Elly Farnell joined the Kenyon math department in 2010 after studying at Whitman College and Colorado State University. Her primary research area is pattern analysis and algorithm development, with a focus on high dimensional data sets in the context of image analysis. She uses a variety of tools from the field of linear algebra in her research and enjoys integrating related projects in her teaching. In her spare time, Elly enjoys playing volleyball and tennis, reading, finding logic puzzles and trivia to challenge her students, and playing games.

### Areas of Expertise

Pattern Analysis, algorithm development, high dimensional data sets

### Education

2010 — Doctor of Philosophy from Colorado State University

2006 — Master of Science from Colorado State University

2004 — Bachelor of Arts from Whitman College, cum laude

### 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, will also 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. Students who have credit for MATH 110Y-111Y may not take this course.

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

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. Offered every fall.

MATH 324

## Linear Algebra II

#### MATH 324

This course builds on the concepts that arise in MATH 224. Topics will vary and will likely include some of the following: abstract vector spaces, inner product spaces, linear mappings and canonical forms, linear models, linear codes, the singular value decomposition, wavelets. Prerequisite: MATH 224. Offered every other year.

MATH 333

## Differential Equations

#### MATH 333

Differential equations arise naturally to model dynamical systems such as often occur in physics, biology, chemistry, and economics, and have given major impetus to other fields in mathematics, such as topology and the theory of chaos. This course covers basic analytic, numerical, and qualitative methods for the solution and understanding of ordinary differential equations. Computer-based technology will be used. Prerequisite: MATH 224 or PHYS 245 or permission of instructor. Offered every other spring.

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 will also 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 347

## Mathematical Models

#### MATH 347

This course introduces students to the concepts, techniques, and power of mathematical modeling. Both deterministic and probabilistic models will be explored, with examples taken from the social, physical, and life sciences. Students engage cooperatively and individually in the formulation of mathematical models and in learning mathematical techniques used to investigate those models. Prerequisite: MATH 106 and MATH 224 or MATH 258 or permission of instructor. Offered every other year.