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

Text:

Note:

The lab meeting times for this section of Math 300 are:

Math 300.01LL Lab Mon 4:00 pm - 4:50 pm
Math 300.01LM Lab Mon 5:30 pm - 6:20 pm
Math 300.01LN Lab Mon 6:30 pm - 7:20 pm

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), 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.

### MATH 300.2: Fundamental Concepts of Mathematics

Alejandro Morales MWF 10:10-11:00

Prerequisites:

Math 132 with a grade of 'C' or better

Note:

The lab meeting times for this section of Math 300 are:

Math 300.02LL Lab Tue 4:00 pm - 4:50 pm
Math 300.02LM Lab Tue 5:30 pm - 6:20 pm
Math 300.02LN Lab Tue 6:30 pm - 7:20 pm

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), 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.

### MATH 300.3: Fundamental Concepts of Mathematics

Luca Schaffler TuTh 11:30-12:45

Prerequisites:

Math 132 with a grade of C or better

Note:

The lab meeting times for this section of Math 300 are:

Math 300.03LL Lab Wed 4:00 pm - 4:50 pm
Math 300.03LM Lab Wed 5:30 pm - 6:20 pm
Math 300.03LN Lab Wed 6:30 pm - 7:20 pm

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), 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.

### MATH 370.1: Writing in Mathematics

Jacob Matherne MWF 1:25-2:15

Prerequisites:

MATH 300 or CS 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. Some goals of the course are to learn how to write documents effectively in LaTeX, learn how to prepare and give good presentations, learn about possible career paths in mathematics, and to attend and listen actively during math talks. Prerequisites: MATH 300 and completion of College Writing (CW) requirement.

### MATH 370.2: Writing in Mathematics

Jacob Matherne MWF 12:20-1:10

Prerequisites:

MATH 300 or CS 250 and completion of 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. Some goals of the course are to learn how to write documents effectively in LaTeX, learn how to prepare and give good presentations, learn about possible career paths in mathematics, and to attend and listen actively during math talks. Prerequisites: MATH 300 and completion of College Writing (CW) requirement.

### MATH 370.3: Writing in Mathematics

Patrick Dragon MWF 9:05-9:55

Prerequisites:

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

Text:

none

Note:

This course satisfies the Junior Year Writing requirement.

Description:

This course will introduce students to technical writing in mathematics. Writing assignments will include proofs, instructional handouts, resumes, cover letters, and a final paper. All assignments will be completed using LaTeX. By the end of the semester, students should be able to clearly convey mathematical ideas through their writing, as geared to a particular audience.

### MATH 370.4: Writing in Mathematics

Patrick Dragon MWF 10:10-11:00

Prerequisites:

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

Text:

none

Note:

This course satisfies the Junior Year Writing requirement.

Description:

This course will introduce students to technical writing in mathematics. Writing assignments will include proofs, instructional handouts, resumes, cover letters, and a final paper. All assignments will be completed using LaTeX. By the end of the semester, students should be able to clearly convey mathematical ideas through their writing, as geared to a particular audience.

### MATH 391T: Seminar- Introduction to K-12 Mathematics Teaching

Michael Hayes Wed 2:40-4:40

Prerequisites:

Open to junior and senior Math majors. MATH 300 and completion of a 400-level or higher Math or Statistics course required. Completion of two or more 400 level or higher Math or Statistics courses strongly recommended.

Description:

This course provides future secondary math teachers an introduction to a range of topics related to the teaching of mathematics in the public schools. The focus will be on increasing the participants' mathematical content knowledge for teaching by exploring the mathematical content and practices of secondary math. Through these explorations, students will have opportunities to gain some familiarity with the Massachusetts Frameworks, Common Core, standardized and local assessments, curriculum resources, and other topics related to secondary math teaching. As part of the course, students will also explore connections between secondary math and the higher-level mathematics courses they have been taking.

### MATH 411.1: Introduction to Abstract Algebra I

Mohammed Zuhair Mullath MWF 10:10-11:00

Prerequisites:

Math 235; Math 300 or CS 250

Text:

Dan Saracino, "Abstract Algebra: A First Course", second edition, Waveland Press.

Description:

Introduction to abstract algebra, focusing primarily on the theory of groups. There will be regular problem assignments and quizzes, as well as midterm and final examinations.

### MATH 411.2: Introduction to Abstract Algebra I

Mohammed Zuhair Mullath MWF 11:15-12:05

Prerequisites:

Math 235; Math 300 or CS 250

Text:

Dan Saracino, "Abstract Algebra: A First Course", second edition, Waveland Press.

Description:

Introduction to abstract algebra, focusing primarily on the theory of groups. There will be regular problem assignments and quizzes, as well as midterm and final examinations.

### MATH 411.3: Introduction to Abstract Algebra I

Paul Gunnells TuTh 10:00-11:15

Prerequisites:

Math 235; Math 300 or CS 250

Description:

The description will be posted soon.

### MATH 412: Introduction to Abstract Algebra II

Ivan Mirkovic MW 2:30-3:45

Prerequisites:

Math 411

Text:

Dan Saracino, "Abstract Algebra: A First Course", Second Edition, Waveland Press.

Description:

This course is a continuation of Math 411. We will study properties of rings and fields. A ring is an algebraic system with two operations (addition and multiplication) satisfying various axioms. Main examples are the ring of integers and the ring of polynomials in one variable. Later in the course we will apply some of the results of ring theory to construct and study fields. At the end we will outline the main results of Galois theory which relates properties of algebraic equations to properties of certain finite groups called Galois groups. For example, we will see that a general equation of degree 5 can not be solved in radicals.

### MATH 421: Complex Variables

William Meeks MWF 9:05-9:55

Prerequisites:

Math 233

Text:

Complex Analysis: A First Course with Applications (Third Edition) by Dennis G. Zill and Patrick D. Shanahan

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

### MATH 425.1: Advanced Multivariate Calculus

Rob Kusner TuTh 11:30-12:45

Prerequisites:

Multivariable calculus (Math 233) and Linear algebra (Math 235)

Text:

J. Marsden and A. Tromba, Vector Calculus [an earlier edition is fine and may be less expensive: W. H. Freeman; Fifth Edition edition (August 1, 2003) ISBN-10: 0716749920; ISBN-13:
978-0716749929]

H. M. Schey, div, grad, curl and all that [ISBN-13: 978-0393925166; ISBN-10: 0393925161]

M. Spivak, Calculus on Manifolds [ISBN-13: 978-0805390216; ISBN-10: 0805390219]

Description:

This course covers the basics of differential and integral calculus in many variables: differentiability, directional and partial derivatives and gradient of functions; critical points without or with constraints (Lagrange-multipliers/tangential-gradient) and the Hessian; vector fields and differential forms; divergence, curl and exterior derivative; line- and surface-integrals; the fundamental theorem of calculus (Gauss/Green/Stokes/Thomson). If time and taste permit, topics from physics (fluids and electromagnetism) and differential geometry (curves and surfaces in space) may also be explored.

### MATH 425.2: Advanced Multivariate Calculus

Dinakar Muthiah TuTh 1:00-2:15

Prerequisites:

Multivariable calculus (Math 233) and Linear algebra (Math 235)

Description:

In this course we will study multivariate differential and integral calculus. We will dive deeper into the material from Math 233 and also cover new material on vector calculus.

The first part of the course will cover with differential calculus of functions of several variables. Then we will study multivariate integration and integration over paths and surfaces. We will conclude with the capstone of this course: vector calculus. Our goal will be to understand Green's theorem, Gauss' theorem, and Stokes' theorems. Some motivation for vector calculus will come from physics, for example Maxwell's equations that describe electromagnetic waves.

Mostly we will focus on functions of three variables but will also sometimes discuss how the theory works with any number of variables.

### MATH 425.3: Advanced Multivariate Calculus

Dinakar Muthiah TuTh 2:30-3:45

Prerequisites:

Multivariable calculus (Math 233) and Linear algebra (Math 235)

Description:

In this course we will study multivariate differential and integral calculus. We will dive deeper into the material from Math 233 and also cover new material on vector calculus.

The first part of the course will cover with differential calculus of functions of several variables. Then we will study multivariate integration and integration over paths and surfaces. We will conclude with the capstone of this course: vector calculus. Our goal will be to understand Green's theorem, Gauss' theorem, and Stokes' theorems. Some motivation for vector calculus will come from physics, for example Maxwell's equations that describe electromagnetic waves.

Mostly we will focus on functions of three variables but will also sometimes discuss how the theory works with any number of variables.

### MATH 455.1: Introduction to Discrete Structures

Annie Raymond TuTh 10:00-11:15

Prerequisites:

Calculus (MATH 131, 132, 233), Linear Algebra (MATH 235), and Math 300 or CS 250. For students who have not taken Math 300 or CS 250, the instructor may permit students with sufficient experience in reading and writing mathematical arguments to enroll.

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, colorings and matchings; the pigeonhole principle, induction and recursion, generating functions, and (if time permits) combinatorial geometry. 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 modeling problem 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 455.2: Introduction to Discrete Structures

Paul Gunnells TuTh 8:30-9:45

Prerequisites:

Calculus (MATH 131, 132, 233), Linear Algebra (MATH 235), and Math 300 or CS 250. For students who have not taken Math 300 or CS 250, the instructor may permit students with sufficient experience in reading and writing mathematical arguments to enroll.

Text:

Harris, Hirst, and Mossinghoff, Combinatorics and Graph Theory, 2nd edition

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, colorings and matchings; the pigeonhole principle, induction and recursion, enumeration, and generating functions. As part of the course, student groups will be assigned and a final project will be presented. This course satisfies the university's Integrative Experience (IE) requirement for math majors.

### MATH 456.1: Mathematical Modeling

Nestor Guillen TuTh 10:00-11:15

Prerequisites:

Math 233 and Math 235 (and Differential Equations, Math 331, is recommended). Some familiarity with a programming language is desirable (Mathematica, Matlab, C++, Python, etc.)

Text:

Richard Durrett. Essentials of Stochastic Processes. 2nd print.

Description:

In this course we will study some of the mathematics used in modern modeling, be it continuous deterministic systems (modeled by differential equations) or discrete stochastic systems (modeled by Markov chains). A first part of the course will involve learning some of the theory in the aforementioned topics, and the second part will involve case studies on real life situations where these models are put to practice. Later on the course, students will team up in small groups (2-3) and do presentations on topics selected from a list made available at the beginning of the semester .

This course satisfies the University's Integrative Experience (IE) requirement for math majors.

### MATH 456.2: Mathematical Modeling

Nestor Guillen TuTh 8:30-9:45

Prerequisites:

Math 233 and Math 235 (and Differential Equations, Math 331, is recommended). Some familiarity with a programming language is desirable (Mathematica, Matlab, C++, Python, etc.)

Description:

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.

### MATH 471: Theory of Numbers

Anna Puskas MWF 1:25-2:15

Prerequisites:

Math 233 and Math 235. Math 300 or CS250 as a co-requisite is not absolutely necessary but highly recommended.

Description:

The course will cover topics from elementary number theory such as the Euclidean Algorithm, congruence arithmetic, prime numbers, the Fundamental Theorem of Arithmetic, quadratic reciprocity, and continued fractions. Applications and connections to cryptography will be explored. To help learn these materials, students will be assigned computational projects using computer algebra software.

### MATH 475: History of Mathematics

Jenia Tevelev TuTh 2:30-3:45

Prerequisites:

Math 233 and Math 300 or CS 250.

Text:

"A History of Mathematics" by V. Katz

Note:

This course satisfies the General Education IE requirement for Mathematics and Statistics.

Description:

This is an introduction to the history of mathematics from ancient civilizations to present day. Students will study major mathematical discoveries in their cultural, historical, and scientific contexts. This course explores how the study of mathematics evolved through time, and the ways of thinking of mathematicians of different eras - their breakthroughs and failures. Students will have an opportunity to integrate their knowledge of mathematical theories with material covered in General Education courses. Forms of evaluation will include a group presentation, class discussions, and a final paper.
The math major is made up of technical courses on the theory of mathematics from Calculus to more complex concepts. This course is unique in providing a humanities-based approach to understanding math. For example, students are required to use primary sources on a weekly basis. Students study examples of how mathematical advances were made in response to or alongside developments in other branches of science - such as Ptolemy’s work in trigonometry being motivated by applications in astronomy, and Newton as the father of both calculus and modern physics. Students also learn to understand mathematicians as people of their times - for example, how Babylonian mathematicians were motivated by the needs of the empire, or how Evariste Galois was both a brilliant mathematician and a passionate French revolutionary. Additionally, many math majors go on to teach mathematics after graduation, and in this course the history of math is is studied in the context of the history of education.
The homework assignments contain a component where students are required to write short compositions. Many of these assignments will ask students to engage in self-reflection on how their study of the history of mathematics in the current course is influenced by the General Education courses they took. This is not limited to courses carrying the Historical Studies designation, but Economics, Sociology, Science and other modes of human thought all present lenses through which one can study the mathematics and mathematicians that are the focus of the course; the homework assignments will invite students to apply all of those lenses to the topics at hand. Students will also be required to write an interdisciplinary 20-page final essay. Essay topics are developed by the student with assistance from the professor. Students draw not only on their mathematics coursework, but also on the knowledge they have accrued from other GenEd curriculum courses they have experienced in their first two years of college. After seeing how mathematical tools have developed in conversation with history, culture, and science, students can better appreciate the uses and possibilities of advanced mathematics.
A substantial fraction of the course grade is based on a class presentation; this is a major change from standard upper division math/stat requirements. Each group, which typically consists of three students, collaboratively researches and presents an interdisciplinary mathematical topic, chosen by the group from the instructor’s list of suggested topics. This is highly unusual for mathematics classes, where problems are typically presented abstracted from their scientific and cultural roots. These presentations are spaced out throughout the semester, and the students’ work is referenced and discussed in a class discussion later in class.
Additionally, highly unusually for a mathematics course, a large part of the course grade is based on participation in class discussions. Each week, the students are assigned readings. A large amount of weekly class time is devoted to a roundtable discussion of the reading - its implications for modern mathematics, how it was understood at the time, mathematical concepts in the reading, and the close-reading of assigned passages.

### MATH 523H: Int. Mod. Analysis I

Sohrab Shahshahani MW 8:40-9:55

Prerequisites:

Math 233, Math 235, and Math 300 or CS 250

Text:

Elementary Analysis: The Theory of Calculus, 2nd edition, Kenneth Ross, Springer-Verlag, 2013

Description:

This is the first part of the introduction to analysis sequence (523 and 524). This course deals with basic concepts of analysis of functions mostly on the real line, and we will try to make many of the concepts one learns in calculus rigorous. Covered topics will include series, and sequences, continuity, differentiability, and integration. Prerequisites for the class are Math 233, Math 235, and Math 300 or CS 250.

### MATH 524: Introduction to Modern Analysis II

Taryn Flock TuTh 1:00-2:15

Prerequisites:

MATH 523H

Description:

Topology of the euclidean space and functions of several variables (implicit function theorem), introduction to Fourier analysis, metric spaces and normed spaces. Applications to differential equations, calculus of variations, and others.

### MATH 534H: Introduction to Partial Differential Equations

Andrea Nahmod TuTh 11:30-12:45

Prerequisites:

Math 233, 235, and 331.
Complex variables (M421) and Introduction to Real Analysis (M523H) are definitely a plus, and helpful, but not absolutely necessary.

Text:

Main Text:
Partial Differential Equations: An Introduction, by Walter Strauss, Wiley, Second Edition.

Optional reference text:
Partial Differential Equations in Action. From modeling to theory; by Sandro Salsa. Springer, Third Edition.

Description:

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

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

### MATH 536: Actuarial Probability

Jinguo Lian MWF 1:25-2:15

Prerequisites:

Math 233 and Stat 515

Text:

ASM Study Manual for Exam P, 1st edition or later edition Author / Publisher: Weishaus / ASM ISBN: 978-1-63588-097-7

Note:

TI BA II Plus calculator is recommended.

Description:

Math 536 serves as a preparation for the first SOA/CAS actuarial exam on the fundamental probability tools for quantitatively assessing risk, known as Exam P (SOA) or Exam 1 (CAS). The course covers general probability, random variables with univariate probability distributions (including binomial, negative binomial, geometric, hypergeometric, Poisson, uniform, exponential, gamma, and normal), random variables with multivariate probability distributions (including the bivariate normal), basic knowledge of insurance and risk management, and other topics specified by the SOA/CAS exam syllabus.

### MATH 537: Intro. to Math of Finance

Eric Sommers TuTh 11:30-12:45

Prerequisites:

Math 233 and Stat 515

Text:

Unpublished e-textbook freely available first week of class.

Recommended Text:

Derivative Markets by Robert L. McDonald, 3rd edition

Note:

A calculator with a cumulative distribution function for the standard normal variable (also known as the Erf" function or normalcdf"). The inverse normalcdf feature is also necessary. The TI-83 or higher, for example, will work.

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

Qian-Yong Chen MWF 10:10-11:00

Prerequisites:

Math 233, Math 235, Math 300

Text:

Linear Algebra and Its Applications, 4th ed., By Gilbert Strang. 2006, Cengage. ISBN-10: 0030105676 ISBN-13: 9780030105678.

Recommended Text:

Elementary Linear Algebra, by Ken Kuttler;
Linear Algebra, by Cherney, Denton and Waldron.

Description:

This is a course in Advanced Linear Algebra and Applications. We will cover LU decom-
position, Vector and Inner Product Spaces, Orthogonality and Least Squares, Determinants and
Eigenvalues, Jordan form, Spectral theorem, symmetric positive definite matrices. Other decom-
positions such as SVD and QR, will be covered as well. If time permitting. basic numerical linear
algebra will be included. There will be elements of proof and computation in the course. No coding
will be taught in the class, but the students will have the option to do a final project instead of the
exam. Homework will be assigned every one or two weeks. Late homework will NOT be accepted.
The final grade will be based on both exams and homework assignments.

### MATH 545.2: Linear Algebra for Applied Mathematics

Robin Young MWF 11:15-12:05

Prerequisites:

Math 233, Math 235, Math 300

Text:

Strang, Linear Algebra and Its Applications,4th ed. I recommend the Indian edition as it is 1/10th of the price of the US edition. Alternative editions are fine because I rarely assign problems directly from the book.

Recommended Text:

There are many alternative texts available online. I will make some available as we go.

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

Yao Li MW 2:30-3:45

Prerequisites:

MATH 233 and 235 as well as knowledge of a scientific programming language, e.g. Matlab, Fortran, C, C++, Python, Java.

Text:

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

Important note about the textbook: UMass has an Institutional SIAM Membership. Thus, a free e-book is available through the UMass library. As a side note, make sure that you are connected to the UMass Amherst Libraries network (if not, you can still have off-campus access by clicking "Off-campus login" and entering your SPIRE ID and password). Of course, a physical copy of the book can be purchased directly from the publisher, SIAM at www.siam.org

Description:

The course will introduce basic numerical methods used for solving problems that arise in different scientific fields. Properties such as accuracy of methods, their stability and efficiency will be studied. Students will gain practical programming experience in implementing the methods. We will cover the following topics (not necessarily in the order listed): Finite Precision Arithmetic and Error Propagation, Linear Systems of Equations, Root Finding, Interpolation, Least squares, Numerical Integration, Numerical Solution of Ordinary Differential Equations.

### MATH 551.2: Intr. Scientific Computing

Mike Sullivan TuTh 2:30-3:45

Prerequisites:

MATH 233 and 235 as well as knowledge of a scientific programming language, e.g. Matlab, Fortran, C, C++, Python, Java.

Text:

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

Important note about the textbook: UMass has an Institutional SIAM Membership. Thus, a free e-book is available through the UMass library. As a side note, make sure that you are connected to the UMass Amherst Libraries network (if not, you can still have off-campus access by clicking "Off-campus login" and entering your SPIRE ID and password). Of course, a physical copy of the book can be purchased directly from the publisher, SIAM at www.siam.org

Description:

The course will introduce basic numerical methods used for solving problems that arise in different scientific fields. Properties such as accuracy of methods, their stability and efficiency will be studied. Students will gain practical programming experience in implementing the methods. We will cover the following topics (not necessarily in the order listed): Finite Precision Arithmetic and Error Propagation, Linear Systems of Equations, Root Finding, Interpolation, least squares, Numerical Integration, Numerical Solution of Ordinary Differential Equations.

### MATH 551.3: Intr. Scientific Computing

Stathis Charalampidis TuTh 11:30-12:45

Prerequisites:

MATH 233 and 235 as well as knowledge of a scientific programming language, e.g. Matlab, Fortran, C, C++, Python, Java.

Text:

A First Course in Numerical Methods, by Ascher & Greif (SIAM)

Description:

This course will introduce basic numerical methods used for solving problems that arise in different scientific fields. The following topics (not necessarily in the order listed) will be covered: finite precision arithmetic and error propagation, linear systems of equations, root finding, interpolation, least squares and numerical integration. Students will gain practical programming experience in implementing such numerical methods using MATLAB. The use of MATLAB for homework assignments will be mandatory, although any other scientific language for solving the homework problems will be accepted. We will also discuss some very important practical considerations of implementing numerical methods using other languages such as fortran, C or C++.

### MATH 552: Applications of Scientific Computing

Stathis Charalampidis TuTh 1:00-2:15

Prerequisites:

Math 551 or equivalent or permission of instructor

Text:

[1] Finite difference methods for ordinary and partial differential equations, by R. J. LeVeque (SIAM).

Recommended Text:

[1] A First Course in Numerical Methods, by U. M. Ascher and C. Greif (SIAM).
[2] A Multigrid tutorial, by W. L. Briggs, V. E. Henson and S. F. McCormick (SIAM).

Description:

This course complements the topics covered in MATH 551. In particular, the following topics (not necessarily in
the order listed) will be covered: finite difference schemes for steady-state boundary value problems, numerical
methods for time-dependent ordinary and partial differential equations, numerical methods for computing eigenvalues,
eigenvectors and singular values as well as fast poisson solvers and the fast Fourier transform (FFT). If time permits,
we will discuss iterative methods for linear systems (including modern methods such as the multigrid method) and parametric
continuation techniques (pseudo-arclength and deflation methods). The use of MATLAB for homework assignments will
be mandatory, although any other scientific language for solving the homework problems will be accepted.

### MATH 563H: Differential Geometry

Weimin Chen MWF 11:15-12:05

Prerequisites:

Very good understanding of Advanced Multivariable Calculus and Linear Algebra (Math 425, 233 and 235).

Text:

Elementary Differential Geometry by Christian B\"{a}r, Cambridge University Press, 2010.

Description:

The course covers standard materials in the theory of curves in the plane and the space, and the theory of surfaces in the space.
Emphasis is given on the interplay between the local numerical quantities such as curvature and the global properties of the geometric objects,
which is a central theme in modern differential geometry.

### STAT 494CI: Cross-Disciplinary Research

John Staudenmayer MWF 9:05-9:55

Prerequisites:

Instructor Consent Required. Note: Previous coursework in probability and statistics, including regression, is required.

Description:

Students will work in teams to collaborate with researchers in other disciplines. Each research project will have a team of two students, one faculty statistician, and one researcher from another discipline. Students will be assigned to teams according to their skills and interests. Each team will work together for one semester and be responsible for its own schedule, work plan, and final report. In addition, the whole class will meet weekly for teams to update each other on their progress and problems. Students will learn about several areas of application and the statistical methods employed by each team. Students in the course will probably learn new statistical methods, a discipline where statistics is applied, how to work collaboratively, how to use R, and how to present oral and written reports.

### STAT 501: Methods of Applied Statistics

Joanna Jeneralczuk TuTh 11:30-12:45

Prerequisites:

Knowledge of high school algebra, junior standing or higher

Text:

TBA

Description:

A non-calculus-based applied statistics course for graduate students and upper level undergraduates with no previous background in statistics who will need statistics in their future studies and their work. The focus is on understanding and using statistical methods in research and applications. Topics include: descriptive statistics, probability theory, random variables, random sampling, estimation and hypothesis testing, basic concepts in the design of experiments and analysis of variance, linear regression, and contingency tables. The course has a large data-analytic component using a statistical computing package.

### STAT 515.1: Statistics I

Eric Hall MWF 9:05-9:55

Prerequisites:

Math 131-132 with a grade of “C” or better in Math 132. Knowledge of multivariable calculus is very useful but necessary concepts will be introduced.

Text:

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

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, the central limit theorem, and sampling distributions. Introduction to basic concepts of estimation (bias, standard error, etc.) and confidence intervals.

### STAT 515.2: Statistics I

Eric Hall MWF 10:10-11:00

Prerequisites:

Math 131-132 with a grade of “C” or better in Math 132. Knowledge of multivariable calculus is very useful but necessary concepts will be introduced.

Text:

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

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, the central limit theorem, and sampling distributions. Introduction to basic concepts of estimation (bias, standard error, etc.) and confidence intervals.

### STAT 515.3: Statistics I

HongKun Zhang TuTh 2:30-3:45

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 required for this course. However, some necessary concepts for multiple integration or partial derivatives will be re-introduced in the course as needed.

Text:

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

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, the central limit theorem, and sampling distributions. Introduction to basic concepts of estimation (bias, standard error, etc.) and confidence intervals.

### STAT 515.4: Statistics I

Markos Katsoulakis TuTh 1:00-2:15

Prerequisites:

Math 131-132 with a grade of “C” or better in Math 132. Knowledge of multivariable calculus is very useful but necessary concepts will be introduced.

Text:

Mathematical Statistics with Applications(7th Edition), D. D. Wackerly, W. Mendenhall and R. L. Schaeffer

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, the central limit theorem, and sampling distributions. Introduction to basic concepts of estimation (bias, standard error, etc.) and confidence intervals.

### STAT 516.1: Statistics II

Vincent Lyzinski TuTh 8:30-9:45

Prerequisites:

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

Text:

Mathematical Statistics with Applications, 7th Edition, by Wackerly, Mendenhall and Scheaffer

Description:

Continuation of Stat 515. Overall objective of the course is the development of basic theory and methods for statistical inference. Topics include: Sampling distributions; General techniques for statistical inference (point estimation, confidence intervals, tests of hypotheses); Development of methods for inferences on one or more means (one-sample, two-sample, many samples - one-way analysis of variance), inference on proportions (including contingency tables), simple linear regression and non-parametric methods (time permitting).

### STAT 516.2: Statistics II

Vincent Lyzinski TuTh 10:00-11:15

Prerequisites:

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

Text:

Mathematical Statistics with Applications, 7th Edition, by Wackerly, Mendenhall and Scheaffer

Description:

Continuation of Stat 515. Overall objective of the course is the development of basic theory and methods for statistical inference. Topics include: Sampling distributions; General techniques for statistical inference (point estimation, confidence intervals, tests of hypotheses); Development of methods for inferences on one or more means (one-sample, two-sample, many samples - one-way analysis of variance), inference on proportions (including contingency tables), simple linear regression and non-parametric methods (time permitting).

### STAT 516.3: Statistics II

Erin Conlon MWF 1:25-2:15

Prerequisites:

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

Text:

Mathematical Statistics with Applications, 7th Edition, by Wackerly, Mendenhall and Scheaffer

Description:

Continuation of Stat 515. Overall objective of the course is the development of basic theory and methods for statistical inference. Topics include: Sampling distributions; General techniques for statistical inference (point estimation, confidence intervals, tests of hypotheses); Development of methods for inferences on one or more means (one-sample, two-sample, many samples - one-way analysis of variance), inference on proportions (including contingency tables), simple linear regression and non-parametric methods (time permitting).

### STAT 525.1: Regression Analysis

Michael Lavine MWF 1:25-2:15

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.

Text:

Applied Linear Regression, 4th Edition, by S Weisberg

Description:

Regression analysis answers questions about the dependence of a response variable on one or more predictors, including prediction of future values of a response, discovering which predictors are important, and estimating the impact of changing a predictor or a treatment on the value of the response. This course focuses on linear regression, which is the basis for many modern, advanced regression techniques, including those used by statisticians, machine learners, and data scientists.

In addition to the usual topics in linear regression, this course also emphasizes (1) graphical methods, because it’s important to visualize your data and not just rely on numerical output from computer packages, and (2) diagnostics, because it’s important to check that any regression analysis accurately represents your data. The course will teach and use the R statistical language.

### STAT 525.2: Regression Analysis

Anna Liu MWF 10:10-11:00

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.

Text:

Applied Linear Regression Models, by Kutner, Nachshem and Neter (4th edition) or Applied Linear Statistical Models by Kutner, Nachtshem, Neter and Li (5th edition). Both published by McGraw-Hill/Irwin. The first 14 chapters of Applied Linear Statistical Models (ALSM) are EXACTLY equivalent to the 14 chapters that make up Applied Linear Regression Models, 4th ed., with the same pagination. The second half of ALSM covers experimental design and the analysis of variance and is used in our ST526 course. (A) Students planning to take STAT 526 should buy the ALSM textbook.

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 526: Design of Experiments

Erin Conlon MWF 11:15-12:05

Prerequisites:

Stat 516 (previous coursework in statistics including knowledge of estimation, hypothesis testing and confidence intervals).

Text:

Applied Linear Statistical Models by Kutner et al. (5th. ed.)

Description:

An applied statistics course on planning, statistical analysis and interpretation of experiments of various types. Coverage includes factorial designs, randomized blocks, incomplete block designs, nested and crossover designs. Computer analysis of data using the statistical package SAS (no prior SAS experience assumed).

### STAT 535: Statistical Computing

Patrick Flaherty MWF 12:20-1:10

Prerequisites:

Prerequisite: Stat 516
Corequisite: Stat 525

Text:
Introduction to Computation and Programming Using Python: With Application to Understanding Data

by John V. Guttag

Recommended Text:
Python for Data Analysis

by Wes McKinney

Introduction to Algorithms

by Cormen, Leiserson, Rivest, and Stein

Description:

The 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 597TS: ST-Time Series (1 Credit)

Peng Wang Tue 4:00-5:15

Prerequisites:

Stat 525 or equivalent

Description:

This one credit undergraduate course aims to introduce basic concepts and modeling techniques for time series data. It emphasizes implementation of the modeling techniques and their practical application in analyzing actuarial and financial data. The open source program R will be used. Chapter 7, 8 and 9 of the textbook will be covered, if time allows. This course satisfies the VEE (Validation by Educational Experience) requirement set by the SOA (Society of Actuaries) in time series of the Applied Statistical Methods topic. Specifically, SOA requires the following educational experience in time series and forecasting:
- Linear time series models
- Moving average, autoregressive and/or ARIMA models
- Estimation, data analysis and forecasting with time series models
- Forecast errors and confidence intervals
This course will cover the above topics and more advanced models such as exponential smoothing, Box-Jenkins and ARCH/GARCH, if time permits.

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

Michael Lavine Thu 11:30-12:45

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 612: Algebra II

Tom Weston TuTh 10:00-11:15

Prerequisites:

Math 611 (or consent of the instructor)

Description:

This fast-paced course is a continuation of Math 611. It 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 621: Complex Analysis

Paul Hacking MWF 9:05-9:55

Prerequisites:

Advanced Calculus. Students are expected to have a working knowledge of complex numbers and functions at the level of M421 for example.

Text:

Complex analysis, by E. Stein and R. Shakarchi, Princeton 2003.

Description:

We will cover the basic theory of functions of one complex variable, at a pace that will allow for the inclusion of some non-elementary topics at the end. Basic Theory: Holomorphic functions, conformal mappings, Cauchy's Theorem and consequences, Taylor and Laurent series, singularities, residues, elliptic functions, other topics as time permits.

### MATH 624: Real Analysis II

Andrea Nahmod TuTh 2:30-3:45

Prerequisites:

Math 523H, Math 524 and Math 623.

Text:

Both of the following texts are required and will be used:

1) Real Analysis: Measure Theory, Integration, and Hilbert Spaces" (Princeton Lecture Series III) by Stein and Shakarchi.

2) Functional Analysis: Introduction to Further Topics in Analysis (Princeton Lecture Series IV), by Elias M. Stein & Rami Shakarchi

Description:

This is the second part of a 2-semester introduction to Real Analysis (namely Math 623 in the Fall, and Math 624 in the Spring) which covers parts of Vol. III and of Vol. IV of Stein&Shakarchi texts.

Math 624 is a continuation of Math 623. We start where Math 623 left off in the Fall and cover in particular the following topics: signed measures; Hilbert spaces and $L^2$ theory; compact operators; the Fourier transform; Banach spaces; elementary operator theory and linear functionals; $L^p$ spaces theory: duality, interpolation, fundamental inequalities and approximation theorems. Time permitting we will discuss some applications in harmonic analysis and some distribution theory.

The prerequisites for this class are Math 623 covering most of chapters 1, 2, 3(part) and 6 (part) of Stein-Shakarchi’s Real Analysis book (Vol. III) (namely having a working knowledge of Measure theory: Lebesgue measure and Integrable functions (Chapter 1); Integration theory: Lebesgue integral, convergence theorems and Fubini theorem (Chapter 2); Differentiation and Integration. Functions of bounded variation (Chapter 3) and some abstract measure theory (first part of Chapter 6) as well as a working knowledge of undergraduate Analysis (as for example taught in classes like M523H and M524H).

### MATH 652: Int. Numerical Analysis II

Yao Li MW 12:20-1:35

Prerequisites:

Math 651 or permission of the instructor.

Recommended Text:

Numerical Solution of Partial Differential Equations, K. W. Morton and D. F. Mayers

Description:

This course will cover two main subjects: numerical methods for partial differential equations and iterative methods for linear/non-linear systems. The following topics will be covered: 1, Finite difference method for partial differential equations (consistency, stability, convergence, and an introduction to the numerical conservation law). 2, Introduction to finite element methods. I will also introduce spectral methods if time permits. 3, Iterative methods for linear and non-linear systems. This includes solving linear and non-linear systems and numerical optimizations. I will also introduce some scientific computing packages (LAPACK, Eigen, Dlib) if time permits. Grades will be based on homework problems (including programming problems) and one final project.

### MATH 672: Algebraic Topology

Prerequisites:

Math 671 (topology), Math 611 (algebra), or equivalent.

Text:

Algebraic Topology, by Allen Hatcher (Cambridge University Press)

Note:

Description:

This course gives an introduction to the basic tools of algebraic topology, which studies topological spaces and continuous maps by producing associated algebraic structures (groups, vector spaces, rings, and homomorphism between them). Emphasis will be placed on being able to compute these invariants, not just on their definitions and associated theorems.

Topics include: Homotopy, fundamental group and covering spaces (reviewed from Math 671), simplical and cell complexes, singular and simplicial homology, long exact sequences and excision, cohomology, Künneth formulas, Poincaré duality.
If time permits, more advanced topics will be discussed at the end, such as higher homotopy groups, sheaf cohomology, the de Rham theorem, and equivariant cohomology.

### MATH 691T: S-Graduate Teaching Seminar

Paul Hacking Mon 4:00-5:15

Description:

The purpose of this seminar is to prepare graduate students to teach their own section of calculus in the Fall. Participants will present portions of calculus lectures in the seminar, observe the presentations of other participants, and provide feedback on the presentations.

### MATH 691Y: Applied Math Project Sem.

Qian-Yong Chen Fri 1:15-2:30

Prerequisites:

Graduate Student in Applied Math MS Program

Text:

None

Description:

Continuation of Project

### MATH 697AM: ST-Appl Math and Math Modeling

Matthew Dobson MWF 10:10-11:00

Prerequisites:

Text:

Lin and Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences.

Description:

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

### MATH 697NA: ST - Numerical Algorithms

Eric Polizzi TuTh 1:00-2:15

Description:

Objectives: Provide a practical understanding of matrix computations for science, engineering and industrial applications;
Provide solid foundations in computational linear algebra; Introduction to parallel computing and programming practices

Contents: Introduction to Scientific Computing- Basic numerical techniques of linear algebra and their applications- Data formats and Practices - Matrix computations with an emphasis on solving sparse linear systems of equations and eigenvalue problems - Parallel architectures and parallel programming with OpenMP, MPI and hybrid -. Numerical parallel algorithms

### MATH 697U: ST-Stochastic Processes and Appl

Luc Rey-Bellet TuTh 10:00-11:15

Prerequisites:

A working knowledge of probability such as STAT 605 or STAT 607.

Text:

No official textbook but see recommended texts.

Recommended Text:

"Introduction to Stochastic processes" by G. Lawler
"Adventures in Stochastic Processes" by S. Rednick
"Essentials of Stochastic Processes" by R. Durrett

Description:

This course is an introduction to stochastic processes and Monte-Carlo methods. And we will use from time to time some more advanced concepts from analysis and linear algebra. One of the main goal in the class is to develop a "probabilist intuition and way of thinking". We will present some proofs and we will skip some others in order to provide a reasonably broad range of topics, concepts and techniques. We emphasize examples both in discrete and continuous time from a wide range of disciplines, for example branching processes, queueing systems, population models, chemical reaction networks and so on. We will also discuss the numerical implementation of Markov chains and discuss the basics of Monte-Carlo algorithms. Among the topics treated in the class are

Simulation of random variables. A first look at Monte-Carlo algorithms.

Markov chains on discrete state spaces (both finite and countable). Definition and basic properties, classification of states (positive recurrence, recurrence and transience), stationary distribution and limit theorems, analysis of transient behavior, applications and examples.

Continuous-Time Markov chains. Definition and basic properties. Poisson Process, Birth and Death Process, Queueing models.

Reversible Markov processes and Monte-Carlo Markov algorithms.

Martingales.

Brownian motion and applications. Elementary stochastic analysis.

### MATH 697WA: ST-Nonlin Waves and Appl/Continua

Panos Kevrekidis TuTh 11:30-12:45

Prerequisites:

Math 532H or equivalent (required) and Math 534H or equivalent (required)

Text:

There will be no required textbook. The course will be based on instructor notes.

Description:

The aim of this course will be to give an overview of the mathematical background, physical applications and numerical computations associated with a number of prototypical wave systems both at the continuum and at the lattice level. We will start from finite dimensional Hamiltonian systems, discuss their symmetries and Lagrangian/Hamiltonian structure, and then extend considerations to infinite dimensional systems of partial differential and differential-difference equations. Prototypical case examples will be the continuum and the discrete nonlinear Schrodinger equation, the Korteweg-de Vries equation, the sine-Gordon equation, and the Fermi-Pasta-Ulam lattice, among others. We will examine the symmetries, conservation laws, solitary wave solutions, linearization spectral properties and dynamics of such equations and attempt to connect them with physical applications from nonlinear optics, fluid mechanics, materials science and atomic physics, as well as develop computational tools (such as bifurcation analysis and time-stepping algorithms) about how to address them. Time permitting, we will also make short excursions to systems of multiple components, higher dimensions or of dissipative character (e.g. reaction-diffusion type) to discuss some similarities and differences with these.

### MATH 705: Symplectic Topology

R. Inanc Baykur TuTh 1:00-2:15

Prerequisites:

671-672, 703, 621, or the consent of the instructor

Text:

"Lectures on Symplectic Geometry" by Ana Cannas da Silva
"Introduction to Symplectic Topology" by Dusa McDuff and Dietmar Salamon

Recommended Text:

"An Introduction to Contact Topology" by Hansjorg Geiges

Description:

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

Student grade will be based on regularly assigned homework (75%) and a final presentation (25%). The latter will consist of an in-class presentation, accompanied by a short expository paper, on a topic in symplectic topology.

### MATH 708: Complex Algebraic Geometry

Eyal Markman MW 2:30-3:45

Prerequisites:

Holomorphic functions of one complex variable (at the level of Math 621),
Differentiable Manifolds and their deRham cohomology (at the level of
Math 703)

Text:

Complex Geometry, an introduction, by Daniel Huybrechts.

Recommended Text:

Principles of Algebraic Geometry, by Phillip Griffiths and Joseph Harris.

Description:

An introductory course to complex algebraic
geometry. The basic techniques of Kahler geometry, Hodge theory, line and
vector bundles, needed for the study of the geometry and topology of complex projective
algebraic varieties, will be introduced and illustrated in basic examples such as Riemann surfaces, algebraic surfaces, abelian
varieties, and Grassmannians.

### MATH 797DC: ST-Derived Categories

Jenia Tevelev TuTh 10:00-11:15

Prerequisites:

Math 612 and any introductory course in Algebraic Geometry

Recommended Text:

Huybrechts, Fourier-Mukai Transforms in Algebraic Geometry

Description:

The course will consist of two parts of roughly the same length. The first part will cover basics of Abelian, triangulated and derived categories using examples from algebra. The only prerequisites for this part are Math 611-612. The second part will be more technical as we will focus on computations and applications of derived categories of coherent sheaves in algebraic geometry. The prerequisite for this part is a standard course in algebraic geometry, either using the language of schemes (Chapter II of Hartschorne will suffice) or complex algebraic geometry (e.g. the first volume of Voisin’s "Hodge theory and complex algebraic geometry”). The course will end with students’ group presentations prepared with the help of the instructor.

The course grade will be based on the homework and the project. There will be 5 biweekly homework sets. Problems will be worth certain number of points depending on their difficulty with a total of 30 points for each homework. To get an A you have to accumulate 100 points by the end of the semester. Homework problems can be either presented during office hours
or written down and submitted.

Many possible reading suggestions for this class can be found here:
http://people.math.umass.edu/~hacking/seminarS13/

### STAT 608: Mathematical Statistics II

Daeyoung Kim TuTh 1:00-2:15

Prerequisites:

STAT 607 or permission of the instructor.

Text:

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

Description:

This is the second part of a two semester sequence on probability and mathematical statistics. Stat 607 covered probability, discrete/continuous distributions, basic convergence theory and basic statistical modelling. In Stat 608 we cover an introduction to the basic methods of statistical inference, pick up some additional probability topics as needed and examine further issues in methods of inference including more on likelihood based methods, optimal methods of inference, more large sample methods, Bayesian inference and Resampling methods. The theory is utilized in addressing problems in parametric/nonparametric methods, two and multi-sample problems, and regression. As with Stat 607, this is primarily a theory course emphasizing fundamental concepts and techniques.

### STAT 691P: Project Seminar

John Staudenmayer MWF 9:05-9:55

Prerequisites:

Permission of instructor.

Text:

None.

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 797L: ST-Mixture Models

Daeyoung Kim TuTh 2:30-3:45

Prerequisites:

Calculus based Probablity and Mathematical Statistics (preferably Stat 607-608 or rough equivalent), and Regression Modeling (preferably Stat 625 (previously Stat 697R) or rough equivalent).
Stat 608 is a co-requisite for this course.

Text:

TBA

Recommended Text:

Titterington, D. M., Smith, A. F. M., and Makov, U. E. (1985) Statistical Analysis of Finite Mixture Distributions;
McLachlan, G.J. and Basford K.E. (1988) Mixture Models. Inference and Applications to Clustering;
Lindsay, B. G. (1995) Mixture Models: Theory, Geometry and Applications;
Bohning, D. (1999) Computer-assisted Analysis of Mixtures and Applications: Meta-analysis, Disease Mapping and Others;
McLachlan, G., and Peel, D. (2000) Finite Mixture Models;
Fruhwirth-Schnatter, S. (2007) Finite Mixture and Markov Switching Models

Description:

A challenging task of statistical analysis is to handle complex data structures including heterogenous data, correlated data, overdispersed data and hierarchically structured data.
Mixture/Latent variable models provide a natural framework to analyze such complex data sets and identify hidden patterns in the data.
This course will focus on introducing the ideas and theories of mixture/latent variable models and their applications in various areas such as biology, medicine, genomics, epidemiology, social and psychological sciences, marketing, economics and finance.
Topics will include the (parametric/finite/nonparametric) mixture/latent variable model specification, identifiability, estimation methods, large sample properties, finite sample properties, computational algorithms, data analysis, and interpretation.
This course will be accessible to students with knowledge of statistics at an intermediate level (i.e., probability, basis for the statistical inference and regression analysis).