2019-2020 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
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)-2014
Associate 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
Wayne M. Dymàček, Ph.D.—(1981)-1993
The Cincinnati Professor of Mathematics
Ph.D., Virginia Polytechnic Institute and State University
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
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
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.
Spring 2020, MATH 100-01: The Art of Mathematical Thinking: Solving Puzzles and Games Using Mathematics (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.
Winter 2020, MATH 100-01: The Art of Mathematical Thinking: Introduction to Codes (3). 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 section, students explore the use of and questions about the numbers and codes which are everywhere. You might have a driver’s license number, a Social Security Number, a student identification number, a telephone number, credit-card numbers–the list goes on and on. If you’re filing out a form and you’re asked for an identification number, will anyone be able to tell right away if you’ve made up a number? If someone is typing your information into a computer, is there a way to make sure they haven’t made any errors? How are credit-card numbers kept safe when we make online purchases? We discuss types of errors, algorithms for checking for errors, and some methods for encrypting information to keep it secure. The only skills needed to enter this course are arithmetic and intellectual curiosity. Students learn how to analyze algorithms and develop problem-solving skills throughout the course. (FM) Finch-Smith.
Fall 2019, MATH 100-01: The Art of Mathematical Thinking: The Mathematics of Politics (3). 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 about politics. What makes this course mathematical is not numbers or formulas but rather reasoning. Students must think about what is possible and what is impossible. Is there a good way to determine winners of elections? Is there a good way to apportion congressional seats? Is there a good way to make decisions in situations of conflict and uncertainty? We begin by carefully and precisely formulating environments in which we can discuss approaches to elections, apportionment, and rational decision-making. We contemplate criteria that capture the notions of goodness within these environments, and see importance of precise definitions and consistent rules of logic in mathematical reasoning. Throughout the course, we pay attention to the way that technical words are defined so that the precise technical meaning is not confused with the ordinary meaning that the word carries in natural language. (FM) Finch-Smith.
Fall 2019, MATH 100-02: The Art of Mathematical Thinking: The Mathematics of Politics (3). 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 about politics. What makes this course mathematical is not numbers or formulas but rather reasoning. Students must think about what is possible and what is impossible. Is there a good way to determine winners of elections? Is there a good way to apportion congressional seats? Is there a good way to make decisions in situations of conflict and uncertainty? We begin by carefully and precisely formulating environments in which we can discuss approaches to elections, apportionment, and rational decision-making. We contemplate criteria that capture the notions of goodness within these environments, and see importance of precise definitions and consistent rules of logic in mathematical reasoning. Throughout the course, we pay attention to the way that technical words are defined so that the precise technical meaning is not confused with the ordinary meaning that the word carries in natural language. (FM) Finch-Smith.
Finch-Smith.
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, and infinite series. 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 122 - Discrete Mathematics II
FDR: SC
Credits: 3
Prerequisite: MATH 121. A continuation of MATH 121. Applications of 121 include probability theory in finite sample spaces and properties of the binomial distribution. This course also includes relations on finite sets, equivalence classes, partial orderings, and an introduction to graph theory and enumeration.
MATH 171 - Mathematics of Cryptography
FDR: SC
Credits: 4
Prerequisite: MATH 101 or 121. The history and application of cryptography. Topics include private-key codes, the ENIGMA machine and other WWII codes, public-key codes, and the RSA system. Appropriate mathematics is introduced, as necessary, to understand the construction and use of these codes. Several assignments are themselves in code, and students must decipher them just to find out what the homework is. Staff.
MATH 180 - FS: First-Year Seminar
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, parametric curves, 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. Instructor consent required. 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 353 - Numerical Analysis
Credits: 4
Prerequisites: MATH 221 and 222. Analysis, implementation, and applications of algorithms for solving equations, fitting curves, and numerical differentiation and integration. Theorems and proofs are complemented by hands-on programming exercises fostering a concrete understanding of accuracy, efficiency and stability, as well as an awareness of potential pitfalls in machine arithmetic. No previous programming experience is required. Staff.
MATH 361 - Graph Theory
Credits: 3
Prerequisite: MATH 122 or 222. Graphs and digraphs, trees, connectivity, cycles and traversability, and planar graphs. Additional topics selected from colorings, matrices and eigenvalues, and enumeration.
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 2020, MATH 383-01:Topic: The Mathematics of Information (4). Prerequisites: MATH 201 and 222 or instructor consent. 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–which, among other things, 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, you will see how these sorts of questions can be formalized and addressed mathematically. Bush.
Spring 2020, MATH 383-03: Topic: Mathematics of Puzzles and Games (4). Prerequisite: MATH 321. An examination of some of the mathematics of the following ten games and puzzles: Rubik’s cube, Sam Lloyd’s 15 puzzle, Sudoku and similar puzzles, poker, blackjack, craps, twister, cribbage, darts, and peg solitaire. Six other games or puzzles chosen by the students are also examined. Dymàček.
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.
Winter 2020, MATH 391A-01: Topic: Functional Analysis (3). Prerequisite: MATH 311. An introduction to the basic topics in functional analysis including metric spaces, normed spaces, and the fundamental examples and topological properties of these spaces. The course is focused on finite- and infinite-dimensional normed spaces, Hilbert spaces, and Banach spaces, with an emphasis on the linear properties and structure of the spaces and the operators on these spaces. Beanland.
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 2020, Math 392A-01: Topic in Abstract Algebra (3). Prerequisite: MATH 321. Rings, including ideals, quotient rings, polynomial rings, and domains. Field of quotients of a domain and enough field theory to prove the impossibility of several classical constructions. Dymàček.
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 2020, Math 393A-01: Topic in Geometry and Topology: Black Holes and Time Warps (3). Prerequisite: MATH 342 or 343. An exploration of the geometry of spacetime primarily by looking at black holes. We also visit other related topics, such as the geometry and topology of the universe, warp drives, worm holes and alternate universes. McRae.
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.
Winter 2020, Math 401-01: Directed Individual Study: FM Prep (1). Prerequisite: Instructor consent. A study of problem-solving techniques in preparation for the Society of Actuaries Exam FM, which covers financial mathematics. Dresden.
Fall 2019, MATH 401-01: Directed Individual Study - Origami (1). Prerequisite: Instructor consent. This course explores the many intersections of origami with mathematics, including geometry, number theory, and combinatorics. McRae.
Fall 2019, MATH 401-02: Directed Individual Study - P Prep (1). Prerequisite: Instructor consent. A study of problem-solving techniques in preparation for the Society of Actuaries Exam P, which covers statistics and probability. Dresden.
Fall 2019, MATH 401-03: Directed Individual Study - Putnam Prep (1). Prerequisite: Instructor consent. This course introduces various problem-solving techniques in preparation for the Virginia Tech and Putnam math contests. Student must participate in these contests to pass the course. Bush.
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.
Winter 2020, Math 402-01: Directed Individual Study: Discrete Models of Financial Markets (2). Prerequisite: MATH 221 and 222. This course explains in simple settings the fundamental ideas of financial market modeling and derivative pricing, using the No Arbitrage Principle. Relatively elementary mathematics leads to powerful notions and techniques–viability, completeness, self-financing and replicating strategies, arbitrage and equivalent martingale measures–which are directly applicable in practice. McRae.
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 421 - Directed Individual Research
Credits: 1
Prerequisite: Permission of the department. Directed independent work in mathematics, especially for honors candidates. May be repeated for degree credit if the topics are different.
MATH 422 - Directed Individual Research
Credits: 2
Prerequisite: Permission of the department. Directed independent work in mathematics, especially for honors candidates. May be repeated for degree credit if the topics are different.
MATH 423 - Directed Individual Research
Credits: 3
Prerequisite: Permission of the department. Directed independent work in mathematics, especially for honors candidates. May be repeated for degree credit if the topics are different.
MATH 426 - Directed Individual Research
Credits: 6
Prerequisite: Permission of the department. Directed independent work in mathematics, especially for honors candidates. 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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