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 on the Tenure and Promotion Committee, 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

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. You can expect essay tests, frequent homework and writing assignments. No prerequisite. Typically 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. 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 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 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. Emphasis will be placed on helping students in reading, writing and understanding 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 semester.

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. Prerequisite: MATH 222 or PHIL 120 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 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.

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. Prerequisite: PHIL 120 or some coursework in mathematics and permission of instructor.

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. Offered every semester. SCMP 118 is crosslisted with mathematics for diversification purposes.

### Academic & Scholarly Achievements

Forthcoming

"Derivability in the Logic of Proofs" is $\Pi^p_2$-complete, To appear, *Annals of Pure and Applied Logic*

2005

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

2003

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

2003

"The Complexity of Predicate Default Logic over a Countable Domain", *Annals of Pure and Applied Logic* 120, 1-3 (April 2003), pp. 151-163.

2001

"Skeptical Reasoning in FC-Normal Logic Programs is $\Pi^1_1$-Complete", *Fundamenta Informaticae* 45, 3 (2001), pp. 237-252.