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 Zhang.

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 Hews.

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. The Department.

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. The Department.

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 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. Omitted 2019-20.

Mathematical modeling is used by scientists to understand the dynamics of a system, make predictions and inform policy. It involves an iterative process of translating a real world problem into a mathematical expression, analyzing it using mathematical tools and numerical simulations, and interpreting the results. This course covers systems thinking and the principles of mathematical modeling in the context of environmental problems. Group projects will be centered on current environmental research questions. Contributions to the projects will be tailored to each student’s level of mathematical preparation and interest. Four class hours per week.

Limited to 24 students. This course and Math 140 cannot both be counted towards the Mathematics major. Spring semester. Professor Hews.

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. Omitted 2019-20. Professor Leise.

(Offered as MATH 205, BLST 309 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. Omitted 2019-20.

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 and spring semesters. The Department.

This course serves as an introduction to mathematical reasoning and pays particular attention to helping students learn how to write proofs. The topics covered may include logic, elementary set theory, functions, relations and equivalence relations, mathematical induction, sequences, and quantifiers. Additional topics may vary from semester to semester. Four class hours per week.

Limited to 25 students. Fall semester: Professor Culiuc. Spring semester: Professor Polanco.

This course is an introduction to proofs and abstract mathematical thinking, serving as a bridge from introductory courses such as calculus to more advanced proof-based courses. The principal goal of this course is to help students develop skills for both reading and writing mathematical proofs. Topics covered may include fundamentals of logic, quantifiers, proof techniques, mathematical induction, elementary set theory, equivalence relations, functions, and the notions of countability and uncountability. Some topics in analysis will also be surveyed, such as open and closed sets in the real line, sequences of real numbers, and limits of functions. Additional topics may vary from year to year.

Four class hours per week. MATH 220 and 221 may not both be taken for credit. Limited to 25 students. Omitted 2019-20. Professor Alvarado.

This course is a mathematical treatment of fractal geometry, a field of mathematics partly developed by Benoit Mandelbrot (1924–2010) that continues to be actively researched in the present day. Fractal geometry is a mathematical examination of the concepts of self-similarity, fractals, and chaos, and their applications to the modeling of natural phenomena. In particular, we will develop the iterated function system (IFS) method for describing fractals, examine Julia sets, Mandelbrot sets, and study the concept of fractal dimension, among other things. Through the teaching of these concepts, the course will also lend itself to familiarizing students with some of the formalisms and rigor of mathematical proofs.

Requisite: MATH 211 or consent of the instructor. Limited to 35 students. Fall Semester. Professor Folsom.

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. Limited to 25 students. Spring semester. Visiting Assistant Professor Polanco.

Many security problems arise when two computers must communicate on a channel with eavesdroppers or malicious attackers. Public-key cryptography applies ideas from number theory and abstract algebra to address these problems. This course concerns the mathematical theory and algorithms needed to construct the most commonly-used public-key ciphers and digital signature schemes, as well as the attacks that must be anticipated when designing such systems. Several topics from number theory, abstract algebra, and algorithms will be introduced, including discrete logarithms, integer factorization algorithms, and elliptic curves. Depending on time and student interest, we may cover some newer systems that are believed to be secure against attacks by quantum computers but not yet commonly implemented in practice. Students will write short programs to implement the systems and to break badly implemented systems. No prior programming experience is expected; basic aspects of programming in Python will be taught in class. Four class hours per week.

Requisite: Experience writing proofs, such as MATH 220/221 or 271/272, or consent of the instructor. Spring semester. Professor Pflueger.

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: MATH 121. Omitted 2019-20.

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. Limited to 25 students. Spring semester. Professor Yacoubou Djima.

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 and spring semesters. Professor Gunturkun.

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. This course 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 Pfueger. Spring semester: Professor Culiuc.

This course is a continuation of the material in MATH 271 and 272, providing more insight into abstract vector spaces and operator theory. Topics may include least squares estimates, singular value decompositions, Jordan canonical forms, inner product spaces, linear functionals and duals, orthogonal polynomials, vector and matrix norms, the spectral theorem, eigenvalue inequalities, and error-correcting codes. Time permitting, applications to graph theory and discrete dynamical systems may be explored. Four class hours per week.

Requisites: MATH 271, MATH 272, or consent of the instructor. Limited to 25 students. Omitted 2019-20. Professor Culiuc.

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 consent of the instructor. MATH 220 or other prior experience with mathematical proofs is recommended. Limited to 30 students. Fall Semester. Professor Contreras Palacios.

This 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 the instructor. Limited to 24 students. Omitted 2019-20.

This course will study numerical techniques for a variety of problems, such as finding roots of polynomials, interpolation, numerical integration, numerical solutions of differential equations, and matrix computations. We will study the underlying theory behind the algorithms, including error analysis, and the algorithms will be implemented using mathematical software to facilitate numerical experimentation.

Requisite: MATH 211 and either 271 or 272, or consent of the instructor. Limited to 24 students. Omitted 2019-20. Professor Yacoubou Djima.

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 121 and 220, or other significant experience with proofs, or by consent of the instructor. Limited to 24 students. Omitted 2019-20. Professor Folsom.

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 Yacoubou Djima.

Network structures and network dynamics are a fundamental modern tool for modeling a broad range of problems from fields like economics, biology, physics, and sociology. Mathematical and machine learning techniques can be used to reveal underlying network structures. The course will use graphs (sets of nodes connected by edges) as a common language to describe networks and their properties. On the theoretical side, the course will cover topics such as basic probability, degree distribution, spectral graph theory (adjacency matrix, graph Laplacian), diffusion geometries, and random graph models. Applications will range over topics such as epidemics, marketing, prediction of new links in a social network, and game theory. The course will also include hands-on experiments and simulations. Three class meetings per week.

Requisite: MATH 271 or MATH 272 or instructor's permission. Limited to 24 students. Spring semester. Professor Yacoubou Djima.

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 Culiuc.

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 211 and either MATH 271 or 272, or consent of the instructor. Students with a grade of B+ or lower in linear algebra are encouraged to take another 200-level course with proofs before taking MATH 350.

Limited to 25 students. Fall semester: Professor Pfueger. Spring semester: Professor Gunturkun.

Completeness of the real numbers; topology of n-space including the Bolzano-Weierstrass and Heine-Borel theorems; sequences, properties of continuous functions on sets; infinite series, uniform convergence. The course may also study the Gamma function, Stirling’s formula, or Fourier series. Four class hours per week.

Requisite: MATH 211 and either MATH 271 or 272, or consent of the instructor. Students with a grade of B+ or lower in linear algebra are encouraged to take another 200-level course with proofs before taking MATH 355.

Limited to 25 students. Fall semester: Professor Alvarado. Spring semester: Professor Zhang.

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, but 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. Four class meetings per week.

Requisite: MATH 350 and MATH 355, or consent of the instructor. Limited to 32 students. Omitted 2019-20. Professor Daniels.

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 the instructor. Limited to 24 students. Omitted 2019-20.

This course builds upon the material in MATH 355 (Introduction to Analysis) in order to rigorously develop basic tools for studying functions of more than one real variable. While the setting in MATH 355 is the real number line, the context for this course will be the n-dimensional Euclidean space. Many facets of analysis on this n-dimensional space will be explored including its topological properties as well as differentiation and Riemann integration in n-variables. The course will cover fundamental results such as the celebrated implicit and inverse function theorems. Time permitting, we may discuss additional topics such as analysis on metric spaces. Four class hours per week.

Requisite: MATH 355 and either MATH 271 or 272; or consent of the instructor. Limited to 25 students. Spring semester. Professor Alvarado.

This 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. Omitted 2019-20.

The study of geometric objects by means of their defining equations dates back to the introduction of coordinates by Descartes in 1637. The advent of computers, along with the increase in their processing speed in the last sixty years, has revolutionized the subject, shaping the fields of computational commutative algebra and computational algebraic geometry.

This course will start by studying the theory of Gröbner bases, introduced in 1965, which make possible the implementation of algorithms that facilitate the manipulation and understanding of algebraic equations. We will also develop a dictionary between algebra and geometry, exploring the structure of ideals in polynomial rings and their quotients. In addition, we will discuss the significance of monomial and binomial ideals. The course will end with student presentations on applications of algebraic geometry to robotics, invariant theory, graph theory, algebraic statistics, and other topics. Four class hours per week, including a weekly one-hour computer lab.

Requisite: MATH 350 or consent of the instructor. Omitted 2019-20.

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. Fall semester. Professor Ching.

Independent reading course.

Fall and spring semesters. The Department.

Lie groups and Lie algebras appear naturally in the study of symmetries of geometric objects. Lie algebras carry local information and can be studied using tools from linear algebra. Finite dimensional Lie groups can be constructed using techniques from calculus and group theory.

This course serves as a first introduction to Lie groups and Lie algebras. We will examine the structure of finite dimensional matrix Lie groups, the exponential and differentiation maps, as well as compact Lie groups. We will also study Lie algebras, ideals and homomorphisms, nilpotent and solvable Lie algebras, Cartan subalgebras, semisimplicity and root systems, and the classification of semisimple Lie algebras. This classification is not only a fundamental result in Lie Theory, but is also an archetype of classifications that appear in other areas of math. More amazingly, this classification is embodied in simple combinatorial pictures called Dynkin diagrams, which underlie surprisingly disparate fields, such as geometric group theory, quiver representation theory, and string theory. Four class meetings per week.

Requisite: MATH 350 or consent of the instructor. Limited to 18 students. Omitted 2019-20. Professor Contreras Palacios.

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. Fall semester. Professor R. Benedetto.

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 Culiuc.

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. Omitted 2019-20. The Department.

This course is an introduction to Analytic Number Theory, a foundational subject in mathematics which dates back to the 1800s and is still a major research area today. The subject generally uses tools and techniques which are analytic in nature to solve problems primarily related to integers. Asymptotic and summation results and methods are of great significance in Analytic Number Theory. Two primary course objectives are to state and prove two major theorems: Dirichlet's Theorem on Primes in Arithmetic Progressions, and the Prime Number Theorem. In particular, we will study Selberg's "elementary" proof of the Prime Number Theorem, 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 the instructor. Prior experience with number theory, such as MATH 250, may be helpful but is not required. Omitted 2019-20.

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