- Introduction
- About Amherst College
- Admission & Financial Aid
- Regulations & Requirements
- Amherst College Courses
- Five College Programs & Certificates
- Honors & Fellowships

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- 01- Bruss Seminar
- 02- Kenan Colloquium
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- 04- Linguistics
- 05- Mellon Seminar
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- 09- Five College Dance

Professors R. Benedetto, Call*, Cox, Horton, and Velleman‡; Associate Professors Folsom*, Leise (Chair) and Wagaman*; Assistant Professors Ching and Liao†; Visiting Assistant Professors Daniels, Juul, Kim, Naqvi, Sosa, and Zhang; Lecturers D. Benedetto and Wang.

*On leave 2015-16.

†On leave fall semester 2015-16.

‡On leave spring semester 2015-16.

The Department offers the major in Mathematics and the major in Statistics, as well as courses meeting a wide variety of interests 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, and three other elective courses in Mathematics numbered 135 or higher. In addition, a major must complete two other courses, each of which is either an elective course in Mathematics numbered 135 or higher or a course from outside Mathematics chosen from among: COSC 201, 301, 401; ECON 300, 301, 361, 420; PHIL 350; any Physics course numbered 116 or higher (excluding PHYS 227); and STAT 230, 235 (formerly 335), 240 (formerly 330), 495. (Note: this requirement can be satisfied by taking two math electives, one math elective and one outside course, or two outside 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.

Students who have taken MATH 130 may count it as an elective for the major, and students who declared their Mathematics major before May 17, 2014 may count toward the major an approved outside course together with a requisite for that course chosen from the same discipline.

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, are excused from taking those courses. Students who place out of Math 111, 121 or 211 do not need to replace these courses. Beginning with the class of 2016, students who place out of MATH 271, 272, 350, or 355 by taking a competency 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).

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 for majors who are not participating in the Honors Program 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. A document describing the comprehensive examination can be obtained from 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. (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.) The examination is described in a document available from the Department website. Before the end of the junior year, an individual thesis topic will be selected by the Honors candidate in conference with a member of the Department. 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 also required to complete MATH 345 and either MATH 450 or 455.

** Statistics**

*Major Program.* The minimum requirements for the Statistics major include MATH 111, 121, 211; MATH 271 or 272; STAT 111 or STAT 135; MATH 140 or COSC 111; STAT 230, 360, 370 (formerly 430), 495; and one additional elective course in Statistics. The additional elective may be STAT 235 (formerly 335), STAT 240 (formerly 330), or another approved elective. 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 Statistics courses numbered 200 or higher, approved Mathematics courses numbered 200 or higher, Computer Science courses numbered 112 or higher, or other 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).

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 355 (Introduction to Analysis) as well as its continuation course, MATH 450 (Functions of a Real Variable).

*Double Majors in Statistics and Mathematics. *Students electing a double major in Statistics and Mathematics may count MATH 111, 121, 211, 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. (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). The qualifying examination covers MATH 211 and 271 or 272, as well as STAT 360. The portion covering MATH 211 and 271 or 272 is equivalent to that portion on the Mathematics honors qualifying examination. The examination is described in a document which can be obtained from the Department website. Before the end of the junior year, an individual thesis topic will be selected by the Honors candidate in conference with a member of the Department. 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 TBA.

MATH 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. Professor TBA.

Basic 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 35 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. Professors TBA.

A 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 consent of the Department. Limited to 35 students per section. Fall and spring semesters. Professor TBA.

Mathematical 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 such as environmental studies and biology (e.g., air pollution, ground water flow, populations of interacting species, social networks). 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. Spring semester. Professor TBA.

Elementary 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 the consent of the instructor. Limited to 35 students per section. Fall and spring semesters. Professors TBA.

This 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. No limit for spring semester. Fall and spring semesters. Professors TBA.

MATH 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 2015-16. Professor Folsom.

(Offered as MATH 240 and BIOL 240.) With new experimental techniques leading to large biological data sets of increased quality, the ability to analyze biological systems using mathematical modeling approaches has become an integral part of modern biology. This course aims to provide students interested in the interface between biology and mathematics with an integrated understanding of some of the mathematical and computational techniques used in this field. The mathematical approaches we will use to study biological systems will include discrete and continuous dynamical models as well as probability models and parameter estimation algorithms.

Requisite: MATH 211 and BIOL 181 or 191, or permission of the instructor. Limited to 24 students. Omitted 2015-16.

An 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. Four class hours per week. Offered in alternate years.

Requisite: MATH 121 or consent of the instructor. Spring semester. Professor TBA.

About 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 Velleman

The 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. Spring semester. Professor TBA.

The 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. Special attention will be 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. Fall and Spring semester.

The 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. Additional topics include ill-conditioned systems of equations, the LU decomposition, covariance matrices, least squares, and the singular value decomposition. Recommended for Economics majors who wish to learn linear algebra. 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. Spring semester.

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 the instructor. Recommended: MATH 220 or other prior experience with mathematical proofs. Limited to 30 students. Spring semester. Professor Sosa.

The 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. Professor TBA.

An 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 TBA.

A 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. Professor TBA. No limit for spring semester. Professor TBA.

Completeness 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 of 25 students fall semester. Professor TBA. No limit for spring semester. Professor TBA.

A 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 TBA.

Most mathematicians consider set theory to be the foundation of mathematics, because everything that is studied in mathematics can be defined in terms of the concepts of set theory, and all the theorems of mathematics can be proven from the axioms of set theory. This course will begin with the axiomatization of set theory that was developed by Ernst Zermelo and Abraham Fraenkel in the early part of the twentieth century. We will then see how all of the number systems used in mathematics are defined in set theory, and how the fundamental properties of these number systems can be proven from the Zermelo-Fraenkel axioms. Other topics will include the axiom of choice, infinite cardinal and ordinal numbers, and models of set theory. Four class hours per week.

Requisite: MATH 220, 271, 272, or 355, or consent of the instructor. Fall semester. Professor TBA.

Mathematicians 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. Omitted 2015-16.

Fall and spring semesters. The Department.

The 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. Professor Ching.

The topic will vary from year to year. The topic for 2014 was computational algebraic geometry.

The study of geometric objects by means of their defining equations dates back to the introduction of coordinates by Descartes in 1637.

This course will introduce algorithmic methods for manipulating and understanding algebraic equations and will develop a dictionary between algebra and geometry. We will also explore the structure of ideals in polynomial rings and the resulting quotient rings. The course will end with student presentations on applications of algebraic geometry to robotics, geometric theorem proving, invariant theory, graph theory, and sudoku. Three class hours per week plus a weekly one-hour computer lab.

Requisite: MATH 350. Limited to 16 students. Omitted 2015-16.

An 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 TBA.

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. Professor TBA.

Open to seniors with the consent of the Department. Fall semester. The Department.

- Five College Courses
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