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

- General Regulations
- Terms and Vacations
- Conduct
- Attendance at College Exercises
- Records and Reports
- Pass/Fail Option
- Examinations and Extensions
- Withdrawals
- Readmission
- Deficiencies
- Housing and Meal Plans
- Degree Requirements
- Course Requirements
- The Liberal Studies Curriculum
- The Major Requirement
- Departmental Majors
- Interdisciplinary Majors
- Comprehensive Requirement
- Degree with Honors
- Independent Scholar Program
- Field Study
- Five College Courses
- Academic Credit from Other Institutions
- Cooperative Doctor of Philosophy
- Engineering Exchange Program with Dartmouth

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- 01- Bruss Seminar
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Professors R. Benedetto‡ (Chair of Mathematics, fall), Call, Ching, Folsom (Chair of Mathematics, spring), Horton, Sidman, Wagaman, and Zurieck-Brown; Associate Professor Daniels*; Assistant Professors Alvarado, Bailey*, Contreras, Correia, Gong, Kraisler, Liao, and Pflueger; Senior Lecturers D. Benedetto and Y. Zhang; Lecturers Donges and Matheson; Postdoctoral Fellow Kuzbary; Visiting Professor Cox (Chair of Statistics); Visiting Assistant Professors Elliott, Moore, and Rasheed; Visiting Lecturer Georgiou.

*On leave the entire academic year‡On leave spring semester

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, 150, and 220, and STAT 111, none of which require a background beyond high school mathematics.

**Mathematics**

*Major Program*. Mathematics majors must complete MATH 111, 121, 211, 271 or 272, 350, and 355. In place of MATH 111, majors may complete MATH 105 and 106. Along with the required courses, a major must complete three elective courses in Mathematics numbered between 135 and 490. 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 CHEM 361; COSC 211, 311, and 401; ECON 224, 300, 301, 331, 361, and 420; GEOL 341; any Physics course numbered 116 or higher; ASTR 200 (which cannot be counted towards the major in addition to MATH/STAT 135); any Astronomy course numbered 226 or higher; 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.

Mathematics majors may not apply a Pass grade from a pass/fail option to any of the core courses required for the mathematics major: MATH 111, 121, 211, 271/272, 350, or 355 (exceptions by petition to the department). Pass grades may be applied to electives.

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 or 121 do not need to replace these courses. Students who place out of MATH 211 or MATH 271/272 must replace them with an additional Mathematics course numbered between 225 and 490. Students who place out of MATH 350 and/or 355 must replace each such course with an additional Mathematics course numbered between 300 and 490.

A student considering a major in Mathematics should consult with a member of the Department as soon as possible, preferably during their 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).

To complement the abstract thinking and mathematical proof-writing skills in the core courses, 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). We also encourage all Mathematics majors to attend department colloquia to learn more about different areas of mathematics and engage in the larger mathematics community.

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 and MATH 121 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 Multivariable Calculus (MATH 211) and Linear Algebra (MATH 271 or 272). 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. *The minimum requirements for the Statistics major include MATH 111 and 121; STAT 111, 135, or 136 (STAT 135 or 136 strongly recommended); STAT 230, 231, 360, 370, and 495; and at least 4 elective courses from among groups A, B, and C below. The combination of elective courses counted towards the Statistics major must be one of the following:

3 courses from group A and 1 course from group B;2 courses from group A and 2 courses from group B;2 courses from group A, 1 course from group B, and 1 course from group C;1 course from group A, 2 courses from group B, and 1 course from group C.

The groups:

Statistics courses numbered 200 or above (statistical breadth);Computer Science courses: COSC 111, COSC 112, COSC 211, COSC 311 (computing requirement);A course numbered 200 or above that has been approved by the department as part of a student’s declared domain application, or a course from the following list of pre-approved courses that supplement the statistics curriculum:

BIOL/CHEM 250, COSC 254, COSC 257, COSC 355, MATH 271, MATH 272, MATH 355, MATH 365.

Statistics majors are strongly encouraged to ensure that their course of study includes depth in a domain application (e.g., astronomy, environmental studies, political science, psychology, or sociology). Majors are encouraged to discuss with their academic advisor which courses might be approved as part of a domain application as described in group C.

Statistics majors may not apply a Pass grade from a pass/fail option to any of the core courses required for the statistics major: STAT 111/135/136, 230, 231, 360, 370, or 495 (exceptions by petition to the department).

Requests for alternative courses must be approved in writing by the Statistics faculty within the Department. Students may submit petitions for such courses via their advisors or by contacting the Department Chair. 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, ideally during the first year. This will facilitate the arrangement of a program best suited to the student’s ability and interests, with plans to accommodate study away, if desired. 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 211, MATH 271 or 272, MATH 355, MATH 450, and additional courses with a focus on computation and algorithmic thinking (often found in computer science).

*Double majors in Mathematics and Statistics.* Note that for double majors, MATH 111, MATH 121, and at most one other course (usually MATH/STAT 360) can be counted towards both the Mathematics and Statistics majors. Aside from this permitted overlap, statistics, mathematics, or computer science courses counted towards the statistics major may not also be counted towards the mathematics major.

*Comprehensive Evaluation. *All Statistics majors will enroll in the capstone course STAT 495 (Advanced Data Analysis). Beginning with the class of 2021, successful completion of this course satisfies the comprehensive evaluation in Statistics.

*Honors Program in Statistics. *For a degree with Honors, a student must have demonstrated the ability to pursue independent work fruitfully and exhibit a strong motivation to engage in research. To apply to the Honors Program, students must have an average grade of B+ or higher in Statistics 230, 231, 360, and 370. Students are also expected to have completed STAT 370 before the first semester of thesis work is undertaken. E students interested in theses should plan accordingly. Students are admitted to the Honors Program on the basis of a thesis proposal which must be accepted by the department. More information about the Honors Program can be found on the Department website.

If a student is accepted to the Honors Program, they will finalize their topic in consultation with their assigned advisor. After intensive study of this topic (one thesis course during each semester of the senior year, or one thesis during the first semester and a double thesis course during the second semester of the senior year), the Honors candidate will write a report in the form of a thesis which should be original in its presentation of material, if not in content, which is then evaluated by the Statistics faculty. In addition, the candidate will give a 30-40 minute presentation of (some of) the material, followed by faculty questions. This presentation occurs during the second semester of the senior year.

Honors students must complete an additional course for the major. The course must be broadly related to the thesis (e.g., a statistics elective with an introduction to the topic; linear algebra for more theoretical theses; an advanced Computer Science course for more computational theses). The thesis advisor must approve the course as “related”. Mathematics and Statistics double majors should remember that only one course beyond calculus can count towards both majors.

MATH 105 and 106 are designed for students whose background and algebraic skills are inadequate for the fast pace of MATH 111. In addition to covering the usual material of beginning calculus, these courses will have an extensive review of algebra and trigonometry. There will be a special emphasis on solving word problems.

MATH 105 starts with a quick review of algebraic manipulations, inequalities, absolute values and straight lines. Then the basic ideas of calculus—limits, derivatives, and integrals—are introduced, but only in the context of polynomial and rational functions. As various applications are studied, the algebraic techniques involved will be reviewed in more detail. When covering related rates and maximum-minimum problems, time will be spent learning how to approach, analyze and solve word problems. Four class meetings per week, one of which is an 80-minute group-work day.

Note: While MATH 105 and 106 are sufficient for any course with a MATH 111 requisite, MATH 105 alone is not. However, students who plan to take MATH 121 should consider taking MATH 105 and then MATH 111, rather than MATH 106. Students cannot register for both MATH 105 and CHEM 151 in the same semester.

Fall semester. The Department.

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.

Requisite: MATH 105. Spring semester. Professor D. Benedetto.

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.

Requisite: Math Placement into 111, or consent of the Department. Limited to 30 students per section. Fall and spring semesters. In the fall semester, the intensive section (Section 01) is open only to students listed as eligible on the Mathematics placement list. The intensive section includes one additional weekly class meeting. The Department.

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.

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.

Limited to 24 students. Spring semester. Professor Moore.

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 2023-24.

Elementary vector calculus; introduction to partial derivatives; multiple integrals in two and three dimensions; line integrals in the plane; Green’s theorem.

Requisite: A grade of C or better in MATH 121 or placement into MATH 211 or consent of the Department. Limited to 30 students per section. Fall and spring semesters. The Department.

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.

Limited to 25 students. Spring and fall semesters. The Department.

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, study the concept of fractal dimension among other theoretical concepts, and examine Julia and Mandelbrot sets (time permitting). Through the teaching of these concepts, the course will also lend itself to familiarizing students with some of the formalisms and rigor of mathematical proofs.

Requisite: MATH 211 or consent of the instructor. Limited to 25 students. Professor Folsom.

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.

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.

Requisite: MATH 121. Fall semester.

This course is a survey of geometry in dimensions 2, 3, 4, and higher. We will consider questions such as: How do we know the angles of a triangle add up to 180 degrees? (Spoiler: usually they don't.) What are the different ways we could tile our kitchen floor? How many tennis balls fit in a bucket? How many regular polyhedra are there in four dimensions? And what shape is the universe? We will examine how mathematicians and scientists, from Euclid to Einstein, have tried to give definitive answers to these questions, and we will explore some of what is still unknown about how the mathematical world reflects what we see around us.

Requisite: MATH 211 or instructor permission. Limited to 25 students. Fall semester. Professor Ching.

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.

Requisite: MATH 211 or consent of the instructor. Limited to 25 students. Fall 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. MATH 271 will feature both proofs and applications, with special attention paid to the theoretical development of the subject.

Requisite: MATH 211 or 220, or consent of the instructor. This course and MATH 272 may not both be taken for credit. Limited to 25 students. Fall and spring semesters. The Department.

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.

Requisite: MATH 211 or 220, or consent of the instructor. This course and MATH 271 may not both be taken for credit. Limited to 25 students. Fall and Spring semester: The Department.

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. Spring semester. Limited to 25 students.

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.

Requisite: MATH 271 or 272 or consent of the instructor. MATH 220 or other prior experience with mathematical proofs is recommended. Limited to 25 students. Fall semester. Professor Contreras.

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. Additional topics may vary.

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

Requisite: MATH 211 and either 271 or 272, or consent of the instructor. Limited to 24 students. Spring semester.

Optimization is a branch of applied mathematics focused on algorithms to determine maxima and minima of functions, often under constraints. Applications range from economics and finance to machine learning and information retrieval. This course will first develop advanced linear algebra tools, and then will study methods of convex optimization. Possible topics include linear, quadratic, second-order cone, and semidefinite models. Several applications will be explored, and algorithms will be implemented using mathematical software to aid numerical experimentation.

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 2021-22.

The central object of study in algebraic geometry is a system of polynomial equations and its solution set. The theory has connections to many other areas in pure mathematics including number theory, representation theory, several complex variables, and combinatorics. Applied algebraic geometry is used in areas including mathematical biology, algebraic statistics, robotics, and computer vision. In this course, we will study algebraic geometry from a computational point of view, beginning with Buchberger's algorithm for computing Groebner bases, and working towards the ideal-variety correspondence. Topics may include projective algebraic geometry, toric varieties, and applications, depending on student interest.

Requisite: MATH 271 or 272, and experience with writing mathematical proofs (such as in MATH 220); or permission of the instructor.Limited to 25 students. Fall semester. The Department.

Pending Faculty Approval

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 2021-22.

What is the shape of a hanging chain? What shape of fixed perimeter encloses the most area? The calculus of variations answers questions such as these through maximizing or minimizing values of functionals over different input functions. Topics covered will include functional differentiation, the Euler-Lagrange equations, necessary and sufficient conditions for extrema, and direct minimization methods. Additionally, there will be applications to physics and optimization, including Hamiltonian mechanics and image processing.

Fall semester. Professor Kraisler

An introduction to analytic functions; complex numbers, derivatives, conformal mappings, integrals. Cauchy’s theorem; power series, singularities, Laurent series, analytic continuation; special functions.

Requisite: MATH 211 and prior experience with mathematical proofs, or consent of the instructor. Limited to 25 students. Fall semester.

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.

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.

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.

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

Limited to 25 students. The Department.

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. Professor Leise.

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. Omitted 2021-22.

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.

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.

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. Fall semester.

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. Prof. Daniels.

Commutative algebra is known as the study of commutative rings and their ideals and modules. Besides being an important branch of algebra for its own sake, commutative algebra has strong ties to other areas, such as algebraic geometry and algebraic number theory, as it provides essential tools for them. This course is an introductory course in commutative algebra. We will explore more about rings (especially polynomial rings) and ideals, which are taught in Math 350. We will also introduce another important algebraic structure, namely modules over rings. Other fundamental topics include Noetherian rings, The Hilbert Basis Theorem, Gröbner bases, localization, primary decompositions, and tensor products.

Requisite: Math 350 or consent of the instructor. Limited to 24 students. Omitted 2023-24. Visiting Assistant Professor Gunturkun.

An elliptic curve is the set of zeros of a cubic polynomial in two variables. If the polynomial has rational coefficients, it is natural to ask for a description of those zeros whose coordinates are either integers or rational numbers. Our study of elliptic curves will focus on this fundamental problem and reveal a fascinating interplay between algebra, geometry, analysis and number theory. Topics discussed will include the geometry and group structure of elliptic curves, the Nagell-Lutz Theorem describing points of finite order, and the Mordell-Weil theorem on the finite generation of the group of rational points. Additional topics may include elliptic curve cryptography, Lenstra's algorithm using elliptic curves to factor large integers, the Thue-Siegel Theorem on the finiteness of the set of integer points, and the crucial role the theory of elliptic curves played in Wiles' proof of Fermat's Last Theorem. By bringing together techniques from a wide range of mathematical disciplines, we plan to illustrate the unity of mathematics and introduce active areas of research. Four class hours per week.

Requisite: MATH 350 or consent of the instructor. Omitted 2021-22.

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.

This course will explore the geometry of curves, surfaces and higher dimensional geometric objects in *n*-dimensional Euclidean space. For curves, the key concepts are curvature and torsion, while for surfaces, the key players are Gaussian curvature, geodesics, and the Gauss-Bonnet Theorem. Other topics covered may include (time permitting) two-dimensional Riemmanian geometry, differential forms and manifolds. MATH 378 and MATH 408 may not both be taken for credit.

Spring semester. Professor Contreras

An introduction to general topology: the topology of Euclidean, metric and abstract spaces, with emphasis on such notions as continuous mappings, compactness, connectedness, completeness, separable spaces, separation axioms, and metrizable spaces. Additional topics may be selected to illustrate applications of topology in analysis or to introduce the student briefly to algebraic topology. Four class hours per week. Offered in alternate years.

Requisite: MATH 355. Spring semester. Prof. Rasheed.

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. Additional topics may include: arithmetic functions, especially their averages, their asymptotics, and related summation formulae; Dirichlet convolutions; characters and Gauss sums; and an introduction to Dirichlet series, such as the Riemann zeta-function and L-functions. Further topics may vary.

Requisite: MATH 355. Limited to 25 students. Fall semester. Professor Folsom.

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

- Five College Courses
- African Studies Certificate
- Asian Pacific American Studies Certificate
- Biomathematics
- Buddhist Studies Certificate
- Coastal and Marine Sciences Certificate
- Culture Health Science Certificate
- Ethnomusicology Certificate
- International Relations Certificate
- Latin American Caribbean Latino Studies Certificate
- Logic Certificate
- Middle Eastern Studies Certificate
- Native American and Indigenous Studies Certificate
- Queer and Sexuality Studies Certificate
- Reproductive Health, Rights and Justice Certificate
- Russian East European Eurasian Studies Certificate
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