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Calculus with Algebra

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 R. Benedetto.

Calculus with Elementary Functions

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. Lecturer D. Benedetto.

Introduction to the Calculus

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 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 semester: TBA. Spring semester: Professor Folsom.

Intermediate Calculus

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 placement into MATH 121 or consent of the Department. Limited to 30 students per section. Fall and spring semesters. Both semesters. Professor Call and Lecturer D. Benedetto.

Introduction to Statistics via Modeling

(Offered as STAT 135 and MATH 135.) Introduction to Statistics via Modeling is an introductory statistics course that uses modeling as a unifying framework. The course provides a basic foundation in statistics with a major emphasis on constructing models from data. Students learn important concepts of statistics by mastering powerful and relatively advanced statistical techniques using computational tools. Topics include descriptive and inferential statistics, probability, study design, ANOVA, and multiple regression. Students who are majoring in mathematics should take this course instead of STAT 111. (Students may not receive credit for both STAT 111 and MATH 135.) Four class hours per week (two will be held in the computer lab).

Requisite: MATH 111. Limited to 24 students. Fall semester: Professor Horton, Lecturer Jeneralczuk and TBA. Spring semester: Professors Horton, Liao, and Matheson.

Mathematical Modeling

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. Omitted 2016-17.

Voting and Elections: A Mathematical Perspective

The 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. Fall semester. Professor Leise.

≠ (Inequality)

(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.

Multivariable Calculus

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 placement into MATH 211 or consent of the Department. Limited to 30 students per section. Fall semester: Professor R. Benedetto and Visiting Assistant Professors Naqvi, Sosa Castillo and Zhang. Spring: Professor Leise and Visiting Professor Zhang.

Discrete Mathematics

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: Visiting Professor Juul. Spring semester: Visiting Professor Naqvi.

Fractal Geometry

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. Fall semester. Professor Folsom.

Mathematical Modeling of Biological Systems

(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 2016-17.

Number Theory

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. Visiting Professor Juul.

Geometry

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. Omitted 2016-17.

Differential Equations

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. Visiting Professor Sosa Castillo.

Linear Algebra

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. 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 semester: Professors Ching and Daniels. Spring semester: Professor Ching.

Linear Algebra with Applications

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. 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. Spring semester. Visiting Professor Naqvi.

Graph Theory

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. Omitted 2016-17.

Numerical Analysis

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. Fall semester. Professor Leise.

Optimization

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 either 271 or 272, or permission of the instructor. Limited to 30 students. Spring semester. Professor Leise.

Introduction to the Theory of Partitions

The 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.* *Fall semester. Professor Folsom.

Wavelet and Fourier Analysis

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. Omitted 2016-17. Professor TBA.

Functions of a Complex Variable

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. Visiting Professor Zhang.

Groups, Rings and Fields

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: Visiting Professor Sosa Castillo. Spring: Visiting Professor Juul.

Introduction to Analysis

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 Call. Spring semester: Professor Cox.

An Introduction to the p-adic Numbers

The 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. Spring semester. Professor Daniels.

Probability

(Offered as STAT 360 and MATH 360.) This course explores the nature of probability and its use in modeling real world phenomena. There are two explicit complementary goals: to explore probability theory and its use in applied settings, and to learn parallel analytic and empirical problem solving skills. The course begins with the development of an intuitive feel for probabilistic thinking, based on the simple yet subtle idea of counting. It then evolves toward the rigorous study of discrete and continuous probability spaces, independence, conditional probability, expectation, and variance. Distributions covered include the binomial, hypergeometric, Poisson, normal, Gamma, Beta, multinomial, and bivariate normal. Other topics include generating functions, order statistics, and limit theorems. Four class hours per week.

Requisite: MATH 121 or consent of the instructor. Limited to 24 students. Fall semester. Professor Wagaman.

Stochastic Processes

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. Omitted 2016-17. Professor TBA.

Theoretical Statistics

(Offered as STAT 370 and MATH 370.) This course examines the theory underlying common statistical procedures including visualization, exploratory analysis, estimation, hypothesis testing, modeling, and Bayesian inference. Topics include maximum likelihood estimators, sufficient statistics, confidence intervals, hypothesis testing and test selection, non-parametric procedures, and linear models. Four class hours per week.

Requisite: STAT 111 or STAT 135 and STAT 360, or consent of the instructor. Limited to 25 students. Spring semester. Professor Horton.

Introduction to Representation Theory

Representation 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. Fall semester. Visiting Professor Naqvi.

Set Theory

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.

Mathematical Logic

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 2016-17.

Special Topics

Fall and spring semesters. The Department.

Galois Theory

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. Omitted 2016-17. Professor Ching.

Topics in Mathematics

The topic will vary from year to year. 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 2016-17.

Functions of a Real Variable

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. Omitted 2016-17. Professor TBA.

Topology

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. Visiting Professor Zhang.

Senior Departmental Honors

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