Judy Holdener joined the Department of Mathematics in 1997, after spending three years at the U.S. Air Force Academy in Colorado Springs, CO. Although her primary research interests are in the areas of algebra and number theory, she has been known to work in other areas when an interesting question arises... especially if the question is accessible to undergraduates.

Judy has collaborated with students on research projects relating to algebra, number theory, dynamical systems, and mathematical biology; their work has culminated in research publications and presentations at national math conferences. In 2008, Judy was awarded the Mathematical Association of America Ohio Section Distinguished Teaching Award and in 2003, she was awarded Kenyon's Tomsich Science Award as well as the Board of Trustees Junior Teaching Award.

### Education

1994 — Doctor of Philosophy from Univ Illinois Urbana

1989 — Master of Science from Univ Illinois Urbana

1987 — Bachelor of Science from Kent State Univ Kent, Phi Beta Kappa

### Courses Recently Taught

MATH 111

## Calculus I

#### MATH 111

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 will study instantaneous rates of change from both a qualitative geometric and a quantitative analytic perspective. We will 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 will be introduced. The course will conclude 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 .5 unit of credit for calculus may not receive credit for MATH 111. Prerequisite: solid grounding in algebra, trigonometry and elementary functions.

MATH 112

## Calculus II

#### MATH 112

The second in a three-semester calculus sequence, this course has two primary foci. The first is integration, including techniques of integration, numerical methods and applications of integration. This study leads into the analysis of differential equations by separation of variables, Euler's method and slope fields. The second focus is the notion of convergence, as manifested in improper integrals, sequences and series, particularly Taylor series. Prerequisite: MATH 111 or permission of instructor. Offered every semester.

MATH 213

## Calculus III

#### MATH 213

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 will be covered as time permits: vector fields, line integrals, flux integrals, curl and divergence. Prerequisite: MATH 112 or permission of instructor.

MATH 222

## Foundations

#### MATH 222

This course introduces students to mathematical reasoning and rigor in the context of set-theoretic questions. The course will cover basic logic and set theory, relations--including orderings, functions and equivalence relations -- and the fundamental aspects of cardinality. The course will emphasize helping students read, write and understand mathematical reasoning. Students will be 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. (Please see a member of the mathematics faculty if you think you might want to do this.) Prerequisite: MATH 213 or permission of instructor. Offered every spring semester.

MATH 224

## Linear Algebra

#### MATH 224

This course will focus on the study of vector spaces and linear functions between vector spaces. Ideas from linear algebra are highly useful in many areas of higher-level mathematics. Moreover, linear algebra has many applications to both the natural and social sciences, with examples arising often in fields such as computer science, physics, chemistry, biology and economics. In this course, we will use a computer algebra system, such as Maple or Matlab, to investigate important concepts and applications. Topics to be covered include methods for solving linear systems of equations, subspaces, matrices, eigenvalues and eigenvectors, linear transformations, orthogonality and diagonalization. Applications will be included throughout the course. Prerequisite: MATH 213. Typically offered three out of four semesters.

MATH 327

## Number Theory Seminar

#### MATH 327

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 will get a glimpse at both the old and the new. In the first two-thirds of the semester, we will 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 will explore some of the current questions and applications of number theory. We will study the famous RSA cryptosystem, and students will read and present some current (carefully chosen) research papers. Prerequisite: MATH 222. Offered every other year.

MATH 335

## Abstract Algebra I

#### MATH 335

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 Abstract Algebra I, 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. In this course, we will 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 will be an emphasis on examples, many of them coming from calculus, linear algebra, discrete math and elementary number theory. There also will be 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 will be a heavy emphasis on the reading and writing of mathematical proofs. Junior standing is recommended. Prerequisite: MATH 222 or permission of instructor. Offered every other fall.

MATH 435

## Abstract Algebra II

#### MATH 435

Abstract Algebra II 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, unique factorization domains; the Gaussian integers; factorization techniques and irreducibility criteria. The final block of the semester will serve 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 will be an emphasis on examples, many of them coming from calculus, linear algebra, discrete math, and elementary number theory. There also will be a heavy emphasis on the reading and writing of mathematical proofs. Prerequisite: MATH 335. Offered every other spring.

MATH 493

## Individual Study

#### MATH 493

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 normally be used to fulfill requirements for the major. Typically, individual study will earn .5 unit or .25 unit of credit.\nTo qualify, a student must identify a member of the Mathematics Department willing to direct the project. The professor, in consultation with the student, will create a tentative syllabus (including a list of readings and/or problems, goals and tasks) and describe 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). Individual studies also require the approval of the department chair. The department expects the student to meet regularly with his or her instructor for at least one hour per week.\nStudents must begin discussion of their proposed individual study well in advance, preferably the semester before the course is to take place. Prerequisite: permission of instructor and department chair.

MATH 498

## Senior Honors

#### MATH 498

This course will consist 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 will culminate 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. Prerequisite: At least one "depth sequence" completed and permission of the department chair.

### Academic & Scholarly Achievements

"Generalized Thue-Morse sequences and the von Koch Curve," to appear in the *International Journal of Pure and Applied Mathematics* (co-authored with Kenyon students Lee Kennard '07 and Matthew Zaremsky '07)

2007

"Abundancy 'outlaws' of the form $(\sigma(N)+t)N$," *The Journal of Integer Sequences*, **10** (2007), Article 07.9.6 (co-authored with Kenyon student William Stanton '07)

2007

"A Cryptographic Scavenger Hunt," *Cryptologia*, **31** (2007) 316-323 (co-authored with Eric Holdener)

2006

"Conditions Equivalent to the Existence of Odd Perfect Numbers," *Mathematics Magazine*, **79**(5) (2006) 389-391

2005

"Product-free Sets in the Card Game Set," *PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies .***15**(4) (2005) 289-297

2005

"Visualizing Patterns in the Integers relating to the Abundancy Index," *Proceedings of the `Art and Math = X' Conference,* University of Colorado, Boulder, CO. (2005)

2005

"When Thue-Morse meets Koch,"* Fractals: Complex Geometry, Patterns, and Scaling in Nature and Society,***13**(2005) 191-206 (co-authored with Kenyon student Jun Ma, Class of 2005)

2004

"Art and Design in Mathematics," *The Journal of Online Mathematics and its Applications,***4** (2004)

2004

"Parametric Plots: A Creative Outlet," *The Journal of Online Mathematics and its Applications,***4** (2004) (co-authored with Keith Howard of Mercer University)

2003

"A Classification of Periodic Turtle Sequences," *The International Journal of Mathematics and Mathematical Sciences,***34** ( 2003) 2193-2201 (co-authored with Kenyon student Amy Wagaman, Class of 2003)

2002

"A Theorem of Touchard and the Form of Odd Perfect Numbers," *The American Mathematical Monthly,***109** (2002) 661-663