2021-2022 University Catalog archived
Mathematics (MATH)
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(Society of the Cincinnati Foundation)
Advanced Standing: In mathematics, qualified first-year students may take a University-administered examination no later than four weeks into fall term to qualify for college credit in MATH 101:Calculus I. Students must be pursuing in the fall term of their first year, and must pass, with a C (2.0) or better, MATH 102: Calculus II. Students should contact the Head of the Mathematics Department head to indicate an interest in this option.
HONORS: An Honors Program in mathematics is offered for well-qualified majors; see department head for details.
Department Head: Alan McRae
Interim Department Head, Fall 2021: Gregory Dresden
Faculty
First date is the year in which the faculty member began service as regular faculty at the University. Second date is the year of appointment to the present rank.
Aaron D. Abrams, Ph.D.—(2012)-2020
Professor of Mathematics
Ph.D., University of California, Berkeley
Kevin J. Beanland, Ph.D.—(2013)-2018
Professor of Mathematics
Ph.D., University of South Carolina, Columbia
Michael R. Bush, Ph.D.—(2012)-2016
Associate Professor of Mathematics
Ph.D., University of Illinois
Cory Colbert, Ph.D.—(2018)-2018
Assistant Professor of Mathematics
Ph.D., University of Texas, Austin
Elizabeth J. Denne, Ph.D.—(2012)-2015
Associate Professor of Mathematics
Ph.D., University of Illinois
Gregory P. Dresden, Ph.D.—(1997)-2010
Professor of Mathematics
Ph.D., University of Texas
Nathan S. Feldman, Ph.D.—(1999)-2010
Rupert and Lillian Radford Endowed Professor of Mathematics
Ph.D., University of Tennessee
Carrie E. Finch-Smith, Ph.D.—(2007)-2019
Professor of Mathematics
Ph.D., University of South Carolina
Chawne Monique Kimber, Ph.D.—(2021)-2021
Professor of Mathematics and Dean of The College
Ph.D., University of Florida
Alan McRae, Ph.D.—(1997)-2009
Professor of Mathematics
Ph.D., State University of New York, Stony Brook
Major
Minor
MATH 100 - The Art of Mathematical Thinking: An Introduction to the Beauty and Power of Mathematical Ideas
FDR: FM
Credits: 3 in fall and winter; 3 or 4 in spring.
Topics vary from term to term. Mathematics is a creative process whose artistic outcome is often a powerful tool for the sciences. This course gives you a new perspective into the world of mathematics while also developing your analytical reasoning skills. May be repeated for degree credit if the topics are different.
Spring 2022, MATH 100-01: The Art of Mathematical Thinking: The Mathematics of Puzzles and Games (3). Any high school mathematics class is sufficient preparation for this material. Students gain a new perspective into the world of mathematics while also developing their analytical and creative reasoning skills. In so doing, they gain an understanding of how theoretical results and concepts can be developed, used for problem-solving or for further investigation, and then how to clearly and coherently communicate their ideas and discoveries to others. In this course, we focus on mathematical reasoning to analyze various puzzles and games, which then leads to figuring out how to increase our chances of winning. The springboard for our discussion comes from the answers to the following questions: 1) Is there a good way to predict the winner of a game before the game ends? 2) Is there a strategy that will improve a player’s chance of winning a game? 3) Is the game fair? The answers depend on what we mean by good and fair. We start by carefully and precisely formulating environments in which we can discuss approaches to solving puzzles and playing games. Then, we contemplate criteria that capture the notions of goodness and fairness within these environments. Along the way, students learn the importance of precise definitions and consistent rules of logic in mathematical reasoning. (FM) Finch-Smith.
Spring 2022, MATH 100A-01: The Art of Mathematical Thinking: Mathematical Foundations of Data Science (3). This course is intended to introduce mathematical foundations of data science. We will focus on linear algebra, numerical computation, data science and the programming language. After successfully completing this course, students will develop the necessary mathematical background for data science and will be able to solve a variety of real-world data-based problems. Furthermore, via numerous examples in real life, we learn how to access information logically, understand connection analytically, and model questions mathematically. (FM) Wang.
Winter 2022, MATH 100-02: The Art of Mathematical Thinking: Mathematics of Tilings and Patterns (3). In this course we study tiling and counting proofs for many famous formulas involving the Fibonacci numbers, the Lucas numbers, continued fractions, and binomial coefficients. No prior knowledge is needed. (FM) Dresden.
MATH 101 - Calculus I
FDR: FM
Credits: 3
An introduction to the calculus of functions of one variable, including a study of limits, derivatives, extrema, integrals, and the fundamental theorem. Sections meet either 3 or 4 days a week, with material in the latter presented at a more casual pace. Staff.
MATH 102 - Calculus II
FDR: FM
Credits: 3
Prerequisite: The equivalent of MATH 101 with C grade or better. Note: Students wanting to take this course should add to the waiting list when open; additional sections may be added. A continuation of MATH 101, including techniques and applications of integration, transcendental functions, infinite series, and parametric curves. Staff.
MATH 118 - Introduction to Statistics
FDR: FM
Credits: 3
Elementary probability and counting. Mean and variance of discrete and continuous random variables. Central Limit Theorem. Confidence intervals and hypothesis tests concerning parameters of one or two normal populations.
MATH 121 - Discrete Mathematics I
FDR: FM
Credits: 3
A study of concepts fundamental to the analysis of finite mathematical structures and processes. These include logic and sets, algorithms, induction, the binomial theorem, and combinatorics.
MATH 175 - Art + Math
FDR: FM
Credits: 4
A course in mathematics and art suitable for liberal arts students. Our approach is highly activity and experience based. Our goal is to explore how some of the greatest mathematical ideas had parallel developments in the world of art, all told through a narrative of culture and the history of ideas. McRae.
MATH 180 - FS: First-Year Seminar
FDR: FDR designation varies with topic, as approved in advance.
Credits: 3-4 depending on content
Prerequisite: First-Year standing. First-year seminar.
MATH 195 - Special Topics in Mathematics
FDR: SC
Credits: 3
Selected topics in mathematics. May be repeated if topics are different.
MATH 201 - Bridges to Advanced Mathematics
FDR: SC
Credits: 3
Prerequisites: 6 credits of MATH courses or MATH 221 or 222. The course explores various important mathematical constructions and ideas, with a particular emphasis on mathematical inquiry and reasoning. Topics include: sets, functions, equivalence relations, modular arithmetic, and basic properties of the integers, real numbers, and complex numbers. Staff.
MATH 221 - Multivariable Calculus
FDR: SC
Credits: 3
Prerequisite: The equivalent of MATH 102 with a C grade or better or MATH 201 or 222. Motion in three dimensions, differential calculus of multivariable functions, multiple integrals, line integrals, and Green’s Theorem.
MATH 222 - Linear Algebra
FDR: SC
Credits: 3
Prerequisite: The equivalent of MATH 102 with a C grade or better or MATH 201 or 221. Linear algebra is the backbone of much of mathematics. Students in this course learn to identify and explain the basic principles, terminology, and theories used in linear algebra, and apply quantitative and/or qualitative reasoning skills to solve problems posed in linear algebra, primarily through applications of to both mathematics and the sciences, and also by writing proofs In mathematics.
MATH 239 - Dimensions of Art and Math
(ARTS 239) FDR: HA
Credits: 3
In this studio course, we explore subject matters from the interface of mathematics and art by combining mathematical principles and artistic sensibilities and processes. We explore the potential synergy between the two disciplines through looking at designs, patterns, dimensions, and forms through two separate prisms, and we try to find ways in which one can be applied to the other. Abrams and Tamir.
MATH 270 - Financial and Actuarial Mathematics
Credits: 3
Prerequisite: MATH 102. An introduction to some of the fundamental topics in financial and actuarial mathematics. Possible topics include calculating present and accumulated values for various streams of cash and the theoretical basis of corporate finance and financial models and the application of those models to insurance and other financial risks. Staff.
MATH 291 - Directed Summer Research
Experiential Learning (EXP): Yes
Credits: 1
Graded Satisfactory/Unsatisfactory. Directed individual research in mathematics during the summer months. Each student conducts primary research in partnership with a mathematics faculty member by prior mutual agreement. Consult with individual faculty for a description of current research areas. May be repeated for degree credit with consent of the instructor. Staff.
MATH 292 - Directed Summer Research
Experiential Learning (EXP): Yes
Credits: 2
Graded Satisfactory/Unsatisfactory. Directed individual research in mathematics during the summer months. Each student conducts primary research in partnership with a mathematics faculty member by prior mutual agreement. Consult with individual faculty for a description of current research areas. May be repeated for degree credit with consent of the instructor. Staff.
MATH 293 - Directed Summer Research
Experiential Learning (EXP): Yes
Credits: 3
Graded Satisfactory/Unsatisfactory. Directed individual research in mathematics during the summer months. Each student conducts primary research in partnership with a mathematics faculty member by prior mutual agreement. Consult with individual faculty for a description of current research areas. May be repeated for degree credit with consent of the instructor. Staff.
MATH 301 - Fundamental Concepts of Mathematics
Credits: 4
Prerequisite: Six credits of mathematics or a grade of at least B in MATH 102. Basic analytical tools and principles useful in mathematical investigations, from their beginning stages, in which experimentation and pattern analysis are likely to play a role, to their final stages, in which mathematical discoveries are formally proved to be correct. Strongly recommended for all prospective mathematics majors.
MATH 303 - Complex Analysis
Credits: 3
Prerequisite: MATH 221 or consent of the instructor. Algebra of complex numbers, polar form, powers, and roots. Derivatives and geometry of elementary functions. Line integrals, the Cauchy Integral Theorem, the Cauchy Integral formula, Taylor and Laurent Series, residues, and poles. Applications.
MATH 309 - Probability
Credits: 3
Prerequisite: The equivalent of MATH 221 with C grade or better. Probability, probability density and distribution functions, mathematical expectation, discrete and continuous random variables, and moment generating functions.
MATH 310 - Mathematical Statistics
Credits: 3
Prerequisite: MATH 309. Sampling distributions, point and interval estimation, testing hypotheses, regression and correlation, and analysis of variance.
MATH 311 - Real Analysis
Credits: 3
Prerequisites: MATH 201 (or 301) and 221. A systematic study of concepts basic to calculus, such as topology of the real numbers, limits, differentiation, integration, sequences and series. Additional topics vary by instructor. Staff.
MATH 321 - Abstract Algebra
Credits: 3
Prerequisites: MATH 201 (or 301) and 222. An introduction to basic algebraic structures common throughout mathematics. These include rings, fields, groups, homomorphisms and quotient structures. Additional topics vary by instructor. Staff.
MATH 332 - Ordinary Differential Equations
Credits: 3
Prerequisite: MATH 221 with C grade or better. First and second order differential equations, systems of differential equations, and applications. Techniques employed are analytic, qualitative, and numerical.
MATH 333 - Partial Differential Equations
Credits: 3
Prerequisite: MATH 332. An introduction to the study of boundary value problems and partial differential equations. Topics include modeling heat and wave phenomena, Fourier series, separation of variables, and Bessel functions. Techniques employed are analytic, qualitative, and numerical.
MATH 343 - Geometry
Credits: 3
Prerequisites: MATH 201 (or 301) , 221, and 222. This course is an introduction to geometric techniques through study of Euclidean and non-Euclidean geometries and their transformations. Additional topics vary by instructor. Staff.
MATH 345 - Calculus on Manifolds
Credits: 3
Prerequisites: MATH 221 and 222. This course builds on material from both multivariable calculus and linear algebra. Topics covered include: manifolds, derivatives as linear transformations, tangent spaces, inverse and implicit function theorems, integration on manifolds, differential forms, and the generalized Stokes’ Theorem. Staff.
MATH 361 - Graph Theory
Credits: 3
Prerequisite: MATH 201 or instructor consent. An overview of major topics in graph theory, including graphs and digraphs, trees, connectivity, cycles and traversability, colorings, and planar graphs.
MATH 363 - Combinatorics
Credits: 3
Prerequisite: MATH 122, 201, or 222. Topics include counting methods, permutations and combinations, binomial identities, recurrence relations. generating functions, special sequences, partitions, and other topics as time and student interest permit. Staff.
MATH 365 - Number Theory
Credits: 3
Prerequisite: MATH 201 or consent of the instructor. Topics include prime numbers, Euclidean algorithm, congruences, Chinese Remainder Theorem, Fermat’s Little Theorem, Euler’s Theorem, arithmetic functions, Euler’s phi function, perfect numbers, the quadratic reciprocity law, continued fractions, and other topics as time and student interest permit.
MATH 369 - The Mathematics of Puzzles and Games
Credits: 4
Prerequisites: MATH 321 or instructor consent. The application of mathematics to puzzles and games. A brief survey on the designs of tournaments. The puzzles and games include but are not limited to the Rubik’s Cube, poker, blackjack, and peg solitaire. Staff.
MATH 383 - Topics in Mathematics
Credits: 3 in fall and winter, 4 in spring
Prerequisite: MATH 201, 221 or 222, but may vary with topic. Readings and conferences for a student or students on topics agreed upon with the directing staff. May be repeated for degree credit if the topics are different.
Spring 2022, MATH 383A-01: The Mathematics of Information (3). Prerequisite: MATH 201 and MATH 222. The modern world runs on information. Huge numbers of bits (0s and 1s) are passing invisibly through the wires and air around you right now. These bits encode various types of data including text, pictures, audio/video signals etc. In 1948, a pioneering paper by Claude Shannon founded a new research area: information theory. Among other things, this investigates the process of converting streams of symbols from one form to another and various associated questions that are still the focus of much modern research. For example, what is the most efficient way to go about encoding a stream of data so that it can be transmitted as quickly as possible over some channel or stored using a minimal amount of space? How can one build in redundancy so that errors due to noise (scratches on a CD/DVD, electromagnetic interference etc.) can be detected and corrected? What should you do if privacy/secrecy is important? In this course, we will see how some of these questions can be formalized and addressed mathematically. Bush.
Spring 2022, MATH 383B-01: Configuration Spaces (3). Prerequisite: MATH 201 or MATH 221 or MATH 222. A configuration space is a mathematical object that encodes all the possible states of a system that has multiple moving parts. Examples of such systems include mechanical linkages as well as physical brain-teasers like the Rubik’s cube. The mathematical study of configuration spaces is applied widely in a number of practical fields, such as robotics. In this course we will study the configuration spaces associated to various types of mechanisms, puzzles, and gadgets. We will learn how to analyze the complexity of mechanical linkages and the difficulty of logic-based puzzles such as Sudoku. Students will have the opportunity to design and build their own examples. Abrams.
MATH 391 - Topics in Analysis
Credits: 3
Prerequisite: MATH 311. Topics vary but can include complex analysis, topology, differential equations, differential topology, numerical analysis, functional analysis, measure theory, fractal geometry, Lebesgue integration and Fourier analysis, harmonic analysis, and analytic number theory. May be repeated for degree credit if the topic is different.
Fall 2021, MATH 391A-01: Topics in Analysis: Numerical Mathematics for Data Science (3). Prerequisite: MATH 311. This course is designed to introduce knowledge of numerical computation and analysis, in order to equip students with necessary numerical techniques to address practical questions arising from data science and other fields. We will discuss useful methods to construct mathematical models from given data and powerful algorithms to solve large scale systems of linear equations which are formulated during the creation of mathematical models. Students will also learn computational complexity, accuracy, stability, conditioning, and other mathematical concepts of numerical analysis which are fundamental in developing an efficient numerical algorithm. MATLAB will be the programming language used for this course. Wang.
MATH 392 - Topics in Abstract Algebra
Credits: 3
Prerequisite: MATH 321. Topics vary but can include field and Galois theory, geometric and combinatorial group theory, representation theory, number theory, algebraic number theory, commutative algebra, algebraic geometry, arithmetic geometry, advanced linear algebra, algebraic coding theory and cryptography, algebraic topology, homological algebra, and graph theory, May be repeated for degree credit if the topic is different.
Winter 2022, Math 392A-01: Topics in Abstract Algebra: Rings, Fields, and Galois Theory (3).
Prerequisite: MATH 321. Rings, ring homomorphisms, modules, module homomorphisms, Euclidean domains, PIDs, UFDs, field extensions, degree, algebraic numbers, automorphisms, irreducible polynomials, Galois groups, Galois correspondence. Colbert.
MATH 393 - Topics in Geometry and Topology
Credits: 3
Prerequisite: MATH 342 or 343. Topics vary but can include knot theory, topology and geometry of surfaces, differential geometry, Riemann surfaces, 3-manifolds, tilings, geometric probability, geometry of spacetime, finite geometry, computational geometry, differential topology, and projective geometry. May be repeated for degree credit if the topic is different.
Winter 2022, Math 393A-01: Topics in Geometry and Topology: Differential Topology (3).
Prerequisite: Math 342 or 343. This course builds on material from multivariable calculus, linear algebra and geometry. We cover a range of topics: an introduction to manifolds with boundary, derivatives as linear transformations, tangent spaces, inverse and implicit function theorems, transversality, and intersections of manifolds. Then integration on manifolds, differential forms, and the generalized Stokes’s Theorem. Denne.
Winter 2021, MATH 393A-01: Topics in Geometry and Topology: Experimenting with Geometry (3). Prerequisite: MATH 342 or 343. This course will be run in an experimental format modeled on the notion of a “Geometry Lab.” Students will study unsolved problems in geometry, learning whatever background material is relevant for understanding and approaching the selected problems. Topics are likely to include algebraic geometry, hyperbolic geometry, and projective geometry. Abrams.
MATH 401 - Directed Individual Study
Credits: 1
Prerequisite: Instructor consent unless otherwise noted. Individual conferences. May be repeated for degree credit if the topics are different.
MATH 402 - Directed Individual Study
Credits: 2
Prerequisite: Consent of the department. Individual conferences. May be repeated for degree credit if the topics are different.
MATH 403 - Directed Individual Study
Credits: 3
Prerequisite: Permission of the department. Individual conferences. May be repeated for degree credit if the topics are different.
MATH 493 - Honors Thesis
Credits: 3-3
Prerequisites: Honors candidacy, senior standing and consent of the department. Honors Thesis.
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