Carol Schumacher, Professor of Mathematics, has been at Kenyon since 1988. She has served two terms as chair of the mathematics department and one term as chair of the faculty. She is the recipient of Kenyon’s Trustee Teaching Award and of the Ohio Section Mathematical Association of America’s award for distinguished teaching. She is the author of Closer and Closer: Introducing Real Analysis and Chapter Zero: Fundamental Notions of Abstract Mathematics, 2E.

Professor Schumacher is active in the Mathematical Association of America and is currently serving as the Vice President of the association. She previously served as co-Chair of the MAA Committee for the Undergraduate Program in Mathematics. In recent years she has been asked to address the Project NExT fellows at their summer workshop, and she has been an organizer and presenter in professional development workshops that help college faculty incorporate inquiry into their classrooms. Prof. Schumacher regularly teaches Calculus, Foundations, Real Analysis I and II and Applied Differential Equations. She lives in Gambier.

Areas of Expertise

Primary: Real Analysis. Secondary: Differential Equations.

Education

1989 — Doctor of Philosophy from Univ Texas Austin

1982 — Bachelor of Arts from Hendrix College

Courses Recently Taught

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.

The third in a three-semester calculus sequence, 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 4 or 5 on the BC calculus AP exam or permission of instructor. 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 or permission of instructor. Offered every semester.

The "Elements" of Euclid, written over 2,000 years ago, is a stunning achievement. The "Elements" and the non-Euclidean geometries discovered by Bolyai and Lobachevsky in the 19th century form the basis of modern geometry. From this start, our view of what constitutes geometry has grown considerably. This is due in part to many new theorems that have been proved in Euclidean and non-Euclidean geometry but also to the many ways in which geometry and other branches of mathematics have come to influence one another over time. Geometric ideas have widespread use in analysis, linear algebra, differential equations, topology, graph theory and computer science, to name just a few areas. These fields, in turn, affect the way that geometers think about their subject. Students consider Euclidean geometry from an advanced standpoint but also have the opportunity to learn about non-Euclidean geometries. This counts toward the continuous/analytic (column B) elective requirement for the major. Prerequisite: MATH 222 or permission of instructor. Offered every other year.

Special Topic

This course is a first introduction to real analysis. "Real" refers to the real numbers. Much of our work revolves around the real number system. We start by carefully considering the axioms that describe it. "Analysis" is the branch of mathematics that deals with limiting processes. Thus the concept of distance is also a major theme of the course. In the context of a general metric space (a space in which we can measure distances), we consider open and closed sets, limits of sequences, limits of functions, continuity, completeness, compactness and connectedness. Other topics may be included if time permits. Junior standing is recommended. This counts toward the continuous/analytic (column B) elective requirement for the major. Prerequisite: MATH 213 and 222. Offered every other fall.

This course follows MATH 341. Topics include a study differentiation and (Riemann) integration of functions of one variable, sequences and series of functions, power series and their properties, iteration and fixed points. Other topics may be included as time permits. For example: a discussion of Newton's method or other numerical techniques; differentiation and integration of functions of several variables; spaces of continuous functions; the implicit function theorem; and everywhere continuous, nowhere differentiable functions. This counts toward the continuous/analytic (column B) elective requirement for the major. Prerequisite: MATH 341. Offered every other spring.

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

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.