Noah Aydin

This course presents an introduction to computer programming intended both for those who plan to take further courses in which a strong background in computation is desirable and for those who are interested in learning basic programming principles. The course will expose the student to a variety of applications where an algorithmic approach is natural and will include both numerical and non-numerical computation. The principles of program structure and style will be emphasized. May be paired with COMP 218 or with any mathematics or statistics course to satisfy the natural science diversification requirement. No prerequisite. Offered every semester.

This course is intended as a second course in programming, as well as an introduction to the concept of computational complexity and the major abstract data structures (such as dynamic arrays, stacks, queues, link lists, graphs and trees), their implementation and application, and the role they play in the design of efficient algorithms. Students will be required to write a number of programs using a high-level language. May be paired with COMP 118 or any mathematics or statistics course to satisfy the natural science diversification requirement. Prerequisite: COMP 118, MATH 138 or PHYS 270. Offered every other spring.

The first in a three-semester calculus sequence, this course covers the basic ideas of differential calculus. Differential calculus is concerned primarily with the fundamental problem of determining instantaneous rates of change. In this course, we study instantaneous rates of change from both a qualitative geometric and a quantitative analytic perspective. We cover in detail the underlying theory, techniques and applications of the derivative. The problem of anti-differentiation, identifying quantities given their rates of change, also is introduced. The course concludes by relating the process of anti-differentiation to the problem of finding the area beneath curves, thus providing an intuitive link between differential calculus and integral calculus. Those who have had a year of high school calculus but do not have Advanced Placement credit for MATH 111 should take the calculus placement exam to determine whether they are ready for MATH 112. Students who have 0.5 units of credit for calculus may not receive credit for MATH 111. This counts toward the core course requirement for the major. Prerequisite: solid grounding in algebra, trigonometry and elementary functions. Offered every semester.

This course examines an important and interesting part of the history of mathematics and, more generally, the intellectual history of humankind: the history of mathematics in the Islamic world. Some of the most fundamental notions in modern mathematics have their roots here, such as the modern number system, the fields of algebra and trigonometry, and the concept of algorithm, among others. In addition to studying specific contributions of medieval Muslim mathematicians in the areas of arithmetic, algebra, geometry and trigonometry in some detail, we examine the context in which Islamic science and mathematics arose, and the role of religion in this development. The rise of Islamic science and its interactions with other cultures (e.g., Greek, Indian and Renaissance European) tell us much about larger issues in the humanities. Thus, this course has both a substantial mathematical component (60-65 percent) and a significant history and social science component (35-40 percent), bringing together three disciplines: mathematics, history and religion. The course counts toward the Islamic Civilization and Cultures concentration but does not count toward any math major requirement. Prerequisite: solid knowledge of algebra and geometry.

This course introduces students to mathematical reasoning and rigor in the context of set-theoretic questions. The course covers basic logic and set theory, relations — including orderings, functions and equivalence relations — and the fundamental aspects of cardinality. The course emphasizes helping students read, write and understand mathematical reasoning. Students are actively engaged in creative work in mathematics. Students interested in majoring in mathematics should take this course no later than the spring semester of their sophomore year. Advanced first-year students interested in mathematics are encouraged to consider taking this course in their first year. This counts toward the core course requirement for the major. This course cannot be taken pass/D/fail. Prerequisite: MATH 213. Offered every semester.

Combinatorics is, broadly speaking, the study of finite sets and finite mathematical structures. A great many mathematical topics are included in this description, including graph theory; combinatorial designs; partially ordered sets; networks; lattices and Boolean algebras; and combinatorial methods of counting, including combinations and permutations, partitions, generating functions, recurring relations, the principle of inclusion and exclusion, and the Stirling and Catalan numbers. This course covers a selection of these topics. Combinatorial mathematics has applications in a wide variety of nonmathematical areas, including computer science (both in algorithms and in hardware design), chemistry, sociology, government and urban planning; this course may be especially appropriate for students interested in the mathematics related to one of these fields. This counts toward the discrete/combinatorial (column C) elective requirement for the major. Prerequisite: MATH 112 or a score or 4 or 5 on the BC calculus AP exam. Offered every other year.

Coding theory (the theory of error-correcting codes), and cryptography are two important applications of algebra and discrete mathematics to information and communications systems. While coding theory is concerned with the reliability of communication, the main problem of cryptography is the security and privacy of communication. Applications of coding theory range from enabling the clear transmission of pictures from distant planets to quality of sound in compact disks and wireless communication. Error correcting codes play a key role in quantum computing. With the ever increasing role of digital communication, online transactions, the general dependence on electronic systems in modern life, and the emergence of quantum computers, the importance of these fields grows each day. In this course, students will learn the basic ideas of coding theory and cryptography, understand their mathematical foundations, and learn how mathematical tools can be used to devise useful error correcting codes and cryptographic systems. A selection of topics from these two disciplines will be discussed including basics of block coding, linear codes, Hamming codes, cyclic codes, symmetric-key and public-key cryptography, digital signatures, and code-based cryptography. Other than some basic linear algebra, the necessary mathematical background (mostly abstract algebra) is covered within the course. This counts toward either a discrete/combinatorial (column C) or an algebraic (column A) elective requirement for the major. Prerequisite: MATH 222 and MATH 224. Offered every other year.

Abstract algebra is the study of algebraic structures that describe common properties and patterns exhibited by seemingly disparate mathematical objects. The phrase "abstract algebra" refers to the fact that some of these structures are generalizations of the material from high school algebra relating to algebraic equations and their methods of solution. In this course, we focus entirely on group theory. A group is an algebraic structure that allows one to describe symmetry in a rigorous way. The theory has many applications in physics and chemistry. Since mathematical objects exhibit pattern and symmetry as well, group theory is an essential tool for the mathematician. Furthermore, group theory is the starting point in defining many other more elaborate algebraic structures including rings, fields and vector spaces. We cover the basics of groups, including the classification of finitely generated abelian groups, factor groups, the three isomorphism theorems and group actions. The course culminates in a study of Sylow theory. Throughout the semester there is an emphasis on examples, many of them coming from calculus, linear algebra, discrete math and elementary number theory. There also are a couple of projects illustrating how a formal algebraic structure can empower one to tackle seemingly difficult questions about concrete objects (e.g., the Rubik's cube or the card game SET). Finally, there is a heavy emphasis on the reading and writing of mathematical proofs. Junior standing is recommended. This counts toward the algebraic (column A) elective requirement for the major. Prerequisite: MATH 222. Offered every other fall.

The senior seminar in mathematics provides a structure to aid students in successfully completing the Mathematics and Statistics Capstone requirement. Students who participate in a 3-2 program ordinarily take this course in their junior year. Students with December anticipated graduation dates take the seminar in their sixth semester (in the third semester before graduation). This schedule makes it possible for students who do not succeed in their first try at the capstone to take full advantage of the “second chance” option without delaying their graduation timeline. \n Students who do not successfully complete their capstone in the fall semester will be assigned NG (no grade) at the end of the senior seminar. This designation will change to a CR when the student successfully completes the capstone, usually at the end of the following semester. This course is a core requirement for the mathematics major. Prerequisite: MATH 222 and at least one 300+-level course in Mathematics or Statistics. The course is offered every fall semester., The course is credit/no credit.

This course consists largely of an independent project in which students read several sources to learn about a mathematical topic that complements material studied in other courses, usually an already completed depth sequence. This study culminates in an expository paper and a public or semi-public presentation before an audience consisting of at least several members of the mathematics faculty as well as an outside examiner. Permission of department chair required. Prerequisite: senior standing and the completion of at least one two-semester sequence at the junior-senior level.

This course presents an introduction to computer programming intended both for those who plan to take further courses in which a strong background in computation is desirable and for those who are interested in learning basic programming principles. The course will expose the student to a variety of applications where an algorithmic approach is natural and will include both numerical and non-numerical computation. The principles of program structure and style will be emphasized. SCMP 118 may be paired with SCMP 218 or either may be paired with any mathematics or statistics course to satisfy the natural science diversification requirement. No prerequisite. Offered every semester.

This course is intended as a second course in programming, as well as an introduction to the concept of computational complexity and the major abstract data structures (such as dynamic arrays, stacks, queues, link lists, graphs and trees), their implementation and application, and the role they play in the design of efficient algorithms. Students will be required to write a number of programs using a high-level language. SCMP 218 may be paired with SCMP 118 or either may be paired with any mathematics or statistics course to satisfy the natural science diversification requirement. Prerequisite: SCMP 118, MATH 138 or PHYS 270 or permission of instructor. Offered every other spring.