Judy Holdener

The seminar in contemporary mathematics provides an introduction to the rich and diverse nature of mathematics. Topics covered vary from one semester to the next (depending on faculty expertise) but typically span algebra and number theory, dynamical systems, probability and statistics, discrete mathematics, topology, geometry, logic, analysis and applied math. The course includes guest lectures from professors at Kenyon, a panel discussion with upper-class math majors and opportunities to learn about summer experiences and careers in mathematics. The course goals are threefold: to provide an overview of modern mathematics, which, while not exhaustive, exposes students to some exciting open questions and research problems in mathematics; to introduce students to some of the mathematical research being done at Kenyon; and to expose students to useful resources and opportunities (at Kenyon and beyond) that are helpful in launching a meaningful college experience. This course does not count toward any requirement for the major. Prerequisite or corequisite: MATH 212 (or equivalent) and concurrent enrollment in another MATH, STAT or COMP course. Open only to first- or second-year students. Offered every fall semester.

The first course in the 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. Elementary differential equations and their applications are also introduced, along with the basics of anti-differentiation. Students who have 4 credit hours 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 seven-week course focuses on the concept of convergence of infinite sequences and series of real numbers, with a focus on their applications to Calculus, culminating in an exploration of Taylor series representations of functions. Topics include evaluation of improper integrals, convergence and divergence of sequences, tests for convergence of series, polynomial approximations of functions, power series, and Taylor’s Theorem.\n\nThis course meets for the first seven weeks of the semester and counts toward the core course requirement for the major. Prerequisite: Math 112 or AP score of 5 on Calculus AB exam or an AB sub-score of 5 on the Calculus BC exam. Students with an AP score of 4 on the Calculus AB exam are encouraged to take the Calculus placement exam. Offered every semester.

This course examines differentiation and integration in three dimensions. Topics of study include functions of more than one variable, vectors and vector algebra, partial derivatives, optimization, and multiple integrals. Some of the following topics from vector calculus also are covered as time permits: vector fields, line integrals, flux integrals, curl and divergence. This counts toward the core course requirement for the major. Prerequisite: MATH 112 or a score of 5 on the AB calculus AP exam, or an AB sub-score of 5 on the BC calculus AP exam. Offered every semester.

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.

Patterns within the set of natural numbers have enticed mathematicians for well over two millennia, making number theory one of the oldest branches of mathematics. Rich with problems that are easy to state but fiendishly difficult to solve, the subject continues to fascinate professionals and amateurs alike. In this course, we get a glimpse at both the old and the new. In the first two-thirds of the semester, we study topics from classical number theory, focusing primarily on divisibility, congruences, arithmetic functions, sums of squares and the distribution of primes. In the final weeks, we explore some of the current questions and applications of number theory. We study the famous RSA cryptosystem, and students read and present some current (carefully chosen) research papers. This counts toward either a discrete/combinatorial (column C) or an algebraic (column A) elective requirement for the major. Prerequisite: MATH 222, prerequisite or corequisite: MATH 212. Offered every other year.

This course picks up where MATH 335 ends, focusing primarily on rings and fields. Serving as a good generalization of the structure and properties exhibited by the integers, a ring is an algebraic structure consisting of a set together with two operations — addition and multiplication. If a ring has the additional property that division is well-defined, one gets a field. Fields provide a useful generalization of many familiar number systems: the rational numbers, the real numbers and the complex numbers. Topics to be covered include polynomial rings; ideals; homomorphisms and ring quotients; Euclidean domains, principal ideal domains and unique factorization domains; the Gaussian integers; factorization techniques; and irreducibility criteria. The final block of the semester serves as an introduction to field theory, covering algebraic field extensions, symbolic adjunction of roots, construction with ruler and compass, and finite fields. 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 is a heavy emphasis on the reading and writing of mathematical proofs. This counts toward the algebraic (column A) elective requirement for the major. Prerequisite: MATH 335. Offered every other spring.

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.

Individual study is a privilege reserved for students who want to pursue a course of reading or complete a research project on a topic not regularly offered in the curriculum. It is intended to supplement, not take the place of, coursework. Individual study cannot be used to fulfill requirements for the major. To qualify, a student must identify a member of the mathematics department willing to direct the project. The professor, in consultation with the student, creates a tentative syllabus (including a list of readings and/or problems, goals and tasks) and describes in some detail the methods of assessment (e.g., problem sets to be submitted for evaluation biweekly; a 20-page research paper submitted at the course's end, with rough drafts due at given intervals; and so on). The department expects the student to meet regularly with his or her instructor for at least one hour per week. All standard enrollment/registration deadlines for regular college courses apply. Because students must enroll for individual studies by the end of the seventh class day of each semester, they should begin discussion of the proposed individual study by the semester before, so that there is time to devise the proposal and seek departmental approval. Individual study courses may be counted as electives in the mathematics major, subject to consultation with and approval by the Department of Mathematics and Statistics. Permission of instructor and department chair required. No prerequisite.\n\n