Professors R. Benedetto, Call, Cox and Horton†; Associate Professors Ching, Folsom, Leise (Chair), and Wagaman; Assistant Professors Daniels, Liao, and Pflueger; Senior Lecturer D. Benedetto; Lecturer Kim; Visiting Assistant Professors Juul, Matheson; Naqvi, Sosa-Castillo, and Zhang; Visiting Lecturer Beldenko; Postdoctoral Fellow Yacoubou Djima.
†On leave fall semester 2017-18.
The Department offers the major in Mathematics and the major in Statistics, as well as courses meeting a wide variety of interests in these fields. Non-majors who seek introductory courses are advised to consider MATH 105, 111, 140, and 220 and STAT 111, none of which require a background beyond high school mathematics.
Mathematics
Major Program. The minimum requirements for the Mathematics major include MATH 111, 121, 211, 271 or 272, 350, 355. Along with the required courses, a major must complete three elective courses in Mathematics numbered 135 or higher. For majors declared after May 17, 2017, at least two of these electives must be numbered 200 or higher. In addition, a major must complete two other courses, each of which is either an elective course in Mathematics numbered between 135 and 490 or a course from outside Mathematics, but in a related field, chosen from among COSC 201 or 211, 301 or 311, and 401; ECON 300, 301, 361, and 420; PHIL 350; any Physics course numbered 116 or higher (excluding PHYS 227); and any Statistics course numbered 200 or higher. Statistics courses cross-listed with Mathematics count as Mathematics electives. The additional requirement of two courses can be satisfied by taking two math electives, one math elective and one related-fields course, or two related-fields courses. Requests for alternative courses must be approved in writing by the chair of the Department in consultation with the Mathematics faculty within the Department. Honors students may petition the department to count one math thesis course as an elective toward the major.
Students who have placed out of certain courses, such as calculus, as indicated by a strong performance on an Advanced Placement Exam or other evidence approved by the department, such as a competency exam are excused from taking those courses. Students who place out of MATH 111, 121 or 211 do not need to replace these courses. Students who place out of MATH 271, 272, 350, or 355 exam must replace each such course with an additional Mathematics course numbered 135 or higher.
A student considering a major in Mathematics should consult with a member of the Department as soon as possible, preferably during the first year. This will facilitate the arrangement of a program best suited to the student’s ability and interests. Students should also be aware that there is no single path through the major; courses do not have to be taken in numerical order (except where required by prerequisites).
All students majoring in Mathematics are encouraged to include in their program courses that include concepts and methods from data analysis, mathematical modeling, and computation (e.g., MATH 135/STAT 135, MATH 140, MATH 284, MATH 360/STAT 360, or COSC 111). Students majoring in Mathematics are expected to attend all Mathematics colloquia during their junior and senior years.
For a student considering graduate study, the Departmental Honors program is strongly recommended. Such a student is advised to take the Graduate Record Examination early in the senior year. It is also desirable to have a reading knowledge of a foreign language, usually French, German, or Russian.
Double Majors in Mathematics and Statistics. Students electing a double major in Mathematics and Statistics may count MATH 111, 121, 211, and MATH 271 or 272 towards both majors. A maximum of one additional course taken to complete the Statistics major may be counted towards the Mathematics major.
Comprehensive Examination. A comprehensive examination will be given near the beginning of the spring semester of the senior year. (Those who will complete their studies in the fall semester may elect instead to take the comprehensive examination at the beginning of that semester.) The examination covers MATH 211, MATH 271 or 272, and a choice of MATH 350 or 355. More information about the comprehensive examination, including regulations and study materials, can be found on the Department website.
Honors Program in Mathematics. Students are admitted to the Honors Program on the basis of a qualifying examination given at the beginning of the spring semester of their junior year and the acceptance of a thesis proposal. (Those for whom the second semester of the junior year occurs in the fall may elect instead to take the qualifying examination at the beginning of that semester.) If a student is accepted to the Honors Program, they will finalize a topic in consultation with Mathematics faculty. After intensive study of this topic, the candidate will write a report in the form of a thesis which should be original in its presentation of material, if not in content. In addition, the candidate will report to the departmental colloquium on their thesis work during the senior year. Honors candidates are also required to complete MATH 345 and at least one Mathematics course numbered 400 to 489.
Statistics
Major Program. For majors in the class of 2016 and 2017, and all other majors who declared statistics before May 7, 2016, the minimum requirements for the Statistics major include MATH 111, 121, and 211; MATH 271 or MATH 272; STAT 111 or STAT 135; MATH 140 or COSC 111; STAT 230, 360, 370, and 495; and one additional elective course in Statistics at the 200 level or higher.
For majors in the class of 2018 and beyond declaring statistics after May 7, 2016, the minimum requirements for the Statistics major include MATH 111, and 121; MATH 271 or MATH 272; STAT 111 or STAT 135; STAT 230, 360, 370, and 495; two courses in Computer Science at the level of 111 or higher (typically COSC 111 and COSC 112); and two additional elective courses in Statistics at the 200 level or higher.
Requests for alternative courses must be approved in writing by the chair of the Department in consultation with the Statistics faculty within the Department. Students who have placed out of certain courses, such as calculus, introductory statistics, or introductory computer science, as indicated by strong performance on an Advanced Placement Exam or other evidence approved by the Department, are excused from taking those courses. Statistics majors may place out of up to three courses without having to replace those courses. Students placing out of more than three courses must replace all but three of those courses with additional courses approved by the Department to complete the major.
A student considering a major in Statistics should consult with a member of the Department as soon as possible, preferably during the first year. This will facilitate the arrangement of a program best suited to the student’s ability and interests. Students should also be aware that there is no single path through the major; courses do not have to be taken in numerical order (except where required by prerequisites). Statistics majors are encouraged to ensure that their course of study includes depth in an application area (e.g., sociology, psychology, or environmental science). Students majoring in Statistics are expected to attend all Statistics colloquia during their junior and senior years. Students planning to attend graduate school in statistics are strongly advised to take MATH 211, MATH 355, and MATH 450.
Double Majors in Statistics and Mathematics. Students electing a double major in Statistics and Mathematics may count MATH 111, 121, (211 for statistics majors for which this is required), and MATH 271 or 272 towards both majors. A maximum of one additional course taken to complete the Mathematics major may be counted towards the Statistics major.
Comprehensive Evaluation. In the fall of their senior year, all Statistics majors will enroll in the capstone course STAT 495, and complete a capstone project under faculty supervision. An extension of the capstone project (completed in the spring semester of senior year) will serve as the basis for a comprehensive evaluation of each student’s achievement in the major. Each student’s project will be assessed by the Statistics faculty in the Department to determine if the student has successfully completed the comprehensive evaluation. (Those for whom the second semester of the junior year occurs in the fall should enroll in STAT 495 in that semester in order to complete the extension of the capstone project and satisfy the comprehensive evaluation).
Honors Program in Statistics. Students are admitted to the Honors Program on the basis of a qualifying examination given at the beginning of the spring semester of their junior year and the acceptance of a thesis proposal. (Those for whom the second semester of the junior year occurs in the fall may elect instead to take the qualifying examination at the beginning of that semester.) More information about the qualifying examination, including a detailed set of regulations, can be found on the Department website. If a student is accepted to the Honors Program, they will finalize a topic in consultation with Statistics faculty. After intensive study of this topic, the candidate will write a report in the form of a thesis which should be original in its presentation of material, if not in content. In addition, the candidate will report to the departmental colloquium on her or his thesis work during the senior year. Honors candidates are not required to complete additional coursework in statistics apart from the thesis courses.
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 a two-hour 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. Professor Daniels.
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. Four class meetings per week, one of which is a two-hour group-work day.
Requisite: MATH 105. Spring semester. Lecturer 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. Four class hours per week.
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 replaces one weekly class hour with a 90-to-120-minute group work day. Fall and spring semesters. Professor Folsom.
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. Four class hours per week.
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. Lecturer D. Benedetto and Professor Pflueger.
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. Four class hours per week, with occasional in-class computer labs.
Limited to 24 students. Fall semester. Professor Leise.
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 2017-18.
2023-24: Not offered(Offered as MATH 205, BLST 209 and HIST 209 [US]) This course will look at issues surrounding inequality in K-12 math education. Mathematics has a reputation for being something that either you can do or you can’t: the so-called "geniuses" know all the answers already, whereas for everyone else it is a constant struggle. In addition, math and other STEM fields have traditionally been discouraged as career paths for women and for students from underrepresented groups. At Amherst today, students from those groups are still in the minority in math classes. We’ll ask why this is, whether it can and should be changed, and if so, how.
Our discussions will be guided by some of the following questions: To what extent is math ability an innate talent that you are either born with or not? How and why is variation in accomplishment in mathematics related to race, gender and socio-economic class? What mathematics should we teach in schools and how should those teachers be prepared? What is "math phobia," how does it develop and how can it be treated? How do attitudes towards math in the general public affect student learning?
Limited to 25 students. Admission with consent of the instructor. Spring semester. Professors Ching and Moss.
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; the Taylor development and extrema of functions of several variables; implicit function theorems; Jacobians. Four class hours per week.
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. Visiting Professors Sosa Castillo and Zhang.
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 is an introduction to some topics in mathematics that do not require the calculus. The topics covered include logic, elementary set theory, functions, relations and equivalence relations, mathematical induction, counting principles, and graph theory. Additional topics may vary from year to year. This course serves as an introduction to mathematical thought and pays particular attention to helping students learn how to write proofs. Four class hours per week.
Limited to 25 students. Fall semester: Visiting Professor Juul. Spring semester: Visiting Professor Naqvi.
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 2025MATH 225 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, examine Julia sets, Mandelbrot sets, and study the concept of fractal dimension, among other things. Through the teaching of these concepts, MATH 225 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 35 students. Omitted 2017-18.
Other years: Offered in Spring 2023, Spring 2025About 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. Four class hours per week.
Requisite: Mathematics 121. Fall semester. Professor Sosa Castillo.
2023-24: Not offeredThe 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. The focus of the course will be on connecting the theoretical aspects of differential equations with real-world applications from physics, biology, chemistry, and engineering. Four class hours per week.
Requisite: MATH 211 or consent of the instructor. Limited to 25 students. Spring semester. Visiting Professor Zhang.
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. Four class meetings per week.
Requisite: MATH 121 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. Professor R. Benedetto.
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. MATH 272 will feature both proofs and applications, with special attention paid to applied topics such as least squares and singular value decomposition.
Four class hours per week, with occasional in-class computer labs.
Requisite: MATH 121 or consent of the instructor. This course and MATH 271 may not both be taken for credit. Limited to 25 students. Fall semester: Professor Pflueger. Spring semester: Visiting Professor Naqvi.
A 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. Four class hours per week.
Requisite: MATH 271 or 272 or permission of instructor. MATH 220 or other prior experience with mathematical proofs is recommended. Limited to 30 students. Spring semester. Visiting Professor Sosa Castillo.
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, tableaux, and stable marriage. Additional topics may vary.
Requisite: MATH 121, and MATH 220 or other prior experience with basic mathematical proof techniques (e.g., induction) by consent of instructor. Limited to 24 students. Fall semester. Professor Folsom.
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 permission of the instructor. Limited to 24 students. Omitted 2017-18.
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, including 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.
Requisite: MATH 211 and 271 or 272, or permission of the instructor. Limited to 30 students. Omitted 2017-18.
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 220 and 121, or other significant experience with proofs, or by consent of instructor. Limited to 24 students. Omitted 2017-18.
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. Fall semester. Post-doctoral Fellow Yacoubou Djima.
2023-24: Not offeredAn introduction to analytic functions; complex numbers, derivatives, conformal mappings, integrals. Cauchy’s theorem; power series, singularities, Laurent series, analytic continuation; Riemann surfaces; special functions. Four class hours per week.
Requisite: MATH 211 and prior experience with mathematical proofs, or consent of the instructor. Fall semester. Professor R. Benedetto.
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. Four class hours per week.
Requisite: 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. Fall semester: Visiting Professor Zhang. Spring: Visiting Professor Juul.
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 functions continuous on sets; infinite series, uniform convergence. The course may also study the Gamma function, Stirling’s formula, or Fourier series. Four class hours per week.
Requisite: MATH 211 and prior experience with mathematical proofs (MATH 271 or 272 recommended), 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. Fall semester: Professor Call. Spring semester: Visiting Professor Naqvi.
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 2025The p-adic numbers were first introduced by Kurt Hensel near the end of the nineteenth century. Since their introduction they have become a central object in modern number theory, algebraic geometry, and algebraic topology. These numbers give a new set of fields (one for each prime p) that contain the rational numbers and behave in some ways like the real numbers. While these fields are similar to the real numbers in some respects, they also possess some unique and unexpected properties. The main objective of this course will be to rigorously construct the p-adic numbers and explore some of the algebraic, analytic, and topological properties that make them interesting.
Requisite: MATH 350 or consent of the instructor. Limited to 32 students. Omitted 2017-18.
2023-24: Not offeredA 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 instructor. Limited to 24 students. Spring semester. Professor Leise.
2023-24: Not offeredRepresentation theory concerns the study of groups by expressing their elements as linear transformations of vector spaces. This approach gives us a more concrete way to think about groups via matrices, and it allows us to use tools from linear algebra to study them. Topics covered in this course include group actions, representations and modules, subrepresentations and homomorphisms, irreducibility, characters and character tables, and induced and restricted representations. We will also explore some applications to physics and other areas of mathematics.
Requisite: MATH 350 or consent of the instructor. Limited to 24 students. Priority to pre-registered students, math majors, and juniors and seniors. Omitted 2017-18.
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. Spring semester. Professor Cox.
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. Four class hours per week. Offered in alternate years.
Requisite: MATH 220, 271, 272, or 355, or consent of the instructor. Spring semester. Professor Ching.
Other years: Offered in Spring 2015, Spring 2018, Fall 2019, Spring 2022Fall 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 algebras originally arose as a way of studying certain continuous transformation groups called Lie groups. Lie algebras are simpler objects than Lie groups since they can be studied using tools from linear algebra, yet they still provide a lot of information about their associated Lie groups.
This class serves as a first introduction to the theory of Lie algebras. We will examine the structure of finite dimensional Lie algebras, ideals and homomorphisms, nilpotent and solvable Lie algebras, Cartan subalgebras, semisimplicity, 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.
Requisite: MATH 350 or consent of the instructor. Limited to 18 students. Fall semester. Visiting Professor Naqvi.
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. Omitted 2017-18.
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. Four class hours per week.
Requisite: MATH 355. Spring semester. Professor R. Benedetto.
Other years: Offered in Spring 2012, Spring 2014, Spring 2016An 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. Omitted 2017-18.
MATH 460 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, as well as an analytic proof. 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: At least two among MATH 345, MATH 350, and MATH 355, with MATH 345 preferred; or by consent of instructor. Prior experience with number theory, such as MATH 250, may be helpful but is not required. Spring 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 2025