# 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 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 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

Text:

TBA

Description:

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

### STAT 108: 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:

CS, INFORMATICS, AND MATH & STATS MAJORS ARE NOT ELIGIBLE. STUDENTS 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 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:

How to Prove it, by Daniel J. Velleman, 2nd edition, 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). Other topics will be included as time allows. Math 300 is designed to help students make the transition from calculus courses to the more theoretical junior-senior level mathematics courses.

### MATH 300.2: Fundamental Concepts of Mathematics

Prerequisites:

Math 132 with a grade of 'C' or better

Text:

An Introduction to Mathematical Thinking: Algebra and Number Systems, by William J. Gilbert and Scott A. Vanstone, Pearson Prentice Hall, 2005.

Description:

The goal of this course is to help students learn the language of rigorous mathematics.

Students will learn how to read, understand, devise and communicate proofs of mathematical statements. A number of proof techniques (contrapositive, contradiction, and especially induction) will be emphasized. Topics to be discussed include set theory (Cantor's notion of size for sets and gradations of infinity, maps between sets, equivalence relations, partitions of sets), basic logic (truth tables, negation, quantifiers). Other topics will be included as time allows. Math 300 is designed to help students make the transition from calculus courses to the more theoretical junior-senior level mathematics courses.

### MATH 300.3: Fundamental Concepts of Mathematics

Prerequisites:

Math 132 with a grade of C or better

Text:

TBA

Recommended Text:

TBA

Description:

The goal of this course is to help students learn the language of rigorous mathematics.
Students will learn how to read, understand, devise and communicate proofs of mathematical statements. A number of proof techniques (contrapositive, contradiction, and especially induction) will be emphasized. Topics to be discussed include set theory (Cantor's notion of size for sets and gradations of infinity, maps between sets, equivalence relations, partitions of sets), basic logic (truth tables, negation, quantifiers). Other topics will be included as time allows. Math 300 is designed to help students make the transition from calculus courses to the more theoretical junior-senior level mathematics courses.

### MATH 370.2: Writing in Mathematics

Prerequisites:

MATH 300 or CS 250 and completion of 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 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 370.3: Writing in Mathematics

Prerequisites:

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

Description:

Students will develop skills in writing, oral presentation, and teamwork. The first part of the course will focus on pre-professional skills, such as writing a resume, cover letter, and graduate school essay and preparing for interviews. Subsequent topics will include presenting mathematics to a general audience, the role of mathematics in society, mathematics education, and clear communication of mathematical content. The end of the term will be dedicated to the research process in mathematics and will include grant writing, research paper, and professional presentation.

### MATH 370.4: Writing in Mathematics

Prerequisites:

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

Text:

No text.

Recommended Text:

Instructor will share necessary course material.

Description:

In the year 2000 the Clay Institute listed seven (then) unsolved problems across all areas of mathematics considered the most challenging and important for the new millenium.
So far only one of the problems, the Poincare Conjecture, has been solved by Perleman (who refused to collect the one million dollar prize stating that mathematics should never be done for money).
In this course we shall focus on the, as of yet unsolved, Birch and Swinnerton-Dyer conjecture. This conjecture pertains to the behavior of integer/rational solutions to certain equations. The area of mathematics is number theory and in particular the study of Diophantine equations. We shall go far back in the history of such problems (from Egyptian, Babylonian and Greek mathematics through the Middle Ages into our millenium) and try to grasp some understanding of the issues in easy to handle cases (Pythagorean numbers, canon ball stacking etc). The course is structured around writing assignments on these topics which will be peer reviewed and/or graded by the instructor and the course TA. During the last quarter of the semester there will be group project presentations.
All writing has to be done in the word processing system LaTex, which is the only word processing system capable of producing a professional layout. Templates and some basic tutorials will be provided.
We shall NOT spend time on resume and job application writing, since there is ample opportunity to receive expert help from the career center (a representative of which will give a presentation in class).

### MATH 397C: Mathematical Computing

Prerequisites:

CS 121 or INFO 190S, Math 235, Math 300

Description:

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

### MATH 411.1: Introduction to Abstract Algebra I

Prerequisites:

Math 235; Math 300 or CS 250

Text:

D. Saracino, Abstract Algebra: A First Course, second edition, Waveland Press 2008

Description:

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

### MATH 411.2: Introduction to Abstract Algebra I

Prerequisites:

Math 235; Math 300 or CS 250

Text:

Abstract Algebra with Applications by Audrey Terras.

Note:

Description:

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

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

The following topics will be covered: Group axioms. Examples of groups. Numbers and matrices. Transformation groups. Multiplication tables. The group of units modulo n. Properties of groups. Generators and Cayley graphs. Subgroups. Symmetric, Dihedral and Cyclic groups. Cyclic subgroups of a group. Order of an element. Properties of permutations. Cayley theorem. Isomorphisms.  Rubik's cube and other permutation puzzles. Subgroups of permutations. Sign of a permutation. Cosets. Lagrange Theorem. Normal Subgroups. Quotient group. Homomorphisms. Isomorphism theorems.  Direct product of groups. Classification of finite Abelian groups.  Group actions. Orbit/stabilizer theorem.  Burnside lemma.  Crystallographic groups. Conjugacy classes. Cauchy theorem. Sylow theorems (w/o proof). Matrix groups over Fp. Group-theoretic aspects of public-key cryptography. Simple groups. Classification of groups of small order.

### MATH 411.3: Introduction to Abstract Algebra I

Prerequisites:

Math 235; Math 300 or CS 250

Recommended Text:

A first course in abstract algebra by John B. Fraleigh, 7th edition

Description:

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

### MATH 412: Introduction to Abstract Algebra II

Prerequisites:

Math 411

Text:

"Abstract Algebra" by Saracino, Dan. 2nd Edition. ISBN: 9781577665366

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

Prerequisites:

Math 233

Text:

Complex Analysis for Mathematics and Engineering, by John H. Mathews and Russell W. Howell. (Any edition may be used.)

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 455: Introduction to Discrete Structures

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, colorings and matchings, the pigeonhole principle, induction and recursion, generating functions, and (if time permits) combinatorial geometry. The course integrates mathematical theories with applications to concrete problems from other disciplines using discrete modeling techniques. Small student groups will be formed to investigate a modeling problem independently, and each group will report its findings to the class in a final presentation. Satisfies the Integrative Experience for BS-Math and BA-Math majors.

### MATH 456.1: Mathematical Modeling

Prerequisites:

Math 233 and Math 235. Some familiarity with a programming language is desirable (R studio, Python, etc.). Some familiarity with statistics and probability is desirable.

Recommended Text:

Statistics for Risk management, 3rd or later edition by Abraham Weishaus. You can buy it online at https://www.studymanuals.com/Product/Show/453142456

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 456.2: Mathematical Modeling

Prerequisites:

Math 233 and Math 235. Some familiarity with a programming language is desirable (R studio, Python, etc.). Some familiarity with statistics and probability is desirable.

Recommended Text:

Statistics for Risk management, 3rd or later edition by Abraham Weishaus. You can buy it online at https://www.studymanuals.com/Product/Show/453142456

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 456.3: Mathematical Modeling

Prerequisites:

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

Description:

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

Prerequisites:

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

Text:

An Illustrated Theory of Numbers, by Martin Weissman (see http://illustratedtheoryofnumbers.com/)

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. Some important applications to cryptography will be discussed.

### MATH 475: History of Mathematics

Prerequisites:

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

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. Satisfies the Integrative Experience requirement for BA-MATH and BS-MATH majors.

### MATH 523H: Introduction to Modern Analysis I

Prerequisites:

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

Description:

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

### MATH 524: Introduction to Modern Analysis II

Prerequisites:

MATH 523H, good knowledge of linear algebra (abstract vector spaces, linear maps etc.)

Text:

I will share pdfs of lecture notes from various authors:
Terence Tao: Analysis 2; Spivak: Analysis on Manifolds; Thiele: Analysis 2 (Bonn Univ.); Zorich: Mathematical Analysis 2; Univ. of Vienna lecture notes on Analysis 2.

Recommended Text:

The material of this course is standard. We shall use various chapters from the lecture notes listed above as complementary reading/study material.

Description:

Topology of Euclidean space; metric spaces; normed vector spaces; functions on normed vector spaces: continuity, differentiability; implicit and inverse function theorems; submanifolds; Calculus on submanifolds: integration of differentiable forms; closed and exact forms, deRham cohomology; change of variable formula; Stokes Theorem.

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

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:

Partial Differential Equations: An Introduction, 2nd Edition
by Strauss, Walter A.

Recommended Text:

A useful (but not necessary) additional text to consider: Introduction to Partial Differential Equations
by Peter J. Olver

Description:

The course will start with the study of transport equations, and an introduction to and classification of second-order partial differential equations and their applications. Subsequently, we will proceed to examine in more detail the wave equation, heat equation and Laplace equation. For instance, we will touch upon the D'Alembert solution to the wave equation, the solution of the heat equation, the maximum principle, energy methods, separation of variables, Fourier series and Fourier transform methods, as well as operator eigenvalue problems, starting with one spatial dimension and then considering generalizations of some of these ideas to higher dimensions.

Time-permitting, we will briefly examine numerical methods for partial differential equations and relevant implementation thereof, e.g., in Matlab, as well as some select examples of nonlinear partial differential equations and the traveling or standing wave solutions possible therein.

The final grade will be determined on the basis of attendance/in class participation, homework, an in-class midterm and a final exam.

### MATH 536: Actuarial Probability

Prerequisites:

Math 233 and Stat 515

Recommended Text:

ASM Study Manual for Exam P 5th or later Edition by Weihause. You can buy it online at https://www.studymanuals.com/Product/Show/453148820

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: Intro. to Math of Finance

Prerequisites:

Math 233 and either Stat 515 or MIE 273

Description:

This course is an introduction to the mathematical models used in finance and economics with particular emphasis on models for pricing financial instruments, or "derivatives." The central topic will be options, culminating in the Black-Scholes formula. The goal is to understand how the models derive from basic principles of economics, and to provide the necessary mathematical tools for their analysis.

### MATH 545.1: Linear Algebra for Applied Mathematics

Prerequisites:

Math 233 and 235, 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

Text:

Texts: 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 decomposition, Vector and Inner Product Spaces, Orthogonality and Least Squares, Determinants and Eigenvalues, Jordan form, Spectral theorem, symmetric positive definite matrices. Other decompositions 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 545.3: 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. Homeworks include programming projects.

### MATH 545.4: Linear Algebra for Applied Mathematics

Description:

Basic concepts (over real or complex numbers): vector spaces, basis, dimension, linear transformations and matrices, change of basis, similarity. Study of a single linear operator: minimal and characteristic polynomial, eigenvalues, invariant subspaces, triangular form, Cayley-Hamilton theorem. Inner product spaces and special types of linear operators (over real or complex fields): orthogonal, unitary, self-adjoint, hermitian. Diagonalization of symmetric matrices, applications. Homeworks include programming projects.

### MATH 551.1: Intr. Scientific Computing

Prerequisites:

MATH 233 & 235 and either COMPSCI 121, E&C-ENG 122, PHYSICS 281, or E&C-ENG 242

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

Text:

Elementary Numerical Analysis (Wiley, 3rd ed.) Atkinson & Han

Description:

The course will introduce basic numerical methods used for solving problems
that arise in a variety scientific fields. Properties such as accuracy of
methods, their stability and efficiency will be studied. We will cover
finite precision arithmetic and error propagation, linear systems of
equations, root finding, interpolation, approximation of functions,
numerical integration and numerical methods for differential equations.

Students will gain
practical programming experience in implementing the methods using MATLAB,
which is available for FREE after following the instructions on UMass OIT website.
The use of MATLAB for homework assignments is mandatory.
We will also discuss some important practical considerations of
implementing numerical methods using such languages as FORTRAN or C.

### MATH 551.2: Intr. Scientific Computing

Prerequisites:

MATH 233 & 235 and either COMPSCI 121, E&C-ENG 122, PHYSICS 281, or E&C-ENG 242

Knowledge of a programming language, e.g. C, C++, Java, Julia, or Python.

Text:

Timothy Sauer, Numerical Analysis, Third Edition, 978-0134696454.

Description:

A practical overview of computational methods used in science, statistics, industry, and machine learning. Topics will include: an introduction to python programming and software for scientific computing such as NumPy and LAPACK, numerical linear algebra, optimization and root-finding, approximation of functions by splines and trigonometric polynomials, and the Fast Fourier Transform. Applications may include regression problems in statistics, audio and image processing, and the calculation of properties of molecules. Homework will be assigned frequently. Each assignment will involve both mathematical theory and python programming. There will be no exams. Instead, each student will pursue an open-ended project related to a topic discussed in class.

### MATH 551.3: Intr. Scientific Computing

Prerequisites:

MATH 233 & 235 and either COMPSCI 121, E&C-ENG 122, PHYSICS 281, or E&C-ENG 242

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

Description:

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

### MATH 552: Applications of Scientific Computing

Prerequisites:

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

Knowledge of a programming language, e.g. C, C++, Java, Julia, or Python.

Text:

None required.

Description:

Introduction to the application of computational methods to models arising in science and engineering, concentrating mainly the numerical solution of ordinary, partial differential equations, and stochastic simulations of particle systems. Topics include finite differences, finite elements, spectral methods, boundary value problems, ODE integrators, and fast Fourier transforms. If time permits we will discuss nonlinear optimisation methods with applications to data science.

Applications and examples will be selected from biomedical engineering, cell biology, and population dynamics.

### MATH 557: Linear Optimization and Polytopes

Prerequisites:

Math 235, Math 300 or CS 250

Description:

This proof-based course covers the fundamentals of linear optimization and polytopes and the relationship between them. The course will give a rigorous treatment of the algorithms used in linear optimization. The topics covered in linear optimization are graphical methods to find optimal solutions in two and three dimensions, the simplex algorithm, duality and Farkas' lemma, variation of cost functions, an introduction to integer programming and Chvatal-Gomory cuts. The topics covered simultaneously in polytopes are two- and three-dimensional polytopes, f-vectors, equivalence of the vertex and hyperplane descriptions of polytopes, the Hirsch conjecture, the secondary polytope, and an introduction to counting lattice points of polytopes.

### MATH 563H: Differential Geometry

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.

### MATH 597F: Fourier Methods

Prerequisites:

MATH 235, 300, and 331

Text:

MWF: 11:15 am - 12:05 pm

Recommended Text:

Fourier Analysis by T.W. Korner (Cambridge University Press)

Description:

The course introduces and uses Fourier series and Fourier transform as a tool to understand varies important problems in applied mathematics: linear ODE & PDE, time series, signal processing, etc. We'll treat convergence issues in a non-rigorous way, discussing the different types of convergence without technical proofs. Topics: complex numbers, sin & cosine series, orthogonality, Gibbs phenomenon, FFT, applications, including say linear PDE, signal processing, time series, etc; maybe ending with (continuous) Fourier transform.

### STAT 310: Fundamental Concepts of Statistics

Prerequisites:

Math 132

Description:

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

### STAT 501: Methods of Applied Statistics

Prerequisites:

Knowledge of high school algebra, junior standing or higher

Description:

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

### STAT 515.1: Statistics I

Prerequisites:

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

Text:

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

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

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.

Text:

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

Description:

This course provides a calculus-based introduction to probability (an emphasis on probabilistic concepts used in statistical modeling) and the beginning of statistical inference (continued in Stat516). Coverage includes basic axioms of probability, sample spaces, counting rules, conditional probability, independence, random variables (and various associated discrete and continuous distributions), expectation, variance, covariance and correlation, probability inequalities, the central limit theorem, the Poisson approximation, and sampling distributions. Introduction to basic concepts of estimation (bias, standard error, etc.) and confidence intervals.

### STAT 515.3: Statistics I

Prerequisites:

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

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

### STAT 515.4: Statistics I

Prerequisites:

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

Text:

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

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

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.

Text:

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

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

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.

Text:

WebAssign for Mathematical Statistics with Applications
ISBN: 9781337901185

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

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.

Text:

WebAssign for Mathematical Statistics with Applications
ISBN: 9781337901185

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 516.2: Statistics II

Prerequisites:

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

Description:

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

### STAT 516.3: Statistics II

Prerequisites:

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

Description:

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

### STAT 516.4: Statistics II

Prerequisites:

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

Description:

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

### STAT 525.1: Regression Analysis

Prerequisites:

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

Description:

Regression analysis is the most popularly used statistical technique with application in almost every imaginable field. The focus of this course is on a careful understanding and of regression models and associated methods of statistical inference, data analysis, interpretation of results, statistical computation and model building. Topics covered include simple and multiple linear regression; correlation; the use of dummy variables; residuals and diagnostics; model building/variable selection; expressing regression models and methods in matrix form; an introduction to weighted least squares, regression with correlated errors and nonlinear regression. Extensive data analysis using R or SAS (no previous computer experience assumed).  Requires prior coursework in Statistics, preferably  ST516, and basic matrix algebra.  Satisfies the Integrative Experience requirement for BA-Math and BS-Math majors.

### STAT 525.2: Regression Analysis

Prerequisites:

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

Description:

Regression analysis is the most popularly used statistical technique with application in almost every imaginable field. The focus of this course is on a careful understanding and of regression models and associated methods of statistical inference, data analysis, interpretation of results, statistical computation and model building. Topics covered include simple and multiple linear regression; correlation; the use of dummy variables; residuals and diagnostics; model building/variable selection; expressing regression models and methods in matrix form; an introduction to weighted least squares, regression with correlated errors and nonlinear regression. Extensive data analysis using R or SAS (no previous computer experience assumed). Requires prior coursework in Statistics, preferably ST516, and basic matrix algebra. Satisfies the Integrative Experience requirement for BA-Math and BS-Math majors.

### STAT 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 analysis is the most popularly used statistical technique with application in almost every imaginable field. The focus of this course is on a careful understanding and of regression models and associated methods of statistical inference, data analysis, interpretation of results, statistical computation and model building. Topics covered include simple and multiple linear regression; correlation; the use of dummy variables; residuals and diagnostics; model building/variable selection; expressing regression models and methods in matrix form; an introduction to weighted least squares, regression with correlated errors and nonlinear regression. Extensive data analysis using R or SAS (no previous computer experience assumed). Requires prior coursework in Statistics, preferably ST516, and basic matrix algebra. Satisfies the Integrative Experience requirement for BA-Math and BS-Math majors.

### STAT 526: Design Of Experiments

Prerequisites:

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

Note:

This course may be taken remotely. Please enroll and contact the instructor if you would like to take the course remotely.
This class meets on the Newton Mount Ida Campus of UMass-Amherst.

Description:

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 programming software SAS (no prior SAS experience assumed).

### STAT 535: Statistical Computing

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: Analysis of Discrete Data

Prerequisites:

Stat 525 or equivalent, and consent of instructor.

Text:

TBA

Recommended Text:

1. Friendly, M. and Meyer, D. (2016). Discrete Data Analysis with R: Visualization and Modeling Techniques for Categorical and Count Data. Chapman & Hall.
2. Agresti, A. (2013). Categorical Data Analysis, 3rd ed., NY: Wiley.
3. Agresti, A. (2007). Introduction to Categorical Data Analysis, 2nd ed., NY: Wiley.

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)

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

Prerequisites:

Math 611 (or consent of the instructor).

Description:

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

### MATH 621: Complex Analysis

Prerequisites:

Advanced Calculus. Students are expected to have a working knowledge of complex numbers and functions at the level of Math 421 for example.

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; other topics as time permits.

### MATH 624: Real Analysis II

Prerequisites:

Math 523H, Math 524 and Math 623.

Text:

Books by Stein and Shakarchi Volumes 3 and 4 (Princeton Press)

Recommended Text:

Folland's book

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 L2 theory; compact operators; the Fourier transform; Banach spaces; elementary operator theory and linear functionals; Lp 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) of Stein-Shakarchi’s Real Analysis book (Vol. III) (namely having a working knowledge of Measure theory: Lebesgue measure and Integrable functions; Integration theory: Lebesgue integral, convergence theorems and Fubini theorem; Differentiation theorems, covering lemmas, approximation to the identity, functions of bounded variation, etc as well as a working knowledge of undergraduate Analysis (as for example taught in classes like M523H and M524H).

### MATH 646: Applied Math and Math Modeling

Prerequisites:

Math 523 or equivalent, or permission of the instructor. Knowledge of basic coding is required. Python is preferred, MATLAB is another option.

Recommended Text:

1. The Koopman Operator Theory in Systems and Control: Concepts, Methodologies, and Applications. Eds.: A. Mauroy, I. Mezic, and Y. Susuki. Lectures Notes in Control and Information Sciences, Springer, 2020.

2. Data-Driven Science and Engineering Machine Learning, Dynamical Systems, and Control ,
DOI: https://doi.org/10.1017/9781108380690.008
Publisher: Cambridge University Press
Print publication year: 2019

Description:

This course covers classical methods in applied mathematics and math modeling, including dynamical systems, as well as stochastic processes generated by dynamical systems.
The main topics include: Foundation of modern dynamical systems, I also plan to add an extra topic about data-driven machine learning method of dynamical systems, using Dynamic Mode Decomposition (DMD), Dimensionality Reduction, and deep Koopman Operator method to learn nonlinear dynamical systems. The knowledge of computer programing is recommended but not required. For Machine learning projects, students will be grouped into a team of 2-3, in which at least one of the student will be responsible for coding. The techniques will be applied to data-driven models arising throughout the natural sciences.

### MATH 652: Int Numerical Analysis II

Prerequisites:

Math 651 or permission of the instructor

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. Prerequisites: Math 651, familiarity with partial differential equations.

### MATH 672: Algebraic Topology

Prerequisites:

Math 671, Math 611 or equivalent.

Text:

Allen Hatcher, Algebraic Topology. Available to download as a pdf from the author's web page: https://pi.math.cornell.edu/~hatcher/AT/ATpage.html

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, or equivariant cohomology.

### MATH 690STA: Math Theory of Machine Learning Part II

Prerequisites:

It is assumed that students enrolling in this course have taken MATH 697MA: Mathematical Theory of Machine Learning Part I in Fall 2022.

Note:

This course is not yet loaded into Spire - the department is working on it and will have it in Spire as soon as possible.

Description:

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

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

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

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

The expected outcome of this course is to prepare students with solid mathematical background of modern machine learning, and to get students engaged with new research topics in this area.

This course complements some earlier courses on machine learning and data sciences, such as MATH 697PA: ST-Math Foundtns/ProbabilistAI and STAT 697ML: ST- Stat Machine Learning.

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

Prerequisites:

Graduate Student in Applied Math MS Program

Description:

This course is the group project that is required for the MS program in Applied Mathematics. Each academic year we undertake an in-depth study of select applied science problems, combining modeling, theory, and computation to understand it. The main goal of the course is to emulate the process of teamwork in problem solving, such as is the norm in industrial applied mathematics.

### MATH 697CM: Combinatorial Optimization

Prerequisites:

Math 235, Math 455 or Math 513/CS 575

Text:

There is no mandatory textbook. Lecture notes will be available on moodle.

Recommended Text:

For additional information, I recommend the following textbooks:
Combinatorial Optimization by Cook, Cunningham, Pulleyblank, Schriver
Combinatorial Optimization: Theory and Algorithms by Korte and Vygen
A First Course in Combinatorial Optimization by Lee
Combinatorial Optimization: Polyhedra and Efficiency by Schrijver

Description:

In this course, we will consider maximization and minimization problems in graphs and networks. We will cover a broad range of topics such as matchings in bipartite graphs and in general graphs, assignment problem, polyhedral combinatorics, total unimodularity, matroids, matroid intersection, min arborescence, max flow;min cut, max cut, traveling salesman problem, stable sets and perfect graphs. One of our main tools will be integer programming, and we will also sometimes rely on semidefinite programming. Many of these problems come from real-world applications, so we will also sometimes discuss the algorithms necessary to solve them. This is a rigorous mathematical introduction to combinatorial optimization with proofs.

### MATH 697U: Stochastic Processes and Applications

Prerequisites:

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

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 704: Topics In Geometry II

Description:

This course aims to give an introduction to the fundamental topics in modern differential geometry, as organized in the following five units.

1. Theory of fiber bundles and connections.
2. Characteristics classes via Chern-Weil theory.
3. Riemannian geometry.
4. Hermitian and Kahler geometry.
5. Hodge theory.

We will present the basic concepts and theorems in each unit listed above, illustrated with interesting examples and detailed proofs of some selected results to demonstrate the various basic techniques in these subjects. We also hope to enhance the learning experience with homework assignments/projects, which form the basis of the course grade. Lecture notes will be provided.

### MATH 790STA: Abelian Varieties

Prerequisites:

Math 611 and 612 or equivalent, Math 621 and 671 or equivalent, Math 703 or equivalent, and a graduate level course in algebraic geometry (such as Math 797 or 708 covering divisors, line bundles, differential forms, and the Riemann-Roch theorem for curves). Familiarity with Cech, de Rham, and Dolbeault cohomology will be assumed.

Note:

In Spire, search for Math S790 - it appears alphabetically at the beginning of the Math course listings (before Math 100)

Description:

Topic covered will include: Line bundles on complex tori and their cohomology, theta functions, the Riemann conditions for a complex torus to be an abelian variety, the dual abelian variety and the Poincare line bundle, the Riemann-Roch theorem, Lefschetz theorem and embeddings in projective space, endomorphisms of abelian varieties, curves and their jacobians, Fourier-Mukai transformations.

### MATH 790STB: Probabilistic Methods in Nonlinear Evolution Equations

Prerequisites:

Math 623-624. Basic knowledge of probability such as that taught in Math 605. Basic knowledge of PDEs, including solving via Banach fixed point argument/Picard iterations.
Acquaintance with distributions, Sobolev spaces and weak convergence of measures would also be useful.
Consult with the instructor if needed

Note:

This course is not yet loaded into Spire - the department is working on it and will have it in Spire as soon as possible.

Description:

The study of randomness in partial differential equations (PDEs) goes back more than seventy years and
include examples such as the modeling of random vibrations of strings, or the scattering of waves by objects
that are imbedded in random media.
Nonlinear dispersive PDEs naturally appear as models describing wave phenomena in quantum mechanics,
nonlinear optics, plasma physics, water waves, and atmospheric sciences. Due to their ubiquitousness
they have been at the center of profound research both from the applied community as well as from the
theoretical one. One way in which randomness enters the field of nonlinear dispersive PDE is via the
random data Cauchy initial value problem for (deterministic) equations, such as the nonlinear Schr¨odinger
(NLS) and the nonlinear wave equations (NLW). The interest comes from two fundamental problems:
(1) invariance of measures such as Gibbs measures which are physical equilibria for these systems;
arising naturally in statistical mechanics and closely related also to QFT models such as the  4, and
(2) the study of generic behavior of solutions in the probabilistic sense, and how they are expected to be
better than worst case (exceptional) scenarios.
The study of this subject in the context of dispersive PDEs can be traced back to Lebowitz–Rose–Speer
(1988, 1989) and Bourgain (1994, 1996) concerning the Gibbs measure for NLS. Since then there have been
substantial developments of their ideas by many different researchers, extending them in different directions
(geometric, infinite volume, other dispersive relations). In recent years, especially since 2018, this field
has seen significant progress and many new ideas and methods have been introduced that go beyond the
original ideas of Bourgain, including hyperbolic versions of paracontrolled calculus (Gubinelli–Koch–Oh
and Bringmann), the method of random averaging operators and the theory of random tensor (both by
Deng–Nahmod–Yue). These new methods have led to the resolution of several important open questions in
this field, and are expected to play more important roles in future developments. The aim of this course is to
provide the foundations upon which these recent developments have built upon, and in particular have a
working knowledge of Bourgain’s seminal works in the subject.

### MATH 797W: Algebraic Geometry

Prerequisites:

Math 611/612 or consent of instructor.

Description:

Algebraic geometry is the study of geometric spaces locally defined by polynomial equations. It is a central subject in mathematics with strong connections to differential geometry, number theory, and representation theory. This course will be a fast-paced introduction to the subject with a strong emphasis on examples.

In the algebraic approach to the subject, local data is studied via the commutative algebra of quotients of polynomial rings in several variables. Passing from local to global data is delicate (as in complex analysis) and is either accomplished by working in projective space (corresponding to a graded polynomial ring) or by using sheaves and their cohomology.

Topics will include projective varieties and schemes, singularities, differential forms, line bundles and sheaves, and sheaf cohomology, including the Riemann--Roch theorem and Serre duality for algebraic curves.

Prerequisites: Commutative algebra (rings and modules) as covered in 611-612. Some prior experience of manifolds would be useful (but not essential).

### STAT 608.1: Mathematical Statistics II

Prerequisites:

STAT 607 or permission of the instructor.

Text:

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

Recommended Text:

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

Description:

This is the second part of a two semester sequence on probability and mathematical statistics. Stat 607 covered probability, basic statistical modelling, and an introduction to the basic methods of statistical inference with application to mainly one sample problem. In Stat 608 we 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 decision theoretic approaches. The theory is utilized in addressing problems in nonparametric methods, two and multi-sample problems, and categorial, regression and survival models. As with Stat 607 this is primarily a theory course emphasizing fundamental concepts and techniques.

### STAT 608.2: Mathematical Statistics II

Prerequisites:

STAT 607 or equivalent, or permission of the instructor.

Note:

This course may be taken remotely. Please enroll and contact the instructor if you would like to take the course remotely.
This class meets on the Newton Mount Ida Campus of UMass-Amherst.

Description:

This 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, additional probability topics and examine further issues in methods of inference including likelihood based methods, optimal methods of inference, large sample methods, Bayesian inference and Resampling methods. The theory is utilized in addressing problems in parametric/nonparametric methods, two and multi-sample problems, various hypothesis testing and regression. As with Stat 607, this is primarily a theory course emphasizing fundamental concepts and techniques.

### STAT 610: Bayesian Statistics

Prerequisites:

Graduate students only. A one-year graduate level calculus-based statistical theory course such as STAT 607-608 or the equivalent is required, experience with regression at the level of STAT 625 is required, knowledge of matrix algebra, and prior experience with R including coding and data analysis (for example, at the level of STAT 535). Stat 515-516 is not a sufficient prerequisite for this course.

Text:

Required Textbook: Bayesian Methods for Data Analysis, 3rd Edition, Carlin and Louis (2008), Taylor and Francis/CRC Press. ISBN Number: 9781584886976.

Note:

This course may be taken remotely. Please enroll and contact the instructor if you would like to take the course remotely.
This class meets on the Newton Mount Ida Campus of UMass-Amherst.

Description:

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

### STAT 690STA: Applied Semiparametric Regression

Prerequisites:

Stat 625 or permission of the instructor. We will use R a lot. Highly motivated undergraduates who have taken 525 are welcome too.

Text:

Required book: Semiparametric Regression with R (Harezlak, Ruppert, and Wand)

Note:

This course is not yet loaded into Spire - the department is working on it and will have it in Spire as soon as possible.

Description:

Using data to estimate relationships between predictors and responses is an important task in statistics and data science.

When datasets are large, modern methods have been developed that allow us to estimate those relationships without making strong assumptions about those relationships- i.e we can let the data determine how E(y|x) relates to x. In statistics, these methods are generally referred to as “nonparametric regression.”

This applied graduate course will focus on learning to use nonparametric regression to analyze data. We will read a book, “Semiparametric Regression with R,” and implement / understand the methods in that book. We will address simple and multiple regression data, binary/count data, spatial data, and correlated/time series data. The course will require a substantial project that will be done in a group of size 3 or more.

### STAT 697MV: Applied Multivariate Statistics

Prerequisites:

Open to Graduate Students only. Undergraduates may enroll with permission of instructor.

Prerequisites: Probability and Statistics at a calculus-based level such as Stat 607 and Stat 608 (concurrent) or Stat 515 and Stat 516 (concurrent). Students must have prior experience with a statistical programming language such as R, Python or MATLAB.

Note:

This course may be taken remotely. Please enroll and contact the instructor if you would like to take the course remotely.
This class meets on the Newton Mount Ida Campus of UMass-Amherst.

Description:

This course provides an introduction to the more commonly-used multivariate statistical methods. Topics include principal component analysis, factor analysis, clustering, discrimination and classification, multivariate analysis of variance (MANOVA), and repeated measures analysis. The course includes a computing component.

### STAT 697V: Data Visualization

Prerequisites:

Open to Graduate Students only. Undergraduates may enroll with permission of instructor.

Prerequisites: Probability and Statistics at a calculus-based level such as Stat 607 and Stat 608 (concurrent) or Stat 515 and Stat 516 (concurrent). Students must have prior experience with a statistical programming language such as R, Python or MATLAB.

Note:

This course may be taken remotely. Please enroll and contact the instructor if you would like to take the course remotely.
This class meets on the Newton Mount Ida Campus of UMass-Amherst.

Description:

The increasing production of descriptive data sets and corresponding software packages has created a need for data visualization methods for many application areas. Data visualization allows for informing results and presenting findings in a structured way. This course provides an introduction to graphical data analysis and data visualization. Topics include exploratory data analysis, data cleaning, examining features of data structures, detecting unusual data patterns, and determining trends. The course will also introduce methods to choose specific types of graphics tools and understanding information provided by graphs.