Prior to his 2002 arrival at Kenyon, Bob Milnikel studied at Carleton College and Cornell University and taught at Wellesley College. His research is focused on the mathematical analysis of logic as used in computer science. His teaching also bridges math and CS, including algebra and calculus courses as well as logic and introductory programming.

Currently serving as Chair of the Mathematics and Statistics Department, Bob is also active in several of Kenyon's musical ensembles. His Chicago area roots engendered an enduring fondness for good pizza and hapless baseball teams.

### Education

1999 — Doctor of Philosophy from Cornell University

1996 — Master of Science from Cornell University

1992 — Bachelor of Arts from Carleton College, Phi Beta Kappa

### Courses Recently Taught

INDS 100

## Data Analysis: Seeing w/ Data

#### INDS 100

In this course, students will gain experience analyzing, interpreting, and critiquing quantitative claims and communicating results and conclusions using graphical representations of data. Examples will be drawn from across the natural and social sciences, with context provided for each data set, so that students from any disciplinary background can participate in and benefit from this course. This course has no pre-requisites. It will be taught at a level accessible to all Kenyon students. Excellent preparation for further work on quantitative topics, this course will hone students' ability to apply mathematical techniques including graphing, statistics, linear and non-linear regression, and modeling the graphical behavior of mathematical functions to understanding and interpreting data. Students will practice these skills by engaging in critical reading of primary sources, oral presentation of quantitative data, and expression of analytic ideas in writing. Assessment will be based on in-class assignments, monthly quizzes, and oral reports on data-driven projects selected in consultation with the instructor.

MATH 105

## Surprises at Infinity

#### MATH 105

Our intuitions about sets, numbers, shapes and logic all break down in the realm of the infinite. Seemingly paradoxical facts about infinity are the subject of this course. We will discuss what infinity is, how it has been viewed through history, why some infinities are bigger than others and how a finite shape can have an infinite perimeter. This very likely will be quite different from any mathematics course you have ever taken. This course focuses on ideas and reasoning rather than algebraic manipulation, though some algebraic work will be required to clarify big ideas. The class will be a mixture of lecture and discussion, based on selected readings. Students can expect essay tests, frequent homework and writing assignments. No prerequisite. Generally offered every other year.

MATH 106

## Elements of Statistics

#### MATH 106

This is a basic course in statistics. The topics to be covered are the nature of statistical reasoning, graphical and descriptive statistical methods, design of experiments, sampling methods, probability, probability distributions, sampling distributions, estimation and statistical inference. Confidence intervals and hypothesis tests for means and proportions will be studied in the one- and two-sample settings. The course concludes with inference regarding correlation, linear regression, chi-square tests for two-way tables, and one-way ANOVA. Statistical software will be used throughout the course, and students will be engaged in a wide variety of hands-on projects. No prerequisite. Offered every semester.

MATH 110

## Pre-Calculus

#### MATH 110

This course prepares students for the study of calculus. It is particularly directed to those planning to enter the calculus sequence that begins with MATH 111. Primary emphasis is placed on the study of real valued functions, particularly polynomial, rational, logarithmic, exponential, trigonometric, and inverse trigonometric functions. Conceptual understanding will be emphasized. Computer labs that use graphing programs and a computer algebra system will be employed. Students with 1/2 unit of credit for calculus may not receive credit for MATH 110.

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. Offered every semester.

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 AP score of 4 or 5 on Calculus AB exam or an AB subscore of 4 or 5 on the Calculus BC exam or permission of instructor. Offered every semester.

MATH 118

## Introduction to Programming

#### MATH 118

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. 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 a score of 4 or 5 on the BC Calculus AP exam or permission of instructor. Offered every semester.

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. Students wanting to do so should contact a member of the mathematics faculty. Prerequisite: MATH 213 or permission of instructor. Offered every spring semester.

MATH 258

## Mathematical Biology

#### MATH 258

In biological sciences, mathematical models are becoming increasingly important as tools for turning biological assumptions into quantitative predictions. In this course, students will learn how to fashion and use these tools to explore questions ranging across the biological sciences. We will survey a variety of dynamic modeling techniques, including both discrete and continuous approaches. Biological applications may include population dynamics, molecular evolution, ecosystem stability, epidemic spread, nerve impulses, sex allocation and cellular transport processes. The course is appropriate both for math majors interested in biological applications and for biology majors who want the mathematical tools necessary to address complex, contemporary questions. As science is becoming an increasingly collaborative effort, biology and math majors will be encouraged to work together on many aspects of the course. Coursework will include homework problem-solving exercises and short computational projects. Final independent projects will require the development and extension of an existing biological model selected from the primary literature. This course will build on (but not be limited by) an introductory-level knowledge base in both math and biology. Interested biology and math majors lacking a prerequisite are encouraged to consult with the instructor. Prerequisite: STAT 106 or MATH 111 or 112 (or any math or statistics AP credit of 4 or 5) and either BIOL 115 or 116. Offered every other year.

MATH 322

## Mathematical Logic

#### MATH 322

This course is a mathematical examination of the formal language most common in mathematics: predicate calculus. We will examine various definitions of meaning and proof for this language, and consider its strengths and inadequacies. We will develop some elementary computability theory en route to rigorous proofs of Godel's Incompleteness Theorems. Concepts from model logic, model theory and other advanced topics will be discussed as time permits. Prerequisite: MATH 222 or PHIL 201 or permission of instructor. Offered occasionally.

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 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 be used to fulfill requirements for the major. Individual studies will earn .25 - .50 units of credit. To 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). 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 preferably the semester before, so that there is time to devise the proposal and seek departmental approval before the registrar's deadline. Permission of instructor and department chair required. No prerequisite.\n\n

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.

PHIL 264

## Philosophy of Mathematics

#### PHIL 264

This course covers core issues in the philosophy of mathematics. Why should we believe mathematical claims? Is mathematics really a priori? If so, what do we mean by that? Are mathematical claims truth claims? What is the nature of a mathematical proof? Do numbers exist? How do we make sense of various mathematical concepts such as infinity, imaginary numbers, probability, and so on. We are going to look at primary texts written by mathematicians and philosophers such as Hilbert, Frege, Brouwer, Russell, Putnam, Wittgenstein and others. And we will examine standard philosophical accounts of mathematics such as intuitionism, Platonism, formalism and more. This is a team taught course. This counts toward the epistemology requirement for the major. Prerequisite: PHIL 201 or some coursework in mathematics and permission of instructor.

PHIL 391

PHIL 491

## ST:Philosophy of Mathematics

#### PHIL 491

SCMP 118

## Introduction to Programming

#### SCMP 118

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 a mathematics or statistics course for diversification purposes. Offered every semester.

STAT 106

## Elements of Statistics

#### STAT 106

This is a basic course in statistics. The topics to be covered are the nature of statistical reasoning, graphical and descriptive statistical methods, design of experiments, sampling methods, probability, probability distributions, sampling distributions, estimation and statistical inference. Confidence intervals and hypothesis tests for means and proportions will be studied in the one- and two-sample settings. The course concludes with inference regarding correlation, linear regression, chi-square tests for two-way tables, and one-way ANOVA. Statistical software will be used throughout the course, and students will be engaged in a wide variety of hands-on projects. No prerequisite. Offered every semester.

### Academic & Scholarly Achievements

Forthcoming

"Group Activities for Math Enthusiasts," coauthored with Judy Holdener, *PRIMUS*.

2015

"A New Angle on an Old Construction," *Mathematics Magazine*, 88:4, October 2015.

2014

"The Logic of Uncertain Justifications," *Annals of Pure and Applied Logic*, 165:1, January 2014.

2013

"The Logic of Uncertain Justifications" (preliminary report), Proceedings of the International Symposium on the Logical Foundations of Computer Science, January 2013.

2012

"Conservativity in Logics of Justified Belief: Two Approaches," Annals of Pure and Applied Logic, 163:7, July 2012.

2009

"Conservativity in Logics of Justified Belief," Presented at LFCS '09, Spring Verlag series LNCS 5407, 2009.

2007

"Derivability in the Logic of Proofs is $\Pi^p_2$-complete," *Annals of Pure and Applied Logic*, 145:3, 223-239, March 2007.

2005

"Sequent Calculi for Skeptical Reasoning in Predicate Default Logic and Other Nonmonotonic Systems," *Annals of Mathematics and Artificial Intelligence* 44:1, 1-34, 2005.

2004

"A Sequent Calculus for Skeptical Reasoning in Autoepistemic Logic," Presented at the 10th International Symposium on Nonmonotonic Reasoning, June 2004.

2003

"Embedding Modal Nonmonotonic Logics into Default Logic," *Studia Logica*, 75, 377-382, 2003.

2003

"The Complexity of Predicate Default Logic Over a Countable Domain," *Annals of Pure and Applied Logic,* 120, 151-163, April 2003.

2003

"A Sequent Calculus for Skeptical Reasoning in Predicate Default Logic" (extended abstract), Presented at ECSQARU 2003, Proceedings Springer-Verlag LNCS 2711.

2001

"Skeptical Reasoning in FC-Normal Logic Programs is $\Pi^1_1$-Complete," *Fundamenta Informaticae,* 45, 237-252, 2001.