First Year Seminars

 

177.  What is Mathematics and Why Won't It Go Away?                                                3 hours

         3NS, QPf

This seminar will provide opportunities to engage in various activities (problem-solving, conjecture, and proof) and to explore the nature of mathematical thinking and discourse.   Works of both non-fiction and fiction will be discussed and issues such as problem-solving solving vs. theory-building, the nature of mathematical truth and proof, aesthetic qualities in mathematics, mathematics and madness, cognition and mathematics will be considered.   Intended for students without extensive background beyond high school mathematics. Enrollment Limit: 14.

 

 

179.  From Logic to Persuasion to Propaganda                                               3 hours

         3NS, QPf

Argumentation and persuasion, more formally, the fields of logic and rhetoric will be used as a lens to examine contemporary culture.   Students will learn how to construct arguments using tools from deductive and inductive logic, including the propositional calculus, the predicate calculus and elementary statistics.  These tools plus others from classical rhetoric and the 'new rhetoric', will be used to analyze and synthesize arguments in areas of current political and social controversy.  A final theme is the development of persuasion into a science and an industry, in the form of advertising and propaganda.

 

Introductory Courses

030.  Topics in Contemporary Mathematics                                  3 hours

         3NS, QPf

The interaction of mathematics with the social sciences is the central theme. Topics are drawn from: graph theory, voting systems, discrete models, coding theory, exploratory data analysis, and combinatorics. Applications are given to social choice, decision-making, management and ecological modeling. Prerequisite: A working knowledge of elementary algebra and geometry. Note: This course does not count toward a major in Mathematics. (Not open to any student who has received credit for a mathematics course numbered 131 or higher). It is intended for students who have not satisfied the quantitative proficiency requirement. Enrollment limit: 30.

 

050.  Dots, Lines, and Coin Flips                                                 3 hours

         3NS, QPf

An introduction to two important ways of describing the world mathematically. Graphs model maps and networks—-road, telephone, computer, social. Probability theory describes the order that can lurk in random phenomena. Using both these tools, we will examine questions like: How random is the stock market? How tightly is the World Wide Web connected? Are there just six degrees of separation? Note: This course does not count toward a major in Mathematics. It is intended for students who have not satisfied the quantitative proficiency requirement.   Enrollment Limit: 30.

 

090.  Environmental Mathematics                                                 3 hours

         3NS, QPf

This course focuses on the application of mathematics to problems concerning the environment.  Topics include simulation (models of population growth, predator-prey relationships, and epidemics); optimization (applications to groundwater hydrology, herbivore foraging, and transportation of hazardous wastes); and decision analysis (applications to management of endangered species and resolution of environmental disputes).

Note: This course does not count toward a major in mathematics.  It is intended for students who have not satisfied the quantitative proficiency requirement and is not open to any student who has received credit for a course in mathematics course numbered 131 or higher.  Enrollment Limit: 30.

 

095.  Discrete Mathematical Models                                                  3 hours

         3NS, QPf

An introduction to discrete mathematical models. Mathematical techniques to be discussed include difference equations, iteration, convergence, elementary probability and optimization. Basic financial, population, economic, and physical models will be explored with applications to personal financial decision-making, ecology, population management, and drug selection, as well as other areas. The course will make extensive use of spreadsheet software. Note: This course does not count towards a major in mathematics. It is intended for students who have not satisfied the quantitative proficiency requirement and is not open to any student who has received credit for a mathematics course numbered 113 or higher. Enrollment Limit: 30

 

100.  Elementary Statistics                                                           3 hours

         4NS, QPf

An introduction to the statistical analysis of data. Topics include exploratory data analysis, probability, sampling, estimation, and hypothesis testing. Statistical software is introduced, but no prior computer experience is assumed. This course focuses on statistical ideas and downplays mathematical formulas. Note: MATH 100 does not count toward a mathematics or economics major and is not open to students who have completed a semester of calculus. Students may not receive credit for more than one of MATH 100, MATH 113, and MATH 114. It is intended for students in the social sciences and humanities with minimal mathematical experience who have not satisfied the quantitative proficiency requirement. Enrollment Limit: 30

 

113.  Statistical Methods for the Social and Behavioral Sciences 4 hours

         4NS, QPf

A standard introduction to statistics for students with a good background in mathematics. Topics covered include exploratory data analysis, descriptive statistics, probability, sampling, estimation, and statistical inference. A broad spectrum of examples is employed. Statistical software is introduced, but no prior computer experience is assumed. Consent of instructor required. Prerequisite: An appropriate score on the Statistics Readiness Exam. Note: The statistical content of this course is largely the same as MATH 114; the applications are different. Students may not receive credit for more than one of MATH 100, MATH 113, and MATH 114.   Enrollment Limit:  32.

114.  Statistical Methods for the Biological Sciences                  4 hours

         4NS, QPf

A standard introduction to statistics for students with a good background in mathematics. Topics covered include exploratory data analysis, descriptive statistics, probability, sampling, estimation, and statistical inference. Biological and medical examples are emphasized. Statistical software is introduced, but no prior computer experience is assumed. Prerequisite: An appropriate score on the Statistics Readiness Exam. Note: The statistical content of this course is largely the same as MATH 113; the applications are different. Students may not receive credit for more than one of MATH 100, MATH 113, and MATH 114. Consent of instructor required.  Enrollment Limit: 32.

131.  Calculus Ia: Limits, Continuity and Differentiation            3 hours

         4NS, QPh

A first course in the calculus of functions of one variable including supporting material from algebra and trigonometry. Topics include limits, continuous functions, solution of equations and inequalities, differentiation of real-valued functions of one variable, and the graphical analysis of functions. Prerequisite: An appropriate score on the Calculus Readiness Exam. The two-course sequence MATH 131, MATH 132 is equivalent to the more intensive MATH 133. Consent of instructor required. Enrollment Limit: 32

132.  Calculus Ib: Integration and Applications                           3 hours

         4NS, QPf

Continuation of MATH 131. Topics include integration of real-valued functions of one variable, basic properties of the trigonometric and exponential functions, the fundamental theorems of the calculus, and applications. Prerequisite: MATH 131 or an appropriate score on the Calculus Readiness Exam. Consent of instructor required. Enrollment Limit: 32

133.  Calculus I: Limits, Continuity, Differentiation,                  4 hours
         Integration, and Applications

         4NS, QPf

A standard first course in the calculus of functions of one variable. Topics include limits, continuous functions, differentiation and integration of real-valued functions of one variable, the fundamental theorems of calculus, and applications. Prerequisite: An appropriate score on the Calculus Readiness Exam. This course is equivalent to the two-course sequence MATH 131, MATH 132. Consent of instructor required. Enrollment Limit: 32.

134.  Calculus II: Special Functions, Integration Techniques,      4 hours
         and Power Series

         4NS, QPf

Continuation of the study of the calculus of functions of one variable. Topics include logarithmic, exponential and the inverse trigonometric functions, techniques of integration, polar coordinates, parametric equations, infinite series and applications. Prerequisite: MATH 132 or MATH 133. Note: The course sequences MATH 133, 134 and MATH 131, 132, 134 both provide a standard introduction to single-variaable calculus. Enrollment Limit: 32

                 

Intermediate Courses

220.  Discrete Mathematics                                                        3 hours

         3NS, QPf

An introduction to a wide variety of mathematical ideas and techniques that do not involve calculus. Topics such as graph theory, combinatorics, difference equations, elementary number theory, recursion, mathematical induction, and logic. Prerequisite: MATH 133. Enrollment Limit: 32.

231.  Multivariable Calculus                                                      3 hours

         3NS, QPf

An introduction to the calculus of several variables. Topics considered include vectors and solid analytic geometry, multidimensional differentiation and integration, and a selection of applications. Prerequisite: MATH 134. Enrollment Limit: 32. 

232.  Linear Algebra                                                                    3 hours

         3NS, QPf

An introduction to linear algebra. Topics considered include the algebra and geometry of Euclidean n-space, matrices, determinants, abstract vector spaces, linear transformations, and diagonalization. Prerequisite: MATH 134 or MATH 220. Enrollment Limit: 32.

234.  Differential Equations                                                        3 hours

         3NS, QPf

An introduction to analytic, qualitative and numerical methods for solving ordinary differential equations. Topics include general first order equations, linear first and second order equations, numerical methods (Euler, Runge-Kutta), systems of first order equations, phase plane analysis, and Laplace Transforms. There is emphasis throughout the course on geometric and qualitative interpretations of differential equations, as well as applications to the natural sciences. Prerequisite: MATH 231. Enrollment Limit: 32.

 

236.  Partial Differential Equationsand Applied Complex Analysis              3 hours

         3NS, QPf

An introduction to complex analysis in the context of applications to partial differential equations. Topics to include: analytic functions, complex integration and residue calculus techniques, Fourier series, partial differential equations in rectangular, cylindrical and spherical coordinates, and associated special functions, Fourier and Laplace transfoms. Depending on student interest, numerical methods including finite difference and finite element techniques may be covered. Prerequisite: MATH 231. Enrollment Limit: 32.

 

Advanced Courses

301.  Advanced Calculus                                                              3 hours

         3NS, QPf

A rigorous examination of the basic elements of analysis. The structure of the real number system, continuity, differentiability, uniform continuity, integrability of functions of a single variable, sequences, series, and uniform convergence are typical topics to be explored. Prerequisite: MATH 231. MATH 220 is also highly recommended.

 

302.   Dynamical Systems                                                                      3 hours

3NS, QPf

A first course in discrete and continuous dynamical systems in dimensions one and higher. Topics include phase portraits, periodic orbits, hyperbolicity, bifurcations, symbolic dynamics, chaos and fractals. Prerequisite: MATH 231 and 232. Note: MATH 234 is recommended. Taught in alternate years only.   
                

317.  Number Theory                                                                      3 hours

         3NS, QPf

This course is an introduction to number theory.   Topics include primality, divisibility, modular arithmetic, finite fields, quadratic reciprocity, and elliptic curves.   Emphasis will be placed both on theoretical questions and on algorithms for computation.   Prerequisites : MATH 220 & 232 or consent of the instructor.   Note :   Taught in alternate years only.

327.  Group Theory                                                                      3 hours

         3NS, QPf

A first course in the modern algebraic structures and techniques fundamental to mathematics and useful in many areas of science and engineering. Topics include: groups, subgroups, quotient groups, isomorphism theorems, permutation groups, finite groups, and applications to combinatorics, geometry, symmetry, and crystallography. Prerequisite: MATH 232. Note: MATH 220 is also highly recommended.

328.  Computational Algebra and Algebraic Geometry           3 hours

         3NS, QPf

This course examines connections between the algebra and geometry of the set of solutions to a system of polynomial equations (called a variety) and the use of algorithms to effect concrete calculations.  Topics studied include rings and idels, Grobner bases, resultants and elimination theory, Hilbert's Nullstellensatz, the correspondence between polynomial ideals and algebraic varieties, and applications of the methods to other areas of mathematics.  There will be opportunities for computer experimentation and student projets.  Prerequisites: MATH 231 and MATH 232.  MATH 220 is also highly recommended.  Note: Given in alternate years only.

329. Abstract Algebra: Rings and Fields                                    3 hours

         3NS, QPf

This is one of two courses introducing algebraic structures and techniques fundamental to mathematics and useful in many areas of science and engineering.  Topics include: rings, subrings, ideals, fields, integral domains, polynomial rings, extension fields, finite fields, famous impossible constructions, and Galois theory. Prerequisite:  MATH 327 or consent by instructor.  Note: Taught in alternate years only.

331.  Optimization                                                                      3 hours

         3NS, QPf

An introduction to linear, integer, and nonlinear programming. Emphasis is placed on the theory of mathematical programming and the analysis of optimization algorithms. These are applied to significant problems in the fields of medicine, finance, public policy, transportation, and telecommunications. Prerequisites: MATH 231 and MATH 232.

335.  Probability                                                                         3 hours

         3NS, QPf

An introduction to the mathematical theory of probability and its applications. Topics include discrete and continuous sample spaces, combinatorial problems, random variables, probability densities, probability distributions, limit theorems, and stochastic processes. Prerequisite: MATH 231. MATH 220 is also strongly recommended.

336.  Mathematical Statistics                                                                        3 hours

         3NS, QPf

The theory of probability is applied to problems of statistics. Topics include sampling theory, point and interval estimation, tests of statistical hypotheses, regression, and analysis of variance. Prerequisites : MATH 232, MATH 335. Note : Taught in alternate years only.

337.  Data Analysis                                                                       3 hours

         3NS, QPf

In this course students will be given an introduction to the theory and use of regression graphics. Special regression and graphics software will be used to study relationships among several variables. Topics will include regression smoothing, residual plots, scatterplot matrices, three-dimensional plots, transformations of predictors and of response variables, and added-variable plots.   Note: Taught in alternate years only. Prerequisites: MATH 113 or 114 and MATH 232 or consent of the instructor.

338.  Probability Models and Random Processes                           3 hours

         3NS, QPf

An introduction to operations research models which incorporate methods of probability theory.  Topics will be chosen from inventory theory, queueing theory, decision analysis, game theory, simulation, Markov chains, and project management.  Computer software for selected topics will also be discussed and utilized.  Prerequisite: MATH 335. Note: Taught in alternate years only.

343.  Combinatorics                                                                              3 hours

         3NS, QP

An advanced course in discrete mathematics. Topics covered include enumeration, combinatorial identities, generating functions, partitions, and set systems. Prerequisites : any one of MATH 317, 327, 328, 329, or 335.  

345.  Information Theory                                                                            3 hours

         3NS, QP

An introduction to Information Theory and Coding Theory. Topics include information and entropy, data compression, Shannon theory and noisy channels, error-correcting codes, and applications to statistics, computer science, economics, and the natural sciences.   Prerequisites : MATH 220, 232, or MATH 335, or consent of instructor.

348.  Graphical Models                                                                               3 hours

         3NS, QP

An introduction to graphical models and Bayesian networks. This course will emphasize both the theory and the applications of these probabilistic models. Topics may include: background in probability and graph theory, building models, tuning and updating models, decision making, algorithms to accomplish all of these tasks, and applications to machine learning, biology and medicine, statistics and other fields. Prerequisites: MATH 220.

350.  Geometry                                                                              3 hours

         3NS, QPf

The course takes a modern approach to geometry based on group theory and the Erlangen Programm making possible the survey of a wide spectrum of geometries, Euclidean and non-Euclidean. Geometries treated include Moebius geometry, hyperbolic geometry, elliptic geometry, and absolute geometry. The discovery of these geometries in the 19th centry caused a scientific and philosophical revolution second only to the Copernican revolution.   Prerequisites:  MATH 220 or consent of instructor. 

 

 

353.  Topology                                                                               3 hours

         3NS, QPf

An introduction to point-set and algebraic topology.   The fundamental notion of a topological space is introduced and properties of separation, compactness and connectedness. Topological spaces are also studied by means of algebraic invariants including homotopy and homology. Some of the famous theorems to be proved using these tools include the Brouwer Fixed Point Theorem, Poincare Index Theorem, Classification of Surfaces and the Ham Sandwich Theorem. Prerequisite: Math 301 or 327 or consent of instructor.   Note: Taught in alternate years only.  

356.  Complex Analysis                                                                               3 hours

         3NS, QPf

An introduction to the theory of differentiable functions of a complex variable, including the Cauchy theorems, residues, series expansions, and conformal mappings. Prerequisite : MATH 301. Note : Taught in alternate years only.

358.  Real Analysis                                                                               3 hours

         3NS, QPf

This course presents important generalizations of integration and differentiation developed in the twentieth century.   An introduction to metric spaces, Lebesgue's theory of the integral, and general measure and integration theory.   Prerequisite : MATH 301.  

 

360.  Mathematical Methods for Computational Neuroscience                                                                               3 hours

         3NS, QPf

An introduction to analytical and numerical mathematical methods with application to computational neuroscience.   Methods include numerical solution of nonlinear ordinary differential equations, Markov chains, bifurcation theory, partial differential equations, and information theory.   Neuroscience applications include dynamics of ion channels and nerve membranes, reliability and precision of patterns of action potentials, synaptic transmission and plasticity, and information processing in neural networks.   Students will use the NEURON programming language to develop simulations at various levels of biophysical complexity. Prerequisites: MATH 232 and MATH 234, or consent of instructor.

 

362.  Mathematical Biology                                                                               3 hours

         3NS, QPf

A research-oriented seminar course on the mathematical modeling of biological systems. Topics include applications of ODEs and PDEs, Monte Carlo methods, stochastic differential equations and bifurcation theory. Biological topics include synchronization, population dynamics, patterning in growth & development, reaction-diffusion systems, coordination of movement and chemotaxis. Student research projects will comprise a significant part of the course.   Prerequisites:  MATH 231 and consent of instructor.

 

 

 

383.  Theory of Computer Science                                                                               3 hours

         3NS, QP

 


Identical to CSCI 383

397.  Seminar in Mathematical Modeling                                                                               3 hours

         3NS, QPf

Teams of students will work on projects involving optimization or probability theory. Possible projects include realignment of teams in the National Football League, simulation of the spread of HIV in prostitutes in Central Africa, selection of portfolios of stocks and bonds, analysis of the behavior of cellular automata, graph drawing and image compression. Prerequisites: MATH 331 or MATH 335.

 

398.  Seminar in Mathematical Logic                                                                              3 hours

         3NS, QPf

An introduction to set theory and computability. This seminar will examine both the foundations of mathematics and the limitations of formal reasoning. Student projects, consisting of a presentation and an expository paper, will be based on independent reading. Prerequisite: Two 300-level mathematics course.

 

399.  Seminar in Number Systems                                                  3 hours

         3NS, QPf

This seminar explores alternatives to the classical real numbers. They systems to be studied are the complexes, the quaternions, the constructive reals, the hyperreals, and the surreals. These include multidimensional numbers, infinitely small and infinitely large numbers, and numbers that represent positions in games. Some systems are old; some are new; several represent conflicting philosophies of mathematics. Prerequisite: One 300-level mathematics course.

401.  Honors                                                  2-4 hours

         2-4NS

Consent of instructor required

550, 551.  Research                                                  1-2 hours

         3NS, QPf

Projects for original investigation. Interested students are encouraged to talk to individual faculty members about possible projects. Prerequisite: Consent of the chair of the department.