Course Descriptions

Lower Division Courses

MATH 011: Elementary Algebra

See Preregistration guide for instructors and times

Description:

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

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

MATH 100: Basic Math Skills for the Modern World

See Preregistration guide for instructors and times

Description:

Topics in mathematics that every educated person needs to know to process, evaluate, and understand the numerical and graphical information in our society. Applications of mathematics in problem solving, finance, probability, statistics, geometry, population growth. Note: This course does not cover the algebra and pre-calculus skills needed for calculus.

MATH 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.

Description:

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

MATH 102: Analytic Geometry and Trigonometry

See Preregistration guide for instructors and times

Prerequisites:

Math 101

Description:

Second semester of the two-semester sequence MATH 101-102. Detailed treatment of analytic geometry, including conic sections and exponential and logarithmic functions. Same trigonometry as in MATH 104.

MATH 103: Precalculus and Trigonometry

See Preregistration guide for instructors and times

Prerequisites:

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

Description:

The trigonometry topics of MATH 104.

MATH 104: Algebra, Analytic Geometry and Trigonometry

See Preregistration guide for instructors and times

Prerequisites:

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

Description:

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

MATH 113: Math for Elementary Teachers I

See Preregistration guide for instructors and times

Prerequisites:

MATH 011 or satisfaction of R1 requirement.

Description:

Fundamental and relevant mathematics for prospective elementary school teachers, including whole numbers and place value operations with whole numbers, number theory, fractions, ratio and proportion, decimals, and percents. For Pre-Early Childhood and Pre-Elementary Education majors only.

MATH 114: Math for Elementary Teachers II

See Preregistration guide for instructors and times

Prerequisites:

MATH 113

Description:

Various topics that might enrich an elementary school mathematics program, including probability and statistics, the integers, rational and real numbers, clock arithmetic, diophantine equations, geometry and transformations, the metric system, relations and functions. For Pre-Early Childhood and Pre-Elementary Education majors only.

MATH 121: Linear Methods and Probability for Business

See Preregistration guide for instructors and times

Prerequisites:

Working knowledge of high school algebra and plane geometry.

Description:

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

MATH 127: Calculus for Life and Social Sciences I

See Preregistration guide for instructors and times

Prerequisites:

Proficiency in high school algebra, including word problems.

Description:

Basic calculus with applications to problems in the life and social sciences. Functions and graphs, the derivative, techniques of differentiation, curve sketching, maximum-minimum problems, exponential and logarithmic functions, exponential growth and decay, and introduction to integration.

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

See Preregistration guide for instructors and times

Prerequisites:

Proficiency in high school algebra, including word problems.

Description:

Honors section of Math 127.

MATH 128: Calculus for Life and Social Sciences II

See Preregistration guide for instructors and times

Prerequisites:

Math 127

Description:

Continuation of MATH 127. Elementary techniques of integration, introduction to differential equations, applications to several mathematical models in the life and social sciences, partial derivatives, and some additional topics.

MATH 131: Calculus I

See Preregistration guide for instructors and times

Prerequisites:

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

Description:

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

MATH 131H: Honors Calculus I

See Preregistration guide for instructors and times

Prerequisites:

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

Description:

Honors section of Math 131.

MATH 132: Calculus II

See Preregistration guide for instructors and times

Prerequisites:

Math 131 or equivalent.

Description:

The definite integral, techniques of integration, and applications to physics, chemistry, and engineering. Sequences, series, and power series. Taylor and MacLaurin series. Students expected to have and use a Texas Instruments 86 graphics, programmable calculator.

MATH 132H: Honors Calculus II

See Preregistration guide for instructors and times

Prerequisites:

Math 131 or equivalent.

Description:

Honors section of Math 132.

MATH 233: Multivariate Calculus

See Preregistration guide for instructors and times

Prerequisites:

Math 132.

Description:

Techniques of calculus in two and three dimensions. Vectors, partial derivatives, multiple integrals, line integrals.

MATH 233H: Honors Multivariate Calculus

See Preregistration guide for instructors and times

Prerequisites:

Math 132.

Description:

Honors section of Math 233.

MATH 235: Introduction to Linear Algebra

See Preregistration guide for instructors and times

Prerequisites:

Math 132 or consent of the instructor.

Description:

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

MATH 235H: Honors Introduction to Linear Algebra

See Preregistration guide for instructors and times

Prerequisites:

Math 132 or consent of the instructor.

Description:

Honors section of Math 235.

MATH 331: Ordinary Differential Equations for Scientists and Engineers

See Preregistration Guide for instructors and times

Prerequisites:

Math 132 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

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

Luca Schaffler 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

Owen Gwilliam MWF 9:05-9:55

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 basic logic (truth tables, negation, quantifiers), set theory (Cantor's notion of size for sets and gradations of infinity, maps between sets, equivalence relations, partitions of sets), and elementary 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

Owen Gwilliam MWF 10:10-11:00

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 basic logic (truth tables, negation, quantifiers), set theory (Cantor's notion of size for sets and gradations of infinity, maps between sets, equivalence relations, partitions of sets), and elementary 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 300.4: Fundamental Concepts of Mathematics

Luca Schaffler TuTh 11:30-12:45

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 370.1: Writing in Mathematics

Mark Wilson MWF 12:20-1:10

Prerequisites:

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

Description:

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

MATH 370.2: Writing in Mathematics

Mark Wilson MWF 1:25-2:15

Prerequisites:

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

Description:

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

MATH 370.3: Writing in Mathematics

Franz Pedit TuTh 11:30-12:45

Prerequisites:

MATH 300 or CS 250 and completion of the College Writing (CW) requirement.
In addition, a solid background of the Calculus cycle, specially Calc III, and Linear Algebra is needed.

Recommended Text:

The Poincare Conjecture by Donal O'Shea.
Flatland: A Romance of Many Dimensions by Edwin A. Abbott.
Texts assigned during the semester.

Description:

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

MATH 411.1: Introduction to Abstract Algebra I

R. Inanc Baykur TuTh 10:00-11:15

Prerequisites:

Math 235; Math 300 or CS 250.

Description:

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

MATH 411.2: Introduction to Abstract Algebra I

Laura Colmenarejo MWF 11:15-12:05

Prerequisites:

MATH 235; MATH 300 or CS 250

Text:

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

Description:

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

MATH 421: Complex Variables

Eyal Markman MWF 11:15-12:05

Prerequisites:

Math 233

Text:

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

Description:

An introduction to functions of a complex variable. Topics include: Complex numbers, functions of a complex variable and their derivatives (Cauchy-Riemann equations). Harmonic functions. Contour integration and Cauchy's integral formula. Liouville's theorem, Maximum modulus theorem, and the Fundamental Theorem of Algebra. Taylor and Laurent series. Classification of isolated singularities. Evaluation of Improper integrals via residues. Conformal mappings.

MATH 425.1: Advanced Multivariate Calculus

Jonathan Simone MWF 9:05-9:55

Prerequisites:

Multivariable Calculus (MATH 233) and Linear Algebra (MATH 235).

Text:

Vector Calculus by Marsden and Tromba, 5th Ed., W. H. Freeman ISBN-10: 0716749920; ISBN-13: 978-0716749929

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

Rob Kusner MW 2:30-3:45

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

Jonathan Simone MWF 10:10-11:00

Prerequisites:

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

Text:

Vector Calculus by Marsden and Tromba, 5th Ed., W. H. Freeman ISBN-10: 0716749920; ISBN-13: 978-0716749929

Description:

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

MATH 437: Actuarial Financial Math

Jinguo Lian MWF 1:25-2:15

Prerequisites:

Math 131 and 132 or equivalent courses with C or better

Text:

ASM Study Manual for Exam FM, 13th or later Edition by Cherry & Shaban ISBN: 978-1-63588-452-4.

Note:

A BA II plus Calculator is required for the course.

Description:

This 3 credit hours course serves as a preparation for SOA's second actuarial exam in financial mathematics, known as Exam FM or Exam 2. The course provides an understanding of the fundamental concepts of financial mathematics, and how those concepts are applied in calculating present and accumulated values for various streams of cash flows as a basis for future use in: reserving, valuation, pricing, asset/liability management, investment income, capital budgeting, and valuing contingent cash flows. The main topics include time value of money, annuities, loans, bonds, general cash flows and portfolios, immunization, interest rate swaps and determinants of interest rates etc. Many questions from old exam FM will be practiced in the course.

MATH 455: Introduction to Discrete Structures

Tom Braden MW 2:30-3:45

Prerequisites:

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

Text:

Harris, Hirst, and Mossinghoff, Combinatorics and Graph Theory, 2nd edition (can be downloaded as a pdf from the UMass library)

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

Qian-Yong Chen TuTh 8:30-9:45

Prerequisites:

Math 233 and Math 235 (and Differential Equations, Math 331, is recommended). Some familiarity with a programming language is desirable (Mathematica, Matlab or similar)

Text:

Not following any particular book.

Recommended Text:

Some reference book:

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 456.2: Mathematical Modeling

Annie Raymond MWF 12:20-1:10

Prerequisites:

Math 233 and Math 235 (and Differential Equations, Math 331, is recommended). Some familiarity with a programming language is desirable (Mathematica, Matlab or similar)

Description:

This course is an introduction to mathematical modeling. The main goal of the class is to learn how to translate real-world problems into quantitative terms for interpretation, suggestions of improvement and future predictions. Since this is too broad of a topic for one semester, this class will focus on linear and integer programming to study such problems. The course will culminate in a final modeling project that will involve optimizing different logistical aspects of the ValleyBike Share service.

MATH 461: Affine and Projective Geometry

Paul Hacking MWF 10:10-11:00

Prerequisites:

Math 235 and Math 300

Text:

J. Stillwell, The Four Pillars of Geometry, Springer Verlag, 2005 (available for free download from UMass libraries.)

Description:

There are three types of surfaces which look the same at every point and in every direction: the plane, the sphere, and the hyperbolic plane. The hyperbolic plane is a remarkable surface in which the circumference of a circle grows exponentially as the radius increases; it was only discovered in the 18th century. We will begin by studying the geometry of the plane and the sphere and their symmetries. Then we will describe and study the hyperbolic plane. The emphasis will be on developing our geometric intuition in each case.

MATH 471: Theory of Numbers

Tom Weston TuTh 8:30-9:45

Prerequisites:

Math 233 and 235. Math 300 or CS 250 as a co-requisite is not absolutely necessary but highly recommended.

Text:

Number Theory, A lively Introduction with Proofs, Applications, and Stories, by James Pommersheim, Tim Marks and Erica Flapan.

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 490A: Intro Abstract Algebra for Future Teachers

George Avrunin TuTh 1:00-2:15

Prerequisites:

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

Text:

TBD

Description:

Abstract algebra forms a key part of the ideas behind high school mathematics and is the basis for several parts of the Massachusetts Test for Educator Licensure for secondary school math teachers. This course will cover the parts of abstract algebra most important for building a deep understanding of the ideas of high school mathematics and their interconnections. It will focus on the properties of rings (especially the integers and polynomial rings over fields), and fields. During the course, we will be making connections between these topics and high school mathematics.

MATH 491A: Seminar: Putnam Exam Preparation (1 credit)

TBD Wed 4:00-4:50

Prerequisites:

One variable Calculus, Linear Algebra

Description:

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

MATH 491P: S - Problem Seminar (1 credit)

TBD Fri 2:55-4:00

Prerequisites:

Required Prerequisites: Math 233, 235, and 300. Suggested Prerequisites: Math 331 (completed or currently taking); Math 411 or Math 523H (completed or currently taking)

Note:

The GRE Mathematics subject exam is offered every April, September, and October. We will meet 6-7 times in September and October, with the course ending before the October exam date.

Students should have already completed, or be currently taking Math 331.
Students should have already completed, or be currently taking Math 411 or Math 523H.

Description:

This one-credit course is designed to help students review and prepare for the GRE Mathematics subject exam, which is a required exam for entrance into many PhD programs in mathematics. Students should have completed the three courses in calculus, a course in linear algebra, and have some familiarity with differential equations. The focus will be on solving problems based on the core material covered in the exam. Students are expected to do practice problems before each meeting and discuss the solutions in class.

MATH 523H: Introduction to Modern Analysis

Siddhant Agrawal TuTh 10:00-11:15

Prerequisites:

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 532H: Nonlinear Dynamics

Jeremiah Birrell TuTh 1:00-2:15

Prerequisites:

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

Text:

Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, 2nd edition by Steven H. Strogatz, Westview Press, 2014

Description:

This course provides an introduction to systems of differential equations and dynamical systems, and will touch upon chaotic dynamics, while providing a significant set of connections with phenomena modeled through these approaches in Physics, Chemistry and Biology. From the mathematical perspective, geometric and analytic methods of describing the behavior of solutions will be developed and illustrated in the context of low-dimensional systems, including behavior near fixed points and periodic orbits, phase portraits, Lyapunov stability, Hamiltonian systems, bifurcation phenomena, and concluding with chaotic dynamics. In addition to the theoretical component, a self-contained computational component towards addressing these systems will be developed with the assistance of Matlab.

MATH 537.1: Intro to Mathematics of Finance

Mike Sullivan MWF 12:20-1:10

Prerequisites:

Single-variable calculus (Math 131, 132), Probability with calculus (Stats 515), multi-variable calculus up to the level of the chain rule for partial derivatives (Math 233).

Text:

Free set of course notes.

Description:

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

MATH 537.2: Intro to Mathematics of Finance

Mike Sullivan MWF 11:15-12:05

Prerequisites:

Single-variable calculus (Math 131, 132), Probability with calculus (Stats 515), multi-variable calculus up to the level of the chain rule for partial derivatives (Math 233).

Text:

Free set of class notes.

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

Noriyuki Hamada MWF 11:15-12:20

Prerequisites:

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

Description:

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

MATH 545.2: Linear Algebra for Applied Mathematics

Noriyuki Hamada MWF 12:20-1:10

Prerequisites:

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

Description:

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

MATH 551.1: Intr. Scientific Computing

Robin Young MWF 12:20-1:10

Prerequisites:

MATH 233 and 235 and CS 121 or knowledge of a scientific programming language, such as Java, C, Python, or Matlab.

Text:

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

Note: The textbook is free from SIAM for students.

Description:

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

MATH 551.2: Intr. Scientific Computing

Hans Johnston MWF 1:25-2:15

Prerequisites:

MATH 233 and 235 and CS 121 or knowledge of a scientific programming language, such as Java, C, Python, or MATLAB.

Text:

Elementary Numerical Analysis (3rd Edition) by Kendall Atkinson and Weimin Han
(ISBN-13: 978-0471433378)

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 increasingly complex codes over the term, with examples in class and students homework assignments. From time to time we will also discuss practical considerations of implementing these methods on modern computer architectures using C, C++ or Fortran.

MATH 551.3: Intr. Scientific Computing

Hans Johnston MWF 2:30-3:20

Prerequisites:

MATH 233 and 235 and CS 121 or knowledge of a scientific programming language, such as Java, C, Python, or MATLAB.

Text:

Elementary Numerical Analysis (3rd Edition) by Kendall Atkinson and Weimin Han
(ISBN-13: 978-0471433378)

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 increrasingly complex codes over the term, with examples in class and students homework assignments. From time to time we will also discuss practical considerations of implementing these methods on modern computer architectures using C, C++ or Fortran.

Math 591CF: S-Cybersecurity Lecture Series (1 cr)

TBD Wed 1:25-2:15

Description:

This course is a one-credit seminar on security research across departments at UMass. Each presentation will cover an active research topic at UMass in a way that assumes only a basic background in security. External speakers may also be invited. Note that this course is not intended to be an introduction to cybersecurity, and will not teach the fundamentals of security in a way that would be useful as a foundation for future security coursework. The intended audience is graduate and advanced undergraduate students, as well as faculty. Meets with COMPSCI 591CF and E&C-ENG 591CF. May be taken repeatedly for credit up to 2 times. This course does not count toward any requirements for the Math major or minor.

STAT 501: Methods of Applied Statistics

Joanna Jeneralczuk TuTh 11:30-12:45

Prerequisites:

Knowledge of high school algebra, junior standing or higher.

Recommended Text:

Introduction to Probability and Statistics, by Mendenhall, Beaver and Beaver, 14th edition, Publishers: Brooks/Cole

Description:

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

STAT 515.1: Introduction to Statistics I

Jiayu Zhai TuTh 10:00-11:15

Prerequisites:

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

Description:

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

STAT 515.2: Introduction to Statistics I

Panagiota Birmpa MWF 9:05-9:55

Prerequisites:

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

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: Introduction to Statistics I

Panagiota Birmpa MWF 10:10-11:00

Prerequisites:

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

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: Introduction to Statistics I

Yao Li TuTh 1:00-2:15

Prerequisites:

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

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 515.5: Introduction to Statistics I

Yao Li TuTh 2:30-3:45

Prerequisites:

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

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 515.6: Introduction to Statistics I

Jiayu Zhai TuTh 11:30-12:45

Prerequisites:

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

Description:

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

STAT 516.1: Statistics II

TBD 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

TBD TuTh 1:00-2:15

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 516.3: Statistics II

TBD TuTh 2:30-3:45

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

Zheni Utic 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. Stat 515 is NOT sufficient background for this course. Familiarity with basic matrix notation and operations is helpful.

Text:

Applied Linear Regression Models, by Kutner, Nachtsheim and Neter (4th edition) or Applied Linear Statistical Models by Kutner, Nachtsheim, Neter and Li (5th edition). Both published by McGraw-Hill/Irwin. The first 14 chapters of Applied Linear Statistical Models (ALSM) are EXACTLY equivalent to the 14 chapters that make up Applied Linear Regression Models, 4th ed., with the same pagination.

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

Zheni Utic TuTh 10:00-11:15

Prerequisites:

Stat 516 or equivalent: Previous coursework in Probability and Statistics, including knowledge of estimation, confidence intervals, and hypothesis testing and its use in at least one and two sample problems. Stat 515 is NOT sufficient background for this course. Familiarity with basic matrix notation and operations is helpful.

Text:

Applied Linear Regression Models, by Kutner, Nachtsheim and Neter (4th edition) or Applied Linear Statistical Models by Kutner, Nachtsheim, Neter and Li (5th edition). Both published by McGraw-Hill/Irwin. The first 14 chapters of Applied Linear Statistical Models (ALSM) are EXACTLY equivalent to the 14 chapters that make up Applied Linear Regression Models, 4th ed., with the same pagination.

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 535: Statistical Computing

Patrick Flaherty MWF 1:25-2:15

Prerequisites:

Stat 516 and CS 121

Description:

This course provides an introduction to fundamental computer science concepts relevant to the statistical analysis of large-scale data sets. Students will collaborate in a team to design and implement analyses of real-world data sets, and communicate their results using mathematical, verbal and visual means. Students will learn how to analyze computational complexity and how to choose an appropriate data structure for an analysis procedure. Students will learn and use the python language to implement and study data structure and statistical algorithms.

STAT 597P: ST - R Programming for Data Analytics and Visualization (1 cr)

Peng Wang Tue 4:00-5:15

Prerequisites:

STATISTC 240, 515, or 516. Open to Seniors, Juniors and Graduate students only. Previous programming experience (e.g. Matlab or Python) helpful, but not required.

Text:

None

Description:

This is an introductory class to R programming language designed for senior undergrads and junior graduate students. It lays out all the fundamentals of R, such as syntax, data structure, function design, package design, etc. Some important advanced features and packages will also be introduced, including dplyr, ggplot2, Rmarkdown and shiny. The objective is to give a comprehensive view about R, to provide as many guidance for future studying and practicing, and at the same time to distribute as many raw materials as possible for a jump start.

STAT 597R: ST - Introduction to Survey Sampling

TBD MW 8:40-9:55

Prerequisites:

STAT 525

Description:

This course is an introduction to modern theories and methods of survey sampling. Students will learn and practice techniques in order to become an informed contributor to a complex survey project on which they might work, to conduct a more limited survey on their own, and to appropriately analyze survey data that has already been collected. Students will learn enough to help you decide whether a survey was properly carried out or properly analyzed, and whether its conclusions are credible. Throughout the course, we will use data from current sources to illustrate the applicability of the concepts of the course to other fields of study.

STAT 598C: Statistical Consulting Practicum (1 cr)

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

Prerequisites:

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

Text:

None

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.

Graduate Courses

MATH 611: Algebra I

Paul Hacking MWF 12:20-1:10

Prerequisites:

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

Text:

Dummit and Foote, Abstract Algebra, 3rd ed.

Description:

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

I. Group Theory.

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 621: Complex Analysis

Sohrab Shahshahani MW 8:40-9:55

Prerequisites:

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

Text:

Complex Analysis (Princeton Lectures in Analysis, No. 2), by E. Stein and R. Shakarchi, Princeton University Press, 2003, ISBN 9780691113852

Description:

Complex number field, elementary functions, holomorphic functions, integration, power and Laurent series, harmonic functions, conformal mappings, applications.

MATH 623: Real Analysis I

Robin Young MWF 10:10-11:00

Prerequisites:

Math 523 or equivalent
(Undergraduate Analysis (calculus with proofs), also basics of metric spaces and linear algebra)

Text:

H.L. Royden, Real Analysis, 3rd Edition, 1988 ISBN-13: 978-0024041517 ISBN-10: 0024041513

Note: The 3rd edition is officially out of print but in my opinion better that the 4th edition (Royden & Fitzpatrick).

Description:

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

MATH 645: ODE and Dynamical Systems

Matthew Dobson TuTh 10:00-11:15

Prerequisites:

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

Text:

Differential Dynamical Systems, Revised Edition, by James D. Meiss; SIAM, 2017

Description:

Classical theory of ordinary differential equations and some of its modern developments in dynamical systems theory. Topics to be chosen from: Linear systems and exponential matrix solutions; Well-posedness for nonlinear systems; Floquet theory for linear periodic systems. Qualitative theory: limit sets, invariant sets and manifolds. Stability theory: linearization about equilibria and periodic orbits, Lyapunov functions. Numerical simulations will be used to illustrate the behavior of solutions and to motivate the theoretical discussion.

MATH 651: Numerical Analysis I

Brian Van Koten MWF 1:25-2:15

Prerequisites:

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

Text:

Quarteroni, Sacco, and Saleri, Numerical Mathematics, Springer, ISBN 978-3-540-34658-6

Description:

The analysis and application of the fundamental numerical methods used to solve a common body of problems in science. Linear system solving: direct and iterative methods. Interpolation of data by function. Solution of nonlinear equations and systems of equations. Numerical integration techniques. Solution methods for ordinary differential equations. Emphasis on computer representation of numbers and its consequent effect on error propagation.

MATH 671: Topology I

Alexei Oblomkov TuTh 8:30-9:45

Prerequisites:

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

Text:

Introduction to Topological Manifolds by John M. Lee, 2nd Edition, 2011, Springer, ISBN-13: 978-1441979391

Description:

Topological spaces. Metric spaces. Compactness, local compactness. Product and quotient topology. Separation axioms. Connectedness. Function spaces. Fundamental group and covering spaces.

MATH 691T: S-Teachng In Univ C

Prerequisites:

Open to Graduate Teaching Assistants in Math and Statistics

Text:

None

Description:

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

MATH 691Y: Applied Math Project Seminar

Matthew Dobson Fri 2:30-3:45

Prerequisites:

Graduate Student in Applied Math MS Program

Text:

None

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 697SG: Symmetric functions and representation theory of the symmetric group

Alejandro Morales TuTh 10:00-11:15

Prerequisites:

Undergraduate level Abstract Algebra (Math 411) and either Discrete Mathematics (Math 455) or Combinatorics (Math 513 / ComSci 575).

Text:

Sagan, Bruce E. The symmetric group: representations, combinatorial algorithms, and symmetric functions. Vol. 203. Springer, 2nd edition, 2013

Description:

Representation theory of the symmetric group can be approached using general theory of group representations or with symmetric functions. This area goes back to foundational work of Frobenius, Schur and Yong and a new approach from Okounkov and Vershik. This course will cover representation theory of the symmetric group using these two approaches. The course will also cover the ring of symmetric functions and its relation to the representation theory of the symmetric group and its applications to enumeration. Topics will include the Littlewood-Richardson rule, formulas for characters, and Schur-Weyl duality.

MATH 703: Topics in Geometry I

Mike Sullivan MW 2:30-3:45

Prerequisites:

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

Text:

Introduction to Smooth Manifolds, by John M. Lee

Description:

Topics to be covered: smooth manifolds, smooth maps, tangent vectors, vector fields, vector bundles (in particular, tangent and cotangent bundles), submersions,immersions and embeddings, sub-manifolds, Lie groups and Lie group actions, Whitney's theorems and transversality, tensors and tensor fields, differential forms, orientations and integration on manifolds, The De Rham Cohomology, integral curves and flows, Lie derivatives, The Frobenius Theorem.

MATH 713: Intr Algebraic Number Theory

Siman Wong MWF 11:15-12:05

Prerequisites:

Math 611 and 612, or equivalence.

Text:

Daniel A. Marcus, "Number Fields", 2nd ed. Springer-Verlag, 2018.

Description:

Algebraic number theory applies algebraic techniques to study
arithmetic problems. Much of the basic theory was originally developed
to understand and study classical diophantine equations, such as
Pell's equation x^2 - d y^2 = 1 and the Fermat equation x^n + y^n = z^n.
Under the local and global fields formulation, it unifies the
study of algebraic numbers and algebraic curves. And with the
introduction of advanced tools from representation theory and
algebraic geometry, we are led to contemporary research in Arithmetic
Geometry.

In this course we will develop foundational results in algebraic
number theory. The goal is study equidistribution theorems via
L-functions, both for their beautiful applications and (if time
permits) as an introduction to Class Field Theory. Concrete examples
and calculation (by hand and with computer software) will
play a crucial role in the course. We will take as a starting point
basic results from Galois theory and commutative algebra (such as
integral closure and localization) as developed e.g. in Math
611-612. There will be regular homeworks.

MATH 718: Lie Algebras

Eric Sommers TuTh 11:30-12:45

Prerequisites:

Math 611 required, Math 612 recommended

Description:

The goal of the course is the classification of semisimple Lie algebras over the complex numbers and an introduction to their representation theory. The classification is essentially the same as that for simply-connected compact Lie groups, which are central objects in mathematics and physics. Topics covered include representations of sl(2), Jordan decomposition, structure theorems, Weyl groups, roots systems, Dynkin diagrams, complete reducibility, finite-dimensional representations and their characters formulas. Students with a good background in both linear algebra and group theory may contact the instructor to enroll without the Math 611 prerequisite.

MATH 731: Partial Differential Equations I

Andrea Nahmod TuTh 2:30-3:45

Prerequisites:

The course assumes that the student has familiarity with the elementary methods of solution of linear ODEs and PDEs.
Modern Real Analysis (Measure Theory, Hilbert Spaces, L^p-theory, etc) at the first-year graduate level is assumed.
Math 623 and Math 624 (or equivalents) are a prerequisite for this class.

Text:

Partial Differential Equations: Methods and Applications by Robert McOwen. Second Edition. Prentice Hall (2003). ISBN-13: 978-0130093356

Note:

We will also be using material L. Evans Book: Partial Differential Equations ( AMS Grad. Textbook) —not required—

Description:

This course introduces the modern methods of analysis in partial differential equations. Emphasis is placed on the theory of existence, uniqueness and stability of solutions to boundary-value problems and initial-value problems. This theory is developed in the context of the prototypical PDEs arising in mathematical physics -- The Laplace/Poisson equation, the Heat/Diffusion equation, and the Wave equation. The theory of distributions is presented along with some functional analysis. Elliptic problems are studied in Hilbert-Sobolev spaces, using the variational formulation. Parabolic problems are considered in the setting of the analytic theory of semigroups. Hyperbolic problems are treated with energy methods.

We will cover most but not all of chapters 1-6 and selected topics from 8 if time permits.

STAT 605: Probability Theory I

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

Prerequisites:

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

Text:

Probability Essentials by Jean Jacod and Philip Protter

Recommended Text:

Other useful texts are
1) A first look at rigorous probability by Jeffrey Rosenthal
2) Probability: Theory and Examples by Rick Durrett
3) A probability path by Sidney Resnick

Description:

This class introduces the fundamental concepts in probability. Prerequisite are a solid working knowledge of undergraduate probability and analysis. Measure theory is not a prerequisite.
Among the topics covered are

1)Axioms of probability and the construction of probability spaces.

2) Random variables, integration, convergence of sequences of random variables, and the law of large numbers.

3) Gaussian random variables, characteristic and moment generating functions, and the central limit theorem.

4) Conditional expectation, the Radon--Nikodym theorem, and martingales.

STAT 607: Mathematical Statistics I

Daeyoung Kim TuTh 11:30-12:45

Prerequisites:

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

Text:

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

Description:

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

STAT 610: ST- Bayesian Statistics

John Staudenmayer MWF 12:20-1:10

Prerequisites:

STAT 607 and 608 or permission of the instructor.

Text:

Bayesian Data Analysis (3rd edition)
Authors: Gelman et al.
ISBN-13: 978-1439840955

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, credible intervals, diagnostics, etc.), and illustrate these objects in simple models. We will then develop Bayesian approaches to more complicated models, including hierarchical models and nonparametric / spatial regression. The course will introduce Markov chain Monte Carlo methods, and weekly in-class hands-on computing sessions will focus on applied Bayesian analyses with R and Stan. The course will have weekly problem sets, a midterm, a final, and a group project.

STAT 625: Regression Modeling

Krista J Gile TuTh 2:30-3:45

Prerequisites:

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

Description:

Regression is the most widely used statistical technique. In addition to learning about regression methods this course will also reinforce basic statistical concepts and introduce students to "statistical thinking" in a broader context. This is primarily an applied statistics course. While models and methods are written out carefully with some basic derivations, the primary focus of the course is on the understanding and presentation of regression models and associated methods, data analysis, interpretation of results, statistical computation and model building. Topics covered include simple and multiple linear regression; correlation; the use of dummy variables; residuals and diagnostics; model building/variable selection, regression models and methods in matrix form. With time permitting, further topics include an introduction to weighted least squares, regression with correlated errors and nonlinear (including binary) regression.

STAT 697ML: ST- Stat Machine Learning

Patrick Flaherty MW 2:30-3:45

Prerequisites:

Students should have taken Stat 515 as a prerequisite or Stat 607 as a co-requisite. Students must have an understanding of linear algebra at the level of Math 235. Students must be comfortable with a high-level programming language such as MATLAB, R or python.

Description:

This course is intended to provide a foundation in statistical machine learning with emphasis on statistical methodology as it applies to large-scale data applications using the graphical model framework. At the end of this course, students will be able to build and test a latent variable statistical model with companion inference algorithm to solve real problems in a domain of their interest including genomics, recommendation systems, topic modeling, and other high-dimensional data problems. Course topics include: exponential families, sufficiency and conjugacy, graphical model framework and approximate inference methods such as expectation-maximization, variational inference, and sampling-based methods. Additional topics may include: nonparametric Bayesian methods, cross-validation, bootstrap, empirical Bayes, and deep learning networks.

STAT 705: Linear Models I

Anna Liu TuTh 11:30-12:45

Prerequisites:

Calculus based Probablity and Mathematical Statistics (preferably Stat 607-608 or rough equivalent), matrix theory and linear algebra.

Text:

TBA

Description:

Linear models are at the heart of many statistics techniques (linear regression and design of experiments), are closely related to many other important areas (multivariate analysis, time series, econometrics, etc.) and form the basis for many more modern techniques dealing with mixed and hierarchical models, both linear and nonlinear. Coverage includes i) a brief review of important definitions and results from linear and matrix algebra and then what is assumed to be some new topics (idempotency, generalized inverses, etc.) in linear algebra; ii) Random vectors, multivariate distribution, the multivariate normal, linear and quadratic forms including an introduction to non-central t, chi-square and F distributions; iii) development of basic theory for inferences (estimation, confidence intervals, hypothesis testing, power) for the general linear model with "application" to both full rank regression and correlation models as well as some treatment of less than full rank models arising in the analysis of variance (one and some two-factor models). The applied part of the course is not directed towards extensive data analysis (which is available in many other applied courses). Instead, the emphasis with applications is on understanding and using the models and on some computational aspects, including understanding the documentation and methods used in some of the computing packages.