# 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 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 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 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 190F: Foundations of Data Science

See SPIRE for instructors and times

Prerequisites:

Completion of the R1 General Education Requirement (or a score of 20 or higher on the Math Placement Exam, Part A) or one of the following courses: Math 101 & 102, Math 104, 127, 128, 131, or 132.

Note:

CROSS-LISTED WITH COMPSCI 190F. CS, INFORMATICS, AND MATH & STATS MAJORS ARE NOT ELIGIBLE. LAB SECTIONS COMPSCI/STAT 190F-01LQ PROVIDE COMPUTER WORKSTATIONS FOR STUDENT USE DURING LABS. STUDENTS REGISTERED FOR OTHER LAB SECTIONS WILL NEED TO BRING A LAPTOP WITH A REASONABLY UP-TO-DATE WEB BROWSER.

Description:

The field of Data Science encompasses methods, processes, and systems that enable the extraction of useful knowledge from data. Foundations of Data Science introduces core data science concepts including computational and inferential thinking, along with core data science skills including computer programming and statistical methods. The course presents these topics in the context of hands-on analysis of real-world data sets, including economic data, document collections, geographical data, and social networks. The course also explores social issues surrounding data analysis such as privacy and design.

### 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

George Avrunin TuTh 1:00-2:15

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), 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

Kuan-Wen Lai MWF 10:10-11:00

Prerequisites:

Math 132 with a grade of 'C' or better

Text:

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

Weimin Chen TuTh 11:30-12:45

Prerequisites:

Math 132 with a grade of C or better

Text:

Title: How to Prove it: A structured Approach, 2nd edition
Author: Daniel J. Velleman, by Cambridge Univ. Press

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

Andrew Havens TuTh 11:30-12:45

Prerequisites:

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

Description:

While the mathematicians of the pre-internet age often spread their mathematical ideas within the community via written letters prior to publication, modern mathematical correspondence and exposition is rapidly facilitated by a variety of digital tools. Of great importance to the publishing process in mathematical sciences is the LaTeX markup language, used to typeset virtually all modern mathematical publications, even at the pre-print stage. In this course we will develop facility with LaTeX, and develop a variety of writing practices important to participation in the mathematical community. There will be regular written assignments completed in LaTeX, as well as collaborative writing assignments, owing to the importance of collaborative writing in mathematical research. Writing topics may include proofs, assignment creation, pre-professional writing (resumes/cover letters, research and teaching statements), expository writing for a general audience, recreational mathematics, and the history of mathematics. Short writing assignments on such topics will be assigned in response to regular assigned readings from a variety of accessible/provided sources. Towards the end of the semester groups will complete a research paper of an expository nature and craft a seminar style presentation. This course meets the junior year writing requirement.

### MATH 370.2: Writing in Mathematics

Patrick Dragon MWF 10:10-11:00

Prerequisites:

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

Description:

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

### MATH 370.3: Writing in Mathematics

Patrick Dragon MWF 12:20-1:10

Prerequisites:

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

Description:

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

### MATH 411.1: Introduction to Abstract Algebra I

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

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 412: Introduction to Abstract Algebra II

Laura Colmenarejo TuTh 1:00-2:15

Prerequisites:

Math 411

Text:

Abstract Algebra: A First Course, Dan Saracino (2nd 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

Jonathan Simone MWF 1:25-2:15

Prerequisites:

Math 233

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. Evaluation of Improper integrals via residues. Conformal mappings.

### MATH 425.1: Advanced Multivariate Calculus

Ivan Mirkovic TuTh 2:30-3:45

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 (2003) ISBN-10: 0716749920, ISBN-13: 978-0716749929; or Sixth Edition (2012) ISBN-10: 1-4292-9411-6, ISBN-13: 978-1-4292-9411-9]

Description:

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

### MATH 425.2: Advanced Multivariate Calculus

Zahra Sinaei MWF 10:10-11:00

Prerequisites:

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

Recommended Text:

J. Marsden and A. Tromba, Vector Calculus [an earlier edition is fine and may be less expensive: W. H. Freeman, Fifth Edition edition (2003) ISBN-10: 0716749920, ISBN-13: 978-0716749929; or Sixth Edition (2012) ISBN-10: 1-4292-9411-6, ISBN-13: 978-1-4292-9411-9]

H. M. Schey, div, grad, curl and all that [ISBN-13: 978-0393925166; ISBN-10: 039 3925161]

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

Description:

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

### MATH 425.3: Advanced Multivariate Calculus

Zahra Sinaei MWF 11:15-12:05

Prerequisites:

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

Recommended Text:

J. Marsden and A. Tromba, Vector Calculus [an earlier edition is fine and may be less expensive: W. H. Freeman, Fifth Edition edition (2003) ISBN-10: 0716749920, ISBN-13: 978-0716749929; or Sixth Edition (2012) ISBN-10: 1-4292-9411-6, ISBN-13: 978-1-4292-9411-9]
H. M. Schey, div, grad, curl and all that [ISBN-13: 978-0393925166; ISBN-10: 039 3925161]
M. Spivak, Calculus on Manifolds [ISBN-13: 978-0805390216; ISBN-10: 0805390219]

Description:

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

### MATH 455: Introduction to Discrete Structures

George Avrunin TuTh 2:30-3:45

Prerequisites:

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

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; 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

Leili Shahriyari 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, Java, C++, Python, etc.). Some familiarity with statistics and probability is desirable.

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. Prerequisites: Calculus (Math 131, 132, 233), required; Linear Algebra (Math 235) and Differential Equations (Math 331), or permission of instructor required

### MATH 456.2: Mathematical Modeling

Qian-Yong Chen MWF 9:05-9:55

Prerequisites:

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

Note:

Not following any particular book. General mathematical references:

Topics in Mathematical Modeling, by K. K. Tung, Princeton University Press, 2007;

Introduction to Probability by Charles M. Grinstead and J. Laurie Snell. American Mathematical Society.

Game theory by James N. Webb. Springer Undergraduate Series.

Description:

Math 456 is an introduction to mathematical modeling, and is one of the Integrated Experience courses approved by the General Education Council. The main goal of the class is to learn how to translate problems from "real-life" into a mathematical model and how to use mathematics to solve the problem. We will learn how to build, use and critique mathematical models. In the beginning, we'll focus on differential equation based models. For the second half, we will study a number of topics from games and gambling, economics, social sciences, for which we will use elementary tools from probability, game theory, information theory, and optimization.

In addition, each student will join a group of 3 students, and the group will investigate a modeling problem. Each group will give a final presentation at the end of the semester. Each student will write an individual report on the group project at the end of the course. After discussion/consultation with the instructor, the choice of modeling topics will be determined by the interests and background of the enrolled students, and the mathematical methods applied will draw upon whatever the students have already learned.

### MATH 456.3: Mathematical Modeling

Mark Wilson MWF 11:15-12:05

Prerequisites:

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

Description:

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

### MATH 471: Theory of Numbers

Tom Weston TuTh 10:00-11:15

Prerequisites:

Math 233 and Math 235. Co-requisite: Math 300 or CS250.

Description:

This course is a proof-based introduction to elementary number theory. We will quickly review basic properties of the integers including modular arithmetic and linear Diophantine equations covered in Math 300 or CS250. We will proceed to study primitive roots, quadratic reciprocity, Gaussian integers, and some non-linear Diophantine equations. Several important applications to cryptography will be discussed.

### MATH 475: History of Mathematics

Prerequisites:

Math 131, 132, 233 and Math 300 or CS 250.

Text:

"A History of Mathematics" by V. Katz, 3d edition

Description:

This course is an introduction to the history of mathematics from ancient civilizations to present day. Students will study major mathematical discoveries in their cultural, historical, and scientific contexts. This course explores how the study of mathematics evolved through time, and the ways of thinking of mathematicians of different eras - their breakthroughs and failures. Students will have an opportunity to integrate their knowledge of mathematical theories with material covered in General Education courses. Students are required to use primary sources on a weekly basis. Students will study examples of how mathematical advances were made in response to or alongside developments in other branches of science, how these advances were understood at the time, and their implications for modern mathematics. Since many math majors go on to teach mathematics after graduation, in this course the history of math is also studied in the context of the history of education.

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

### MATH 513: Combinatorics

Paul Gunnells TuTh 10:00-11:15

Prerequisites:

CompSci 250 or Math 455 with a grade of 'B' or better.

Mathematical maturity, calculus, linear algebra, discrete mathematics course such as CompSci 250 or Math 455. Math 411 recommended but not required.

Note:

5 seats reserved for CompSci majors

Description:

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.

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

Siddhant Agrawal TuTh 8:30-9:45

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.

### MATH 524: Introduction to Modern Analysis II

Sohrab Shahshahani TuTh 8:30-9:45

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.

Recommended Text:

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

Reference text (optional): Partial Differential Equations in Action: From Modelling to Theory by Sandro Salsa, (UNITEXT; Springer) 3rd ed. 2016 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

Recommended Text:

ASM Study Manual for Exam P, 3rd or later Edition by Weishaus, you can buy 6-month digital license at https://www.studymanuals.com/Product/Show/453140947

Description:

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

### MATH 537.1: Intro. to Math of Finance

Jinguo Lian MWF 10:10-11:00

Prerequisites:

Math 233 and either Stat 515 or MIE 273

Recommended Text:

Derivative Markets by Robert L. McDonald, 3rd edition.
ASM Exam IFM Study Manual 1st or later Edition by Abraham Weishaus.

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 Math of Finance

Jinguo Lian MWF 11:15-12:05

Prerequisites:

Math 233 and Stat 515

Recommended Text:

Derivative Markets by Robert L. McDonald, 3rd edition.
ASM Exam IFM Study Manual 1st or later Edition by Abraham Weishaus.

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, both with grade of 'C' or better, 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 545.2: Linear Algebra for Applied Mathematics

Prerequisites:

Math 233 and 235, both with grade of 'C' or better, 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 545.3: Linear Algebra for Applied Mathematics

Jeremiah Birrell TuTh 1:00-2:15

Prerequisites:

Math 233 and 235, both with grade of 'C' or better, Math 300 or CS 250

Description:

This is a second course in Linear Algebra, building upon the concepts and techniques introduced in Math 235. We will study the decomposition of matrices, particularly the LU, QR, and singular value decompositions. We also study vector spaces and linear transformations, inner product spaces, orthogonality, and spectral theory. We will emphasize applications of these techniques to various problems including, as time permits: solutions of linear systems, least-square fitting, search engine algorithms, error-correcting codes, fast Fourier transform, dynamical
systems.The coursework will be a mix of proof and computation. For the latter, we will often use MATLAB.

### MATH 551.1: Intr. Scientific Computing

Hans Johnston MWF 12:20-1:10

Prerequisites:

MATH 233 and 235 and one of CS 121, E&C-Eng 242, or Physics 281.

Knowledge of a scientific programming language, e.g. MATLAB, Fortran, C, C++, Python, Java.

Text:

Elementary Numerical Analysis (3rd edition), by Atkinson & Han (Wiley: ISBN 9780471433378)

Description:

The course will introduce foundational numerical methods used for problems that arise in many scientific fields. Properties such as accuracy of methods, their stability and efficiency will be considered. Students will gain practical programming experience in implementing the methods using MATLAB, which will be taught through incrasingly complex codes over the term. We will also discuss practical considerations of implementing numerical methods on modern computer architectures using C, C++ or Fortran. Today's average smartphone can computationally crush a 1990's era Cray C90, which cost $10 million at the time ($18 million in today's $) for sixteen 244Mhz vector processors and 8GB of RAM. Topics covered: Finite Precision Arithmetic and Error Propagation, Root Finding, Linear Systems of Equations, Interpolation and Approximation of Functions, Numerical Differentiation and Integration. ### MATH 551.2: Intr. Scientific Computing Hans Johnston MWF 1:25-2:15 Prerequisites: MATH 233 and 235 and one of CS 121, E&C-Eng 242, or Physics 281. Knowledge of a scientific programming language, e.g. MATLAB, Fortran, C, C++, Python, Java. Text: Elementary Numerical Analysis (3rd edition), by Atkinson & Han (Wiley: ISBN 9780471433378) Description: The course will introduce foundational numerical methods used for problems that arise in many scientific fields. Properties such as accuracy of methods, their stability and efficiency will be considered. Students will gain practical programming experience in implementing the methods using MATLAB, which will be taught through incrasingly complex codes over the term. We will also discuss practical considerations of implementing numerical methods on modern computer architectures using C, C++ or Fortran. Today's average smartphone can computationally crush a 1990's era Cray C90, which cost$10 million at the time ($18 million in today's$) for sixteen 244Mhz vector processors and 8GB of RAM.

Topics covered: Finite Precision Arithmetic and Error Propagation, Root Finding, Linear Systems of Equations, Interpolation and Approximation of Functions, Numerical Differentiation and Integration.

### MATH 551.3: Intr. Scientific Computing

Maria Correia TuTh 2:30-3:45

Prerequisites:

MATH 233 and 235 and one of CS 121, E&C-Eng 242, or Physics 281.

Knowledge of a scientific programming language, e.g. MATLAB, Fortran, C, C++, Python, Java.

Description:

This course provides an introduction to computational techniques used in science and industry. Topics selected from root finding, interpolation, data fitting, linear systems of equations, approximation of functions, numerical integration, numerical methods for differential equations and error analysis.

### MATH 552: Applications of Scientific Computing

Hans Johnston MWF 10:10-11:00

Prerequisites:

Math 233, Math 235, Math 331 or permission of instructor, Math 551 (or equivalent) or permission of instructor.

Knowledge of scientific programming language is required.

Text:

Elementary Numerical Analysis (3rd edition), by Atkinson & Han (Wiley: ISBN 9780471433378)

Description:

This course is the second half of a sequence in Scientific Computing, the first being Math 551, a prerequisite for this course. We will cover the following foundational topics in Numerical Methods for ODEs and PDEs: Discrete Time Stepping Methods; Advanced Topics in Numerical Linear Algebra; Fast Transforms and the FFT; Finite Difference and Spectral Methods. While we will use MATLAB for the 'numeric laboratory,' we will also discuss practical considerations of implementing these methods on modern computer architectures
using C, C++ or Fortran.

Grading Policy: Homework (25%), midterm exam (40%) and final exam (35%). Homework assignments will be given out every week or so.

### MATH 563H: Differential Geometry

Rob Kusner MW 2:30-3:45

Prerequisites:

Very good understanding of Advanced Multivariable Calculus and Linear Algebra (Math 425, 233 and 235). Math 331 highly recommended.

Recommended Text:

Curves and Surfaces: Second Edition
Sebastián Montiel and Antonio Ros
Publication Year: 2009
ISBN-10: 0-8218-4763-5
ISBN-13: 978-0-8218-4763-3
AMS Graduate Studies in Mathematics, vol. 69.R

Description:

This course is an introduction to differential geometry, where we apply theory and computational techniques from linear algebra, multivariable calculus and differential equations to study the geometry of curves, surfaces and (as time permits) higher dimensional objects; global and variational aspects of geometry will be a central theme of the course.

### 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:

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

### STAT 501: Methods of Applied Statistics

Joanna Jeneralczuk TuTh 11:30-12:45

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.

### STAT 515.1: Statistics I

Qian-Yong Chen MWF 11:15-12:05

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: Statistics I

Jiayu Zhai MW 4:00-5:15

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, 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

Jiayu Zhai MW 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 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, 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

Luc Rey-Bellet TuTh 1:00-2:15

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: Statistics I

Markos Katsoulakis TuTh 11:30-12:45

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 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: Statistics I

Panagiota Birmpa MWF 1:25-2:15

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 515.7: Statistics I

Eric Knyt TuTh 4:00-5:15

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.

Note:

This section of Stat 515 is open to Biomedical Engineering majors only.

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

Mike Sullivan TuTh 10:00-11:15

Prerequisites:

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

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

Prerequisites:

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

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

Ted Westling MWF 10:10-11:00

Prerequisites:

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

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

Maryclare Griffin TuTh 8:30-9:45

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 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 525.2: Regression Analysis

Krista J Gile TuTh 2:30-3:45

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 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 525.3: 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 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

Zijing Zhang Tuesdays 6:00-8:30 PM

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.)

Note:

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

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:

Stat 516 and CompSci 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 597T: ST-Analysis of Discrete Data

Daeyoung Kim TuTh 1:00-2:15

Prerequisites:

Stat 525 or equivalent, and consent of instructor.

Text:

TBA

Description:

Discrete/Categorical data are prevalent in many applied fields, including biological and medical sciences, social and behavioral sciences, and economics and business. This course provides an applied treatment of modern methods for visualizing and analyzing broad patterns of association in discrete/categorical data. Topics include forms of discrete data, visualization/exploratory methods for discrete data, discrete data distributions, correspondence analysis, logistic regression models, models for polytomous responses, loglinear and logit Models for contingency tables, and generalized linear models. This is primarily an applied statistics course. While models and methods are written out carefully with some basic mathematical derivations, the primary focus of the course is on the understanding of the visualization and modeling techniques for discrete data, presentation of associated models/methods, data analysis, interpretation of results, statistical computation and model building.

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

Krista J Gile and Anna Liu Thursdays 1:00-2:15

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

Paul Hacking TuTh 1:00-2:15

Prerequisites:

Math 611 (or consent of the instructor).

Text:

Abstract Algebra, by Dummit and Foote, 3rd edition

Recommended Text:

M. Atiyah and I. MacDonald, Introduction to commutative algebra.

Description:

A continuation of Math 611. Topics covered will include field theory, Galois theory, and commutative algebra.

Syllabus of Math 611 - Math 612:

I. Group 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.

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 624: Real Analysis II

Robin Young TuTh 10:00-11:15

Prerequisites:

Math 523H, Math 524 and Math 623.

Text:

https://www.math.ucdavis.edu/~bxn/applied_analysis.pdf

Applied Analysis,by John Hunter and Bruno Nachtergaele, World Scientific Press, 2001.

Description:

Continuation of Math 623. Introduction to functional analysis; elementary theory of Hilbert and Banach spaces; functional analytic properties of Lp-spaces, applications to Fourier series and integrals; interplay between topology, and measure, Stone-Weierstrass theorem, Riesz representation theorem. Further topics depending on instructor.

### MATH 646: Applied Math and Math Modeling

Matthew Dobson MWF 9:05-9:55

Text:

David Logan, Applied Mathematics

Recommended Text:

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

Description:

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

### MATH 652: Int Numerical Analysis II

Brian Van Koten MW 2:30-3:45

Prerequisites:

Math 651

Description:

Presentation of the classical finite difference methods for the solution of the prototype linear partial differential equations of elliptic, hyperbolic, and parabolic type in one and two dimensions. Finite element methods developed for two dimensional elliptic equations. Major topics include consistency, convergence and stability, error bounds, and efficiency of algorithm.

### MATH 672: Algebraic Topology

Alexei Oblomkov TuTh 8:30-9:45

Prerequisites:

Math 671, Math 611 or equivalent.

Text: Description:

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.

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

Matthew Dobson Fridays 2:30-3:45

Prerequisites:

Graduate Student in Applied Math MS Program

Description:

Continuation of Project

### MATH 697SS: ST-Sums of Squares

Annie Raymond MW 8:40-9:55

Prerequisites:

Math 235 and Math 455, and ideally either a course on polytopes or linear programming, though a motivated student will be able to pick up the notions needed from these topics as we go along.

Description:

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

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

Luc Rey-Bellet TuTh 11:30-12:45

Prerequisites:

Stat 605 or Stat 607. A good working knowledge of linear algebra and analysis.

Text:

Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues by Pierre Bremaud

Description:

This course is an introduction to stochastic processes. The course will cover Monte Carlo methods, Markov chains in discrete and continuous time, martingales, and Brownian motion. Theory and applications will each play a major role in the course. Applications will range widely and may include problems from population genetics, statistical physics, chemical reaction networks, and queueing systems, for example.

### MATH 705: Symplectic Topology

R. Inanc Baykur TuTh 11:30-12:45

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 symplectic topology, along with its connections to differential, complex algebraic, and contact geometry and topology.

### MATH 708: Complex Algebraic Geometry

Eyal Markman TuTh 10:00-11:15

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.

Outline of topics. Complex Manifolds, Sheaf cohomology, Kahler manifolds, Hodge
Theory, the Hard Lefschetz Theorem,
Holomorphic Vector bundles, Kodaira's Vanishing and Embedding theorems, other topics as time permits.

Homework will be assigned regularly and graded.

### MATH 725: Intro Functional Analysis

Andrea Nahmod TuTh 2:30-3:45

Prerequisites:

Math 623 and M624 (M624 concurrently is allowed with permission)

Description:

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

### MATH 797D: ST-Top/Geom Singular Spaces

Paul Gunnells TuTh 8:30-9:45

Description:

Singular spaces arise naturally in many contexts, including algebraic geometry and representation theory. Any singular space has a decomposition into manifolds (a "stratification"), and so the study of them is a mixture of topology, geometry, and combinatorics. The goal of this course is to present the basics of stratified spaces and to illustrate the general theory with some important examples. Potential topics: general material, including stratifications, Whitney conditions, local structures, intersection homology; isolated singularities of complex hypersurfaces (the Milnor fibration and related topics), including connections to knot theory; and compactifications of locally symmetric spaces.

### MATH 797DS: ST-Infinite Dimensional Integral Sys

Franz Pedit MW 2:30-3:45

Prerequisites:

Basic knowledge of manifolds, symplectic geometry, Lie algebras, Riemann surfaces.

Description:

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

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

### MATH 797P: ST-Stochastic Calculus

Yao Li TuTh 1:00-2:15

Prerequisites:

Statistics 605

Text:

Stochastic Differential Equations: An Introduction with Applications by Bernt Oksendal, 6th edition

Description:

This course is largely based on Statistics 605. Classical topics including martingale theory, Brownian motion, stochastic integration, stochastic differential equation, and connections to partial differential equations will be covered. I will also introduce some applications to mathematical finance and some numerical simulation methods. Final grades are based on several homework assignments and one take home final exam.

### STAT 608.1: Mathematical Statistics II

Daeyoung Kim TuTh 2:30-3:45

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 608.2: Mathematical Statistics II

Hyunsun Lee Mondays 6:00-8:30 PM

Prerequisites:

Statistc 607 or equivalent, or permission of the instructor.

Text:

All of Statistics, by Larry Wasserman. Publisher: Springer, 2010.

Note:

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

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.

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 697DS: Statistical Methods for Data Science

Hyunsun Lee Wednesdays 6:00-8:30 PM

Prerequisites:

Probability and Statistics at a calculus-based level such as Stat 607 and Stat 608 (concurrent) or Stat 515 and Stat 516 (concurrent), and knowledge of regression at the level of Stat 525 or Stat 625. Students must have an understanding of linear algebra at the level of Math 235. Students must have prior experience with a statistical programming language such as R, Python or MATLAB.

Text:

“An Introduction to Statistical Learning: with Applications in R” by Gareth James, Daniela Witten, Trevor Hastie and Robert Tibshirani, 2013, Springer. ISBN-13: 978-1461471370.

Note:

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

Description:

This course provides an introduction to the statistical techniques that are most applicable to data science. Topics include regression, classification, resampling, linear model selection and regularization, tree-based methods, support vector machines and unsupervised learning. The course includes a computing component using statistical software.

### STAT 697L: ST-Categorical Data Analysis

Anna Liu TuTh 11:30-12:45

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. 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.

### STAT 697TS: ST-Time Series Analysis and Appl

Maryclare Griffin TuTh 10:00-11:15

Prerequisites:

STAT 607/608 for familiarity with maximum likelihood estimation. STAT 625 or 705 for familiarity with linear algebra, specifically in the context of regression, recommended but not required.

Note:

Open to Graduate Math and Statistics students only.

Description:

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

### STAT 797S: ST-Estimation/SemiNonParametMD

Ted Westling MWF 12:20-1:10

Prerequisites:

STAT 607/608 or permission of instructor.

Description:

Statistical inference in parametric models is generally well-understood, but parametric assumptions are unrealistic in many settings. Semiparametric and nonparametric models provide more flexible alternatives that may better reflect our knowledge of the problem at hand, but statistical inference in these models is often challenging. In this course, we will introduce the statistical theory and methods underlying targeted inference of Euclidean parameters in semiparametric and nonparametric models. We will begin by discussing aspects of semiparametric efficiency theory. We will then introduce several general-purpose methods of targeted estimation in these models. Finally, we will provide an overview of tools for analyzing the behavior of such estimators, emphasizing the role that modern machine learning methods can play. Throughout the course, we will illustrate these methods using problems from causal inference, survival analysis, and missing data. Grades will be based on regular homework assignments and a final project.