MATH 105 and 106 are designed for students whose background and algebraic skills are inadequate for the fast pace of MATH 111. In addition to covering the usual material of beginning calculus, these courses will have an extensive review of algebra and trigonometry. There will be a special emphasis on solving word problems.
MATH 105 starts with a quick review of algebraic manipulations, inequalities, absolute values and straight lines. Then the basic ideas of calculus—limits, derivatives, and integrals—are introduced, but only in the context of polynomial and rational functions. As various applications are studied, the algebraic techniques involved will be reviewed in more detail. When covering related rates and maximum-minimum problems, time will be spent learning how to approach, analyze and solve word problems. Four class meetings per week, one of which is an 80-minute group-work day.
Note: While MATH 105 and 106 are sufficient for any course with a MATH 111 requisite, MATH 105 alone is not. However, students who plan to take MATH 121 should consider taking MATH 105 and then MATH 111, rather than MATH 106. Students cannot register for both MATH 105 and CHEM 151 in the same semester.
Fall semester. The Department.
Other years: Offered in Fall 2011, Fall 2012, Fall 2013, Fall 2014, Fall 2015, Fall 2017, Fall 2018, Fall 2019, Fall 2020, Fall 2021, Fall 2022, Fall 2023, Fall 2024MATH 106 is a continuation of MATH 105. Trigonometric, logarithmic and exponential functions will be studied from the point of view of both algebra and calculus. The applications encountered in MATH 105 will reappear in problems involving these new functions. The basic ideas and theorems of calculus will be reviewed in detail, with more attention being paid to rigor.
Requisite: MATH 105. Spring semester. Professor D. Benedetto.
Other years: Offered in Spring 2012, Spring 2013, Spring 2014, Spring 2015, Spring 2016, Spring 2017, Spring 2018, Spring 2019, Spring 2020, Spring 2021, Spring 2022, Spring 2023, Spring 2025Basic concepts of limits, derivatives, anti-derivatives; applications, including max/min problems and related rates; the definite integral, simple applications; trigonometric functions; logarithms and exponential functions.
Requisite: Math Placement into 111, or consent of the Department. Limited to 30 students per section. Fall and spring semesters. In the fall semester, the intensive section (Section 01) is open only to students listed as eligible on the Mathematics placement list. The intensive section includes one additional weekly class meeting. The Department.
Other years: Offered in Fall 2011, Spring 2012, Fall 2012, Spring 2013, Fall 2013, Spring 2014, Fall 2014, Spring 2015, Fall 2015, Spring 2016, Fall 2022, Spring 2023, Fall 2023, Fall 2024, Spring 2025A continuation of MATH 111. Inverse trigonometric and hyperbolic functions; methods of integration, both exact and approximate; applications of integration to volume and arc length; improper integrals; l’Hôpital’s rule; infinite series, power series and the Taylor development; and polar coordinates.
Requisite: A grade of C or better in MATH 111 or placement into MATH 121 or consent of the Department. Limited to 30 students per section. Fall and spring semesters.The Department.
Other years: Offered in Fall 2011, Spring 2012, Fall 2012, Spring 2013, Fall 2013, Spring 2014, Fall 2014, Spring 2015, Fall 2015, Spring 2016, Fall 2022, Spring 2023, Fall 2023, Fall 2024, Spring 2025Mathematical modeling is the process of translating a real world problem into a mathematical expression, analyzing it using mathematical tools and numerical simulations, and then interpreting the results in the context of the original problem. Discussion of basic modeling principles and case studies will be followed by several projects from areas including biology and the social sciences (e.g., flocking and schooling behavior, disease spread in populations, generation of artificial societies). This course has no requisites; projects will be tailored to each student’s level of mathematical preparation.
Limited to 24 students. Spring semester. Professor Moore.
Other years: Offered in Fall 2012, Fall 2014, Spring 2016, Fall 2017, Spring 2019, Spring 2021, Spring 2022, Spring 2023, Spring 2025The outcomes of many elections, whether to elect the next United States president or to rank college football teams, can displease many of the voters. How can perfectly fair elections produce results that nobody likes? We will analyze different voting systems, including majority rule, plurality rule, Borda count, and approval voting, and assess a voter’s power to influence the election under each system, for example, by calculating the Banzhaf power index. We will prove Arrow’s Theorem and discuss its implications. After exploring the pitfalls of various voting systems through both theoretical analysis and case studies, we will try to answer some pressing questions: Which voting system best reflects the will of the voters? Which is least susceptible to manipulation? What properties should we seek in a voting system, and how can we best attain them?
Limited to 24 students. Omitted 2023-24.
2023-24: Not offeredElementary vector calculus; introduction to partial derivatives; multiple integrals in two and three dimensions; line integrals in the plane; Green’s theorem.
Requisite: A grade of C or better in MATH 121 or placement into MATH 211 or consent of the Department. Limited to 30 students per section. Fall and spring semesters. The Department.
Other years: Offered in Fall 2011, Spring 2012, Fall 2012, Spring 2013, Fall 2013, Spring 2014, Fall 2014, Spring 2015, Fall 2015, Spring 2016, Fall 2022, Spring 2023, Fall 2023, Fall 2024, Spring 2025This course serves as an introduction to mathematical reasoning and pays particular attention to helping students learn how to write proofs. The topics covered may include logic, elementary set theory, functions, relations and equivalence relations, mathematical induction, sequences, and quantifiers. Additional topics may vary from semester to semester.
Limited to 25 students. Spring and fall semesters. The Department.
Other years: Offered in Spring 2012, Spring 2013, Spring 2014, Spring 2015, Fall 2015, Spring 2016, Spring 2017, Fall 2022, Spring 2023, Fall 2023, Fall 2024, Spring 2025This course is a mathematical treatment of fractal geometry, a field of mathematics partly developed by Benoit Mandelbrot (1924–2010) that continues to be actively researched in the present day. Fractal geometry is a mathematical examination of the concepts of self-similarity, fractals, and chaos, and their applications to the modeling of natural phenomena. In particular, we will develop the iterated function system (IFS) method for describing fractals, study the concept of fractal dimension among other theoretical concepts, and examine Julia and Mandelbrot sets (time permitting). Through the teaching of these concepts, the course will also lend itself to familiarizing students with some of the formalisms and rigor of mathematical proofs.
Requisite: MATH 211 or consent of the instructor. Limited to 25 students. Professor Folsom.
Other years: Offered in Spring 2023, Spring 2025An introduction to the theory of rational integers; divisibility, the unique factorization theorem; congruences, quadratic residues. Selections from the following topics: cryptology; Diophantine equations; asymptotic prime number estimates; continued fractions; algebraic integers.
Other years: Offered in Fall 2012, Spring 2014, Spring 2015, Spring 2016, Spring 2023, Spring 2025Many security problems arise when two computers must communicate on a channel with eavesdroppers or malicious attackers. Public-key cryptography applies ideas from number theory and abstract algebra to address these problems. This course concerns the mathematical theory and algorithms needed to construct the most commonly-used public-key ciphers and digital signature schemes, as well as the attacks that must be anticipated when designing such systems. Several topics from number theory, abstract algebra, and algorithms will be introduced, including discrete logarithms, integer factorization algorithms, and elliptic curves. Depending on time and student interest, we may cover some newer systems that are believed to be secure against attacks by quantum computers but not yet commonly implemented in practice. Students will write short programs to implement the systems and to break badly implemented systems. No prior programming experience is expected; basic aspects of programming in Python will be taught in class. Four class hours per week.
Requisite: Experience writing proofs, such as MATH 220/221 or 271/272, or consent of the instructor. Spring semester. Professor Pflueger.
2023-24: Not offeredAbout 2300 years ago, Euclid introduced the axiomatic method to mathematics in his geometry textbook, the Elements. In this book, Euclid deduced the theorems of geometry from a small number of simple axioms about points, lines, and circles. Among his axioms is the parallel axiom, which asserts that if we are given a line and a point not on the line, then there is a unique line through the given point that is parallel to the given line.
Over 2000 years after Euclid, mathematicians discovered that by replacing Euclid's parallel axiom with its negation, we can develop a different kind of geometry in which we still have geometric objects like triangles and circles, but many of the theorems and formulas are different. For example, the sum of the angles of a triangle will always be less than 180 degrees, and this sum will determine the area of the triangle.
In this course we will study both Euclidean and non-Euclidean geometry. We will also consider the fascinating history of how non-Euclidean geometry was discovered.
Requisite: MATH 121. Fall semester.
2023-24: Not offeredThis course is a survey of geometry in dimensions 2, 3, 4, and higher. We will consider questions such as: How do we know the angles of a triangle add up to 180 degrees? (Spoiler: usually they don't.) What are the different ways we could tile our kitchen floor? How many tennis balls fit in a bucket? How many regular polyhedra are there in four dimensions? And what shape is the universe? We will examine how mathematicians and scientists, from Euclid to Einstein, have tried to give definitive answers to these questions, and we will explore some of what is still unknown about how the mathematical world reflects what we see around us.
Requisite: MATH 211 or instructor permission. Limited to 25 students. Fall semester. Professor Ching.
Other years: Offered in Fall 2023The study of differential equations is an important part of mathematics that involves many topics, both theoretical and practical. The course will cover first- and second-order ordinary differential equations, basic theorems concerning existence and uniqueness of solutions and continuous dependence on parameters, long-term behavior of solutions and approximate solutions.
Requisite: MATH 211 or consent of the instructor. Limited to 25 students. Fall semester.
Other years: Offered in Spring 2013, Spring 2014, Spring 2015, Spring 2016, Spring 2017, Spring 2018, Spring 2019, Spring 2020, Spring 2021, Spring 2022, Spring 2023, Fall 2023, Fall 2024, Spring 2025The study of vector spaces over the real and complex numbers, introducing the concepts of subspace, linear independence, basis, and dimension; systems of linear equations and their solution by Gaussian elimination; matrix operations; linear transformations and their representations by matrices; eigenvalues and eigenvectors; and inner product spaces. MATH 271 will feature both proofs and applications, with special attention paid to the theoretical development of the subject.
Requisite: MATH 211 or 220, or consent of the instructor. This course and MATH 272 may not both be taken for credit. Limited to 25 students. Fall and spring semesters. The Department.
Other years: Offered in Fall 2011, Fall 2012, Fall 2013, Fall 2014, Fall 2015, Spring 2016, Spring 2017, Fall 2017, Fall 2022, Spring 2023, Fall 2023, Fall 2024, Spring 2025The study of vector spaces over the real and complex numbers, introducing the concepts of subspace, linear independence, basis, and dimension; systems of linear equations and their solution by Gaussian elimination; matrix operations; linear transformations and their representations by matrices; eigenvalues and eigenvectors; and inner product spaces. This course will feature both proofs and applications, with special attention paid to applied topics such as least squares and singular value decomposition.
Requisite: MATH 211 or 220, or consent of the instructor. This course and MATH 271 may not both be taken for credit. Limited to 25 students. Fall and Spring semester: The Department.
Other years: Offered in Spring 2012, Spring 2013, Spring 2014, Spring 2015, Spring 2016, Spring 2017, Fall 2017, Spring 2018, Fall 2018, Spring 2019, Fall 2019, Spring 2020, Spring 2021, Fall 2021, Spring 2022, Fall 2022, Spring 2023, Fall 2023, Fall 2024, Spring 2025This course is a continuation of the material in MATH 271 and 272, providing more insight into abstract vector spaces and operator theory. Topics may include least squares estimates, singular value decompositions, Jordan canonical forms, inner product spaces, linear functionals and duals, orthogonal polynomials, vector and matrix norms, the spectral theorem, eigenvalue inequalities, and error-correcting codes. Time permitting, applications to graph theory and discrete dynamical systems may be explored. Four class hours per week.
Requisites: MATH 271, MATH 272, or consent of the instructor. Spring semester. Limited to 25 students.
2023-24: Not offeredA graph is a collection of points with edges drawn between them. Graph theory was first introduced by Leonhard Euler in his solution to the Königsberg bridge problem in 1736. Since then, graph theory has become an active area of study in mathematics due both to its wide array of real life applications in biology, chemistry, social sciences and computer networking, and to its interactions with other branches of mathematics.
The course will start with an overview of the fundamental concepts and general results in graph theory, followed by explorations of a variety of topics in graph theory and their applications, including: connectivity, planar graphs, directed graphs, greedy algorithms, matchings, vertex and edge colorings. The course will end with the introduction of a more advanced topic.
Requisite: MATH 271 or 272 or consent of the instructor. MATH 220 or other prior experience with mathematical proofs is recommended. Limited to 25 students. Fall semester. Professor Contreras.
Other years: Offered in Fall 2022, Fall 2023, Fall 2024This course emphasizes enumerative combinatorics, a classical subject in mathematics related to the theory of counting. Problems in this area often pertain to finding the number of possible arrangements of a set of objects under some particular constraints. This course incorporates a wide set of problems involving enumerative combinatorics, as well as theory and applications. Topics include the sum and product rules; combinations and permutations; binomial and multinomial coefficients; the principle of inclusion and exclusion; generating functions; recurrence relations; Catalan, Stirling, Bell and Eulerian numbers; partitions. Additional topics may vary.
This course will study numerical techniques for a variety of problems, such as finding roots of polynomials, interpolation, numerical integration, numerical solutions of differential equations, and matrix computations. We will study the underlying theory behind the algorithms, including error analysis, and the algorithms will be implemented using mathematical software to facilitate numerical experimentation.
Requisite: MATH 211 and either 271 or 272, or consent of the instructor. Limited to 24 students. Spring semester.
Optimization is a branch of applied mathematics focused on algorithms to determine maxima and minima of functions, often under constraints. Applications range from economics and finance to machine learning and information retrieval. This course will first develop advanced linear algebra tools, and then will study methods of convex optimization. Possible topics include linear, quadratic, second-order cone, and semidefinite models. Several applications will be explored, and algorithms will be implemented using mathematical software to aid numerical experimentation.
Other years: Offered in Fall 2022, Spring 2025The theory of partitions is a fundamental branch of combinatorics and number theory pertaining to enumerative properties and patterns of the integers. With its mathematical origins tracing back to the seventeenth century, partition theory has evolved through contributions made by many influential mathematicians including Euler, Legendre, Hardy, Ramanujan, Selberg and Dyson, and continues to be an active area of study today. Topics include partition identities and bijections, Ferrers diagrams and Durfee squares, partition generating functions and q-series, the pentagonal number theorem, q-binomial numbers (Gaussian polynomials), and partition congruences.
Requisite: MATH 121 and 220, or other significant experience with proofs, or by consent of the instructor. Limited to 24 students. Omitted 2021-22.
2023-24: Not offeredThe central object of study in algebraic geometry is a system of polynomial equations and its solution set. The theory has connections to many other areas in pure mathematics including number theory, representation theory, several complex variables, and combinatorics. Applied algebraic geometry is used in areas including mathematical biology, algebraic statistics, robotics, and computer vision. In this course, we will study algebraic geometry from a computational point of view, beginning with Buchberger's algorithm for computing Groebner bases, and working towards the ideal-variety correspondence. Topics may include projective algebraic geometry, toric varieties, and applications, depending on student interest.
Requisite: MATH 271 or 272, and experience with writing mathematical proofs (such as in MATH 220); or permission of the instructor.Limited to 25 students. Fall semester. The Department.
Pending Faculty Approval
2023-24: Not offeredThe first half of the course covers continuous and discrete Fourier transforms (including convolution and Plancherel’s formula), Fourier series (including convergence and the fast Fourier transform algorithm), and applications like heat conduction along a rod and signal processing. The second half of the course is devoted to wavelets: Haar bases, the discrete Haar transform in 1 and 2 dimensions with application to image analysis, multiresolution analysis, filters, and wavelet-based image compression like JPEG2000. Three class hours per week plus a weekly one-hour computer laboratory.
Requisite: MATH 211 and 271 or 272. Omitted 2021-22.
2023-24: Not offeredWhat is the shape of a hanging chain? What shape of fixed perimeter encloses the most area? The calculus of variations answers questions such as these through maximizing or minimizing values of functionals over different input functions. Topics covered will include functional differentiation, the Euler-Lagrange equations, necessary and sufficient conditions for extrema, and direct minimization methods. Additionally, there will be applications to physics and optimization, including Hamiltonian mechanics and image processing.
Fall semester. Professor Kraisler
Other years: Offered in Fall 2023An introduction to analytic functions; complex numbers, derivatives, conformal mappings, integrals. Cauchy’s theorem; power series, singularities, Laurent series, analytic continuation; special functions.
Requisite: MATH 211 and prior experience with mathematical proofs, or consent of the instructor. Limited to 25 students. Fall semester.
Other years: Offered in Fall 2011, Fall 2012, Fall 2013, Fall 2014, Fall 2015, Fall 2017, Fall 2018, Fall 2019, Fall 2020, Fall 2021, Fall 2022, Fall 2023, Fall 2024A brief consideration of properties of sets, mappings, and the system of integers, followed by an introduction to the theory of groups and rings including the principal theorems on homomorphisms and the related quotient structures; integral domains, fields, polynomial rings.
Requisite: MATH 211 and either MATH 271 or 272, or consent of the instructor. Students with a grade of B+ or lower in linear algebra are encouraged to take another 200-level course with proofs before taking MATH 350.
Limited to 25 students.
Other years: Offered in Spring 2012, Spring 2013, Spring 2014, Fall 2014, Spring 2015, Fall 2015, Spring 2016, Spring 2017, Fall 2017, Fall 2022, Spring 2023, Fall 2023, Fall 2024, Spring 2025Completeness of the real numbers; topology of n-space including the Bolzano-Weierstrass and Heine-Borel theorems; sequences, properties of continuous functions on sets; infinite series, uniform convergence.
Requisite: MATH 211 and either MATH 271 or 272, or consent of the instructor. Students with a grade of B+ or lower in linear algebra are encouraged to take another 200-level course with proofs before taking MATH 355.
Limited to 25 students. The Department.
Other years: Offered in Spring 2012, Spring 2013, Fall 2013, Spring 2014, Fall 2014, Spring 2015, Fall 2015, Spring 2016, Spring 2017, Fall 2017, Fall 2022, Spring 2023, Fall 2023, Fall 2024, Spring 2025A stochastic process is a collection of random variables used to model the evolution of a system over time. Unlike deterministic systems, stochastic processes involve an element of randomness or uncertainty. Examples include stock market fluctuations, audio signals, EEG recordings, and random movement such as Brownian motion and random walks. Topics will include Markov chains, martingales, Brownian motion, and stochastic integration, including Ito’s formula. Four class hours per week, with weekly in-class computer labs.
Requisite: MATH 360 or consent of the instructor. Limited to 24 students. Professor Leise.
2023-24: Not offeredThis course builds upon the material in MATH 355 (Introduction to Analysis) in order to rigorously develop basic tools for studying functions of more than one real variable. While the setting in MATH 355 is the real number line, the context for this course will be the n-dimensional Euclidean space. Many facets of analysis on this n-dimensional space will be explored including its topological properties as well as differentiation and Riemann integration in n-variables. The course will cover fundamental results such as the celebrated implicit and inverse function theorems. Time permitting, we may discuss additional topics such as analysis on metric spaces. Four class hours per week.
Requisite: MATH 355 and either MATH 271 or 272; or consent of the instructor. Limited to 25 students. Omitted 2021-22.
2023-24: Not offeredThis course will explore the geometry of curves and surfaces in n-dimensional Euclidean space. For curves, the key concepts are curvature and torsion, while for surfaces, the key players are Gaussian curvature, geodesics, and the Gauss-Bonnet Theorem. Other topics covered may include (time permitting) the Four Vertex Theorem, map projections, the Hairy Ball Theorem, and minimal surfaces.
Requisites: MATH 211, MATH 271 or 272, and MATH 355 or consent of the instructor.
2023-24: Not offeredMathematicians confirm their answers to mathematical questions by writing proofs. But what, exactly, is a proof? This course begins with a precise definition specifying what counts as a mathematical proof. This definition makes it possible to carry out a mathematical study of what can be accomplished by means of deductive reasoning and, perhaps more interestingly, what cannot be accomplished. Topics will include the propositional and predicate calculi, completeness, compactness, and decidability. At the end of the course we will study Gödel’s famous Incompleteness Theorem, which shows that there are statements about the positive integers that are true but impossible to prove.
Other years: Offered in Spring 2015, Spring 2018, Fall 2019, Spring 2022Independent reading course.
Fall and spring semesters. The Department.
Other years: Offered in Fall 2011, Spring 2012, Fall 2012, Spring 2013, Fall 2013, Spring 2014, Fall 2014, Spring 2015, Fall 2015, Spring 2016, Fall 2016, Spring 2017, Fall 2017, Spring 2018, Fall 2018, Spring 2019, Fall 2019, Spring 2020, Fall 2020, Spring 2021, Fall 2021, Spring 2022, Fall 2022, Spring 2023, Fall 2023, Fall 2024, Spring 2025Lie groups and Lie algebras appear naturally in the study of symmetries of geometric objects. Lie algebras carry local information and can be studied using tools from linear algebra. Finite dimensional Lie groups can be constructed using techniques from calculus and group theory.
This course serves as a first introduction to Lie groups and Lie algebras. We will examine the structure of finite dimensional matrix Lie groups, the exponential and differentiation maps, as well as compact Lie groups. We will also study Lie algebras, ideals and homomorphisms, nilpotent and solvable Lie algebras, Cartan subalgebras, semisimplicity and root systems, and the classification of semisimple Lie algebras. This classification is not only a fundamental result in Lie Theory, but is also an archetype of classifications that appear in other areas of math. More amazingly, this classification is embodied in simple combinatorial pictures called Dynkin diagrams, which underlie surprisingly disparate fields, such as geometric group theory, quiver representation theory, and string theory. Four class meetings per week.
Requisite: MATH 350 or consent of the instructor. Fall semester.
2023-24: Not offeredThe quadratic formula shows us that the roots of a quadratic polynomial possess a certain symmetry. Galois Theory is the study of the corresponding symmetry for higher degree polynomials. We will develop this theory starting from a basic knowledge of groups, rings, and fields. One of our main goals will be to prove that there is no general version of the quadratic formula for a polynomial of degree five or more. Along the way, we will also show that a circular cake can be divided into 17 (but not 7) equal slices using only a straight-edged knife.
Requisite: MATH 350 or consent of the instructor. Spring semester. Prof. Daniels.
2023-24: Not offeredCommutative algebra is known as the study of commutative rings and their ideals and modules. Besides being an important branch of algebra for its own sake, commutative algebra has strong ties to other areas, such as algebraic geometry and algebraic number theory, as it provides essential tools for them. This course is an introductory course in commutative algebra. We will explore more about rings (especially polynomial rings) and ideals, which are taught in Math 350. We will also introduce another important algebraic structure, namely modules over rings. Other fundamental topics include Noetherian rings, The Hilbert Basis Theorem, Gröbner bases, localization, primary decompositions, and tensor products.
Requisite: Math 350 or consent of the instructor. Limited to 24 students. Omitted 2023-24. Visiting Assistant Professor Gunturkun.
2023-24: Not offeredAn elliptic curve is the set of zeros of a cubic polynomial in two variables. If the polynomial has rational coefficients, it is natural to ask for a description of those zeros whose coordinates are either integers or rational numbers. Our study of elliptic curves will focus on this fundamental problem and reveal a fascinating interplay between algebra, geometry, analysis and number theory. Topics discussed will include the geometry and group structure of elliptic curves, the Nagell-Lutz Theorem describing points of finite order, and the Mordell-Weil theorem on the finite generation of the group of rational points. Additional topics may include elliptic curve cryptography, Lenstra's algorithm using elliptic curves to factor large integers, the Thue-Siegel Theorem on the finiteness of the set of integer points, and the crucial role the theory of elliptic curves played in Wiles' proof of Fermat's Last Theorem. By bringing together techniques from a wide range of mathematical disciplines, we plan to illustrate the unity of mathematics and introduce active areas of research. Four class hours per week.
Requisite: MATH 350 or consent of the instructor. Omitted 2021-22.
2023-24: Not offeredAn introduction to Lebesgue measure and integration; topology of the real numbers; inner and outer measures and measurable set; the approximation of continuous and measurable functions; the Lebesgue integral and associated convergence theorems; the Fundamental Theorem of Calculus.
Other years: Offered in Spring 2012, Spring 2014, Spring 2016This course will explore the geometry of curves, surfaces and higher dimensional geometric objects in n-dimensional Euclidean space. For curves, the key concepts are curvature and torsion, while for surfaces, the key players are Gaussian curvature, geodesics, and the Gauss-Bonnet Theorem. Other topics covered may include (time permitting) two-dimensional Riemmanian geometry, differential forms and manifolds. MATH 378 and MATH 408 may not both be taken for credit.
Spring semester. Professor Contreras
An introduction to general topology: the topology of Euclidean, metric and abstract spaces, with emphasis on such notions as continuous mappings, compactness, connectedness, completeness, separable spaces, separation axioms, and metrizable spaces. Additional topics may be selected to illustrate applications of topology in analysis or to introduce the student briefly to algebraic topology. Four class hours per week. Offered in alternate years.
Requisite: MATH 355. Spring semester. Prof. Rasheed.
2023-24: Not offeredThis course is an introduction to Analytic Number Theory, a foundational subject in mathematics which dates back to the 1800s and is still a major research area today. The subject generally uses tools and techniques which are analytic in nature to solve problems primarily related to integers. Asymptotic and summation results and methods are of great significance in Analytic Number Theory. Two primary course objectives are to state and prove two major theorems: Dirichlet's Theorem on Primes in Arithmetic Progressions, and the Prime Number Theorem. In particular, we will study Selberg's "elementary" proof of the Prime Number Theorem. Additional topics may include: arithmetic functions, especially their averages, their asymptotics, and related summation formulae; Dirichlet convolutions; characters and Gauss sums; and an introduction to Dirichlet series, such as the Riemann zeta-function and L-functions. Further topics may vary.
Requisite: MATH 355. Limited to 25 students. Fall semester. Professor Folsom.
Other years: Offered in Fall 2023Open to seniors with the consent of the Department. Fall semester. The Department.
Other years: Offered in Spring 2012, Spring 2013, Spring 2014, Spring 2015, Spring 2016, Spring 2017, Spring 2018, Spring 2019, Spring 2020, Spring 2021, Spring 2022, Spring 2023, Spring 2025This course will provide a rigorous presentation of fundamental statistical principles and ethics. We will discuss standards for relationships between statisticians and policymakers, researchers, the press, and other institutions, as well as the standards for interactions between statisticians and their employers/clients, colleagues and research subjects. The course will explore how the interplay of institutions (e.g., organizations, systems, laws, codes of professional ethics) and the broader sociopolitical culture affect the production of reliable, high quality statistics. Students will also explore the implications of statistical principles and ethics for the operation of national, regional, and international official statistical systems. In addition, we will investigate the proper place of official statistics within a government system that operates with separate branches. Students will gain a strong foundation in international statistical principles and professional ethics as well as an understanding and the tools to assess the quality of the statistics they use. The course is designed to make students responsible and effective supporters of reliable, high quality statistics in their professions. Students will particularly learn how to assess the quality of official statistics produced by governments and how to identify areas for improvement. Examples, case studies, readings from statistical practice, and discussion will provide a full appreciation of real world applications. The course is also intended for non-majors interested in an introduction to quantitative social science and the use of data in public policy.
Limited to 30 students. Fall semester. Visiting Scholar Andreas Georgiou.
Other years: Offered in Fall 2018, Fall 2019, Spring 2020, Fall 2021, Fall 2022, Fall 2023, Fall 2024Introduction to Statistics provides a basic foundation in descriptive and inferential statistics, including constructing models from data. Students will learn to think critically about data, produce meaningful graphical and numerical summaries of data, apply basic probability models, and utilize statistical inference procedures using computational tools. Topics include basic descriptive and inferential statistics, visualization, study design, and multiple regression. S
Students who have taken a semester of calculus (MATH 111 or higher, or equivalent placement) or who are planning to major in statistics should take STAT 135/MATH 135 or STAT 136 instead of this course. (Students who have taken STAT/MATH 135, STAT136, PSYC 122, or ECON 360/361 may not also receive credit for STAT 111. STAT 111 does not count towards the major in Mathematics.)
Limited to 24 students per section. Permission with consent of the instructor. Fall semester. Professor Matheson.
Other years: Offered in Fall 2014, Spring 2015, Fall 2015, Spring 2016, Fall 2022, Fall 2023, Fall 2024This is an interactive course designed to help students understand inequities in mental health issues via statistics. We will begin the course by examining mental health stigmas and practice self-care exercises to train our “happy muscles” together. We will discover the scientific evidence behind those self-care practices and explore existing racial disparities in mental health care systems, while learning about important statistical concepts and mastering our data analysis skills using R (a popular statistical software package). Statistical topics covered include descriptive statistics, visualization, study design, simulation-based inferences, and multiple regression. Students are expected to play an active role in co-creating the course and co-building an inclusive learning community with their peers and the professor. Course components include weekly reading and discussion, regular self-reflections and problem sets, and collaborative work in groups. We will use an OER (Open-Educational-Resources) textbook in this course. No prior experience with statistical software is expected.
This course is an alternative to STAT135 (Introduction to Statistics via Modeling) with a special focus on mental health issues. Students may not receive credit for both this course and STAT 111, STAT 135, or PSYC122. Limited to 24 students.
Fall semester. Prof. Liao.
Other years: Offered in Fall 2022, Spring 2025The COVID-19 pandemic has illuminated continuing health inequity. This course examines public health research through the critical lens of research methods and statistical analysis. A variety of health-related issues in which race, ethnicity, biological sex, and/or sexual orientation are associated with negative health outcomes will be examined. Students will read current research literature demonstrating health inequity, examine the history of health-related research in that area, and scrutinize uninformed analyses/conclusions that lead to the perpetuation of harmful stereotypes and inadequate medical practice. Examples of intervention programs designed to eliminate health disparities will be reviewed with special attention given to those aspects associated with success and how effective program evaluation is conducted. This sophomore seminar is intended for students who have some experience with statistical methods.
Requisite: STAT 111 or MATH/STAT 135 or STAT 136 or equivalent. Limited to 20 students. Spring semester. Professor Matheson.
Other years: Offered in Fall 2024This course will focus on the use of text analytics to explore the rich history of Holyoke, MA. Holyoke has been a site of rapid industrialization, multiple waves of immigration and migration, urban development, rapid changes in its workforce, and ongoing creativity, activism, and innovation. Students will develop the skills to mine textual data from archives at the Wistariahurst Museum, the Holyoke Public Library, Holyoke Community College, and other repositories to address important questions regarding the development and history of this planned community. Topics include sentiment analysis, regular expressions, document-term-matrices, named entity recognition, and Latent Dirichlet analysis. Requisite:
Student has completed or is in the process of completing: STAT 111 or MATH/STAT 135 or STAT136 or PSYC 122 or has a placement of STAT230.
Recommended requisite: HIST 351 or COSC 111.
Professor Horton.
2023-24: Not offeredThis course is an introduction to nonparametric and distribution-free statistical procedures and techniques. These methods often rely on counting and ranking techniques and will be explored through both theoretical and applied perspectives. One- and two-sample procedures will provide students with alternatives to traditional parametric procedures, such as the t-test. A variety of other topics may be explored in the nonparametric setting depending on the instructor. Potential topics include but are not limited to: nonparametric correlation and regression, resampling techniques (e.g. bootstrapping and permutation procedures), categorical data and contingency tables, density estimation, and the one-way and two-way layouts for analysis of variance. The course will emphasize data analysis (with appropriate use of statistical software) and the intuitive nature of nonparametric statistics.
Requisite: STAT 111 or MATH/STAT 135 or STAT 136 or equivalent. Limited to 24 students. Fall semester. Professor Wagaman.
Other years: Offered in Fall 2023This course is an intermediate applied statistics course that builds on the statistical data analysis methods introduced in STAT 111, STAT 135, or STAT 136. Students will learn how to pose a statistical question, perform appropriate statistical analysis of the data, and properly interpret and communicate their results. Emphasis will be placed on the use of statistical software, data wrangling, model fitting, and assessment. Topics covered will include ethics, experimental design, resampling approaches, analysis of variance models, multiple regression, model selection, and logistic regression. No prior experience with statistical software is expected
Requisite: Student has completed or is in process of completing any of the following course(s): STAT 111 or MATH/STAT 135 or STAT 136 or PSYC 122 or has a STAT 230 placement or has consent of the instructor. Limited to 24 students. Four spots reserved for incoming first-year students in each Fall section. Fall and Spring semester. The Department.
Other years: Offered in Spring 2015, Fall 2015, Spring 2016, Fall 2022, Spring 2023, Fall 2023, Fall 2024, Spring 2025Computational data analysis is an essential part of modern statistics and data science. This course provides a practical foundation for students to think with data by participating in the entire data analysis cycle. Students will generate statistical questions and then address them through data acquisition, cleaning, transforming, modeling, and interpretation. This course will introduce students to tools for data management, wrangling, and databases that are common in data science and will apply those tools to real-world applications. Students will undertake practical analyses of large, complex, and messy data sets leveraging modern computing tools.
Requisite:
STAT 111 or STAT 135 or STAT136 and COSC 111 or consent of the instructor. Limited to 24 students. Fall and Spring semesters. The Department.
Other years: Offered in Fall 2022, Spring 2023, Fall 2023, Fall 2024, Spring 2025Making sense of a complex, high-dimensional data set is not an easy task. The analysis chosen is ultimately based on the research question(s) being asked. This course will explore how to visualize and extract meaning from large data sets through a variety of analytical methods. Methods covered include principal components analysis and selected statistical and machine learning techniques, both supervised (e.g. classification trees and random forests) and unsupervised (e.g. clustering). Additional methods covered may include factor analysis, dimension reduction methods, or network analysis at instructor discretion. This course will feature hands-on data analysis with statistical software, emphasizing application over theory.
The course is expected to include small group work, interactive labs, peer interactions such as peer review and short presentations, and a personal project, to foster student engagement in the course and with each other.
Requisite:
Requisite: STAT 111 or 135 or 136 or PSYC 122. Limited to 24 students. Omitted 2023-24.
2023-24: Not offeredStatistical Communication is an important component of the capacity to "think with data." This course will integrate theoretical and practical aspects of statistics with a focus on communicating results and their implications. Students will gain experience clearly synthesizing and explaining complex data using diverse predictive and explanatory models. Learning objectives include: understanding the role of a statistician, developing communications skills, working collaboratively on group projects, designing studies to collect information, acquiring existing data resources, utilizing publications in statistics, creating reproducible research and developing oral arguments, relevant project reports, and dynamic graphical displays. Emphasis will be placed on the use of statistical software, data management, visual presentation, and oral and written communication skills that are necessary for communicating technical content.
Requisite:
Student has completed or is in the process of completing STAT 230 or has consent of the instructor. Limited to 24 students. Fall semester. Omitted 2021-22. Professor Matheson.
2023-24: Not offeredText analytics is a form of natural language processing that utilizes computational systems to process, find patterns, classify, and model information contained within unstructured text documents. These methods are attractive because they can be applied to large collections of documents that would be infeasible to undertake by hand. In this course, students will interact with a variety of text sources with the goal of finding insights, identifying patterns, extracting meaning, and communicating results. Topics will include data wrangling, tokenization, regular expressions, n-grams, named entity recognition, sentiment analysis, topic modeling, classification, cloud computing, dynamic visualization tools, and ethical considerations.
Students may not receive credit for both STAT210 and STAT325
Spring semester. Professor Horton
Other years: Offered in Spring 2025Time series (TS) data are everywhere in our daily lives but often carry complicated information from which it is difficult to distill useful insights. Some familiar examples include daily stock prices, monthly unemployment rates, global warming trends, and the number of deaths of COVID-19 during the pandemic. In this course, students will gain fundamental understanding of the features and processes used to analyze TS data, learn how to model the patterns in historical values using various TS models, and further utilize the refined information from modeling to make educated predictions or forecasts. Topics covered will include time series regression, smoothing, unit root and differencing, autoregressive integrated moving average (ARIMA) models, spectral analysis, and generalized autoregressive conditionally heteroscedastic (GARCH) models. If time permits, state-space models may also be introduced.
Requisite: STAT 230 or consent of the instructor. Limited to 20 students. Omitted 2021-22. Assistant Professor Liao.
2023-24: Not offeredEpidemiology is the study of the distribution and determinants of disease and health in human populations. It typically involves the analysis of multivariate observational data that pose challenges when trying to make causal conclusions. The course will focus on reasoning about cause and effect, study design, bias and missing data, models and analysis of risk, detection and classification, and modern approaches to confounding and causal inference. Topics include: Measures of disease (incidence and prevalence); Measures of association (relative risk, odds ratio, relative hazard, excess risk, attributable risk); Study designs (exposure and disease base sampling); Assessing significance in a 2x2 table; Assessing significance in a 2x2x2 table; Missing data; Introduction to confounding; Matching; Propensity score adjustment; Unmeasured confounding; Introduction to causal inference and counterfactuals; Causal graphs; and D-separation.
Requisite: STAT-230 (or PSYC 122 and PSYC 200 and consent of the instructor). Omitted 2021-22.. Professor Horton.
2023-24: Not offeredStatistical inference using what are called frequentist methods, where only the data are random, has long dominated the manner in which data are analyzed. The rise of computing power this century has unlocked Bayesian inference, a technique that blends prior knowledge with data, as an increasingly popular and powerful alternative approach. This course will explore the theory behind and application of Bayesian inference including situations where Markov Chain Monte Carlo (MCMC) simulation is employed.
Omitted 2023-24. Professor Donges
2023-24: Not offeredCompetitions, which can include individual and team sports, eSports, tabletop gaming, preference formation, and elections, produce data dependent on interrelated competitors and the decision, league, or tournament format. In this course, students will learn to think about the ways a wide variety of statistical methodologies can be applied to the complex and unique data that emerge through competition, including paired comparisons, decision analysis, rank-based and kernel methods, and spatio-temporal methods. The course will focus on the statistical theory relevant to analyzing data from contests and place an emphasis on simulation and data visualization techniques. Students will develop data collection, wrangling,combination, exploration, analysis, and interpretation skills individually and in groups. Applications may include rating players and teams, assessing shot quality, animating player tracking data, roster construction, comparing alternative voting systems, developing optimal strategies for games, and predicting outcomes. Prior experience with probability such as STAT 360 may be helpful, but is not required.
Requisite: STAT 230 and STAT 231. Limited to 24 students. Omitted 2023-2024. Professor McShane.
2023-24: Not offeredReal world datasets are plagued by missing observations. Statistical software packages often ignore these cases by default, but there are better ways to approach the problem. This course will introduce students to the different missing data mechanisms and explore naive and modern methods for handling missing data. It will prepare students to read the current literature in this area and have broad appreciation for the implications of missing data.
This course is intended for students who have experience with standard statistical methods for complete data and want to extend them to handle missing data in practice.
Requisite:
Student has completed or is in the process of completing STAT 230 and STAT/MATH 370, or has consent of the instructor. Fall semester: Professor Correia.
Other years: Offered in Fall 2022, Fall 2023, Fall 2024The application of statistical methods for categorical data is ubiquitous in the modern world, especially in social sciences, biomedical research, and economics and business. However, many traditional statistical tools (like linear regression models) are not designed to handle observations classified into categories and thus inappropriate for analyzing such data. In this course, students will learn important theories and statistical methods specifically designed for visualizing and modeling categorical and ordinal data, while undertaking meaningful analyses and interpretations of complex real-world data that focus on common good issues of their choices. This course is ideal for students who seek to not only develop new statistical capacities but also apply their analytic skills to work for the greater good.
Requisite: STAT 230 and STAT 360, or equivalent experience by consent of the instructor. Spring semester. Professor Liao
Linear regression and logistic regression are powerful tools for statistical analysis, but they are only a subset of a broader class of generalized linear models. This course will explore the theory behind and practical application of generalized linear models for responses that do not have a normal distribution, including counts, categories, and proportions. We will also delve into extensions of these models for dependent responses such as repeated measures over time.
Requisite:
Student has completed or is in the process of completing: STAT 230 and STAT/MATH 360. Limited to 20 students. Spring semester. Professor Bailey.
2023-24: Not offeredFall and spring semesters. The Department.
Other years: Offered in Fall 2022, Spring 2023, Fall 2023, Fall 2024Our world is awash in data. To allow decisions to be made based on evidence, there is a need for statisticians to be able to make sense of the data around us and communicate their findings. In this course, students will be exposed to advanced statistical methods and will undertake the analysis and interpretation of complex and real-world datasets that go beyond textbook problems. Course topics will vary from year to year depending on the instructor and selected case studies. Potential topics include but are not limited to: visualization techniques to summarize and display high dimensional data, advanced topics in design and linear regression, ethics, selected topics in machine learning and data mining, nonparametric analysis, spatial data, and analysis of network data. Students will enhance their capacity to think and compute with data, undertake and assess analyses, and effectively communicate their results.
Other years: Offered in Fall 2014, Fall 2015, Fall 2022, Fall 2023, Fall 2024Spring semester. The Department.
Other years: Offered in Spring 2025