Undergraduate Courses
MATH 110. Introduction to Mathematical Communication and Software (1)
Mathematical text editors. Mathematical composition and exposition. Posting mathematical material on the Web. Basics of computer symbolic manipulation (Mathematica). Computer vector/matrix manipulation and applications (MATLAB). Basic computer statistical methods (Minitab). Integration of output from computer calculations into text.
MATH 120. Elementary Functions and Analytic Geometry (3)
Polynomial, rational, exponential, logarithmic, and trigonometric functions (emphasis on computation, graphing, and location of roots) straight lines and conic sections. Primarily a precalculus course for the student without a good background in trigonometric functions and graphing and/or analytic geometry. Not open to students with credit for MATH 121 or MATH 125. Prereq: Three years of high school mathematics.
MATH 121. Calculus for Science and Engineering I (4)
Functions, analytic geometry of lines and polynomials, limits, derivatives of algebraic and trigonometric functions. Definite integral, antiderivatives, fundamental theorem of calculus, change of variables. Prereq: Three and one half years of high school mathematics.
MATH 122. Calculus for Science and Engineering II (4)
Continuation of MATH 121. Exponentials and logarithms, growth and decay, inverse trigonometric functions, related rates, basic techniques of integration, area and volume, polar coordinates, parametric equations. Taylor polynomials and Taylor's theorem. Prereq: MATH 121.
MATH 123. Calculus I (4)
Limits, continuity, derivatives of algebraic and transcendental functions, including applications, basic properties of integration. Techniques of integration and applications. Prereq: Placement by the department.
MATH 124. Calculus II (4)
Review of differentiation. Techniques of integration, and applications of the definite integral. Parametric equations and polar coordinates. Taylor's theorem. Sequences, series, power series. Complex arithmetic. Introduction to multivariable calculus. Prereq: MATH 123 and placement by the department.
MATH 125. Mathematics I (4)
Discrete and continuous probability; differential and integral calculus of one variable; graphing, related rates, maxima and minima. Integration techniques, numerical methods, volumes, areas. Applications to the physical, life, and social sciences. Students planning to take more than two semesters of introductory mathematics should take MATH 121. Prereq: Three and one half years of high school mathematics.
MATH 126. Mathematics II (4)
Continuation of MATH 125 covering differential equations, multivariable calculus, discrete methods. Partial derivatives, maxima and minima for functions of two variables, linear regression. Differential equations; first and second order equations, systems, Taylor series methods; Newton's method; difference equations. Prereq: MATH 125.
MATH 150. Mathematics from a Mathematician's Perspective (3)
An interesting and accessible mathematical topic not covered in the standard curriculum is developed. Students are exposed to methods of mathematical reasoning and historical progression of mathematical concepts. Introduction to the way mathematicians work and their attitude toward their profession. Should be taken in freshman year to count toward a major in mathematics. Prereq: Three and one half years of high school mathematics.
MATH 201. Introduction to Linear Algebra (3)
Matrix operations, systems of linear equations, vector spaces, subspaces, bases and linear independence, eigenvalues and eigenvectors, diagonalization of matrices, linear transformations, determinants. Less theoretical than MATH 307. May not be taken for credit by mathematics majors. Only one of MATH 201 or MATH 307 may be taken for credit. Prereq: MATH 122 or MATH 126.
MATH 223. Calculus for Science and Engineering III (3)
Introduction to vector algebra; lines and planes. Functions of several variables: partial derivatives, gradients, chain rule, directional derivative, maxima/minima. Multiple integrals, cylindrical and spherical coordinates. Derivatives of vector valued functions, velocity and acceleration. Vector fields, line integrals, Green's theorem. Prereq: MATH 122.
MATH 224. Elementary Differential Equations (3)
A first course in ordinary differential equations. First order equations and applications, linear equations with constant coefficients, linear systems, Laplace transforms, numerical methods of solution. Prereq: MATH 223.
MATH 227. Calculus III (3)
Vector algebra and geometry. Linear maps and matrices. Calculus of vector valued functions. Derivatives of functions of several variables. Multiple integrals. Vector fields and line integrals. Prereq: MATH 124 or placement by department.
MATH 228. Differential Equations (3)
Elementary ordinary differential equations: first order equations; linear systems; applications; numerical methods of solution. Prereq: MATH 227.
MATH 234. Differential Equations and Dynamical Systems (3)
An introductory course in discrete and continuous dynamics (difference and differential equations). One dimensional differential equations: dynamics; linear equations, separable equations; numerical methods. Systems of differential equations in two dimensions: dynamics of autonomous systems, numerical methods, solution of constant coefficient linear systems, with and without forcing. Laplace transforms and convolution. Discrete dynamics; introduction to chaos, numerical methods as difference equations. Linear difference equations in one and two dimensions, z-transform, convolution. Prereq: MATH 223.
MATH 301. Undergraduate Reading Course (1-3)
Students must obtain the approval of a supervising professor before registration. More than one credit hour must be approved by the undergraduate committee of the department.
MATH 302. Problem Solving Seminar (1)
A seminar devoted to methods of solving problems in various areas of mathematics. Content varies. Students may take this course for credit up to four times.
MATH 303. Elementary Number Theory (3)
Primes and divisibility, theory of congruencies, and number theoretic functions. Diophantine equations, quadratic residue theory, and other topics determined by student interest. Emphasis on problem solving (formulating conjectures and justifying them). Prereq: MATH 122.
MATH 304. Discrete Mathematics (3)
A general introduction to basic mathematical terminology and the techniques of abstract mathematics in the context of discrete mathematics. Topics introduced are mathematical reasoning, Boolean connectives, deduction, mathematical induction, sets, functions and relations, algorithms, graphs, combinatorial reasoning. Prereq: MATH 122 or MATH 126.
MATH 307. Introduction to Abstract Algebra I (3)
First semester of an integrated, two-semester theoretical course in abstract and linear algebra, studied on an axiomatic basis. The major algebraic structures studied are groups, rings, fields, modules, vector spaces, and inner product spaces. Topics include homomorphisms and quotient structures, the theory of polynomials, canonical forms for linear transformations and the principal axis theorem. This course is required of all students majoring in mathematics. Only one of MATH 201 or MATH 307 may be taken for credit. Prereq: MATH 122.
MATH 308. Introduction to Abstract Algebra II (3)
Continuation of MATH 307. Prereq: MATH 307.
MATH 319. Applied Probability and Stochastic Processes for Biology (3)
Applications of probability and stochastic processes to biological systems. Mathematical topics will include: introduction to discrete and continuous probability spaces (including numerical generation of pseudo random samples from specified probability distributions), Markov processes in discrete and continuous time with discrete and continuous sample spaces, point processes including homogeneous and inhomogeneous Poisson processes and Markov chains on graphs, and diffusion processes including Brownian motion and the Ornstein- Uhlenbeck process. Biological topics will be determined by the interests of the students and the instructor. Likely topics include: stochastic ion channels, molecular motors and stochastic ratchets, actin and tubulin polymerization, random walk models for neural spike trains, bacterial chemotaxis, signaling and genetic regulatory networks, and stochastic predator-prey dynamics. The emphasis will be on practical simulation and analysis of stochastic phenomena in biological systems. Numerical methods will be developed using both MATLAB and the R statistical package. Student projects will comprise a major part of the course. Prereq: MATH 224 or MATH 228 or BIOL 300 or BIOL 306. Cross-listed as BIOL 319, BIOL 419, EECS 319
MATH 321. Fundamentals of Analysis I (3)
Abstract mathematical reasoning in the context of analysis in Euclidean space. Introduction to formal reasoning, sets and functions, and the number systems. Sequences and series; Cauchy sequences and convergence. Required for all mathematics majors. Prereq: MATH 223.
MATH 322. Fundamentals of Analysis II (3)
Continuation of MATH 321. Point-set topology in metric spaces with attention to n-dimensional space; completeness, compactness, connectedness, and continuity of functions. Topics in sequences, series of functions, uniform convergence, Fourier series and polynomial approximation. Theoretical development of differentiation and Riemann integration. Required for all mathematics majors. Prereq: MATH 321.
MATH 323. Advanced Calculus (3)
A systematic approach to the differential and integral calculus of functions of several variables. Sets and topology in Euclidean spaces. Continuity. Differentiability. Riemann integration in Euclidean spaces. Inverse and implicit function theorems. Introduction to manifolds. Prereq: MATH 321.
MATH 324. Introduction to Complex Analysis (3)
Properties, singularities, and representations of analytic functions, complex integration. Cauchy's theorems, series residues, conformal mapping and analytic continuation. Riemann surfaces. Relevance to the theory of physical problems. Prereq: MATH 224.
MATH 327. Convexity and Optimization (3)
Introduction to the theory of convex sets and functions and to the extremes in problems in areas of mathematics where convexity plays a role. Among the topics discussed are basic properties of convex sets (extreme points, facial structure of polytopes), separation theorems, duality and polars, properties of convex functions, minima and maxima of convex functions over convex set, various optimization problems. Prereq: MATH 223 or consent.
MATH 330. Scientific Computing: Fundamentals and Applications (3)
An introductory survey to Scientific Computing, from principles to applications. Topics include accuracy and efficiency, conditioning and stability, numerical solution of linear and nonlinear systems, optimization, interpolation, quadrature rules, numerical solutions of ODEs and PDEs. Coreq: MATH 224.
MATH 338. Introduction to Dynamical Systems (3)
Nonlinear discrete dynamical systems in one and two dimensions. Chaotic dynamics, elementary bifurcation theory, hyperbolicity, symbolic dynamics, structural stability, stable manifold theory. Prereq: MATH 223.
MATH 343. Theoretical Computer Science (3)
Introduction to mathematical logic, different classes of automata and their correspondence to different classes of formal languages, recursive functions and computability, assertions and program verification, denotational semantics. MATH/EECS 343 and MATH 410 cannot both be taken for credit. Prereq: MATH 304 and EECS 340. Cross-listed as EECS 343.
MATH 345. Introduction to Applied Mathematics (3)
Mathematical formulation of problems, development of various methods of solution, and interpretation of results, boundary value problems. Sturm-Liouville problems, complex analysis, transform methods. Prereq: MATH 224.
MATH 350. Domain Theoretic Methods for Artificial Intelligence (3)
Resolution for propositional logic and completeness via Zorn's Lemmq, Domain theory and topology through three-value logic. Default reasoning and extensions. Clausal logic for Scott domains and Smyth power domains. Power defaults theory and the semantics of nonmonotonic reasoning and disjunctive logic programming. Prereq: EECS 343, EECS 391, MATH 307, or PHIL 306. Cross-listed as EECS 358.
MATH 363. Knot Theory (3)
An introduction to the mathematical theory of knots and links, with emphasis on the modern combinatorial methods. Reidemeister moves on link projections, ambient and regular isotopies, linking number tricolorability, rational tangles, braids, torus knots, seifert surfaces and genus, the knot polynomials (bracket, X, Jones, Alexander, HOMFLY), crossing numbers of alternating knots and amphicheirality. Connections to theoretical physics, molecular biology, and other scientific applications will be pursued in term projects, as appropriate to the background and interests of the students. Prereq: MATH 223.
MATH 378: Computational Neuroscience (3)
Computer simulations and mathematical analysis of neurons and neural circuits, and the computational properties of nervous systems. Students are taught a range of models for neurons and neural circuits, and are asked to implement and explore the computational and dynamic properties of these models. The course introduces students to dynamical systems theory for the analysis of neurons and neural circuits, as well as a cable theory, passive and active compartmental modeling, numerical integration methods, models of plasticity and learning, models of brain systems, and their relationship to artificial and neural networks. Term project
required. Students enrolled in MATH 478 will make arrangements with the instructor to attend additional lectures and complete additional assignments addressing mathematical topics related to the course. Prereq: MATH 223 & MATH 224 or BIOL 300 & BIOL 306, or consent of department. Cross-listed as EBME 478, EECS 478, MATH 478, NEUR 478
MATH 380. Introduction to Probability (3)
Combinatorial analysis. Permutations and combinations. Axioms of probability. Sample space and events. Equally likely outcomes. Conditional probability. Bayes' formula. Independent events and trials. Discrete random variables, probability mass functions. Expected value, variance. Bernoulli, binomial, Poisson, geometric, negative binomial random variables. Continuous random variables, density functions. Expected value and variance. Uniform, normal, exponential, Gamma random variables. The De Moivre-Laplace limit theorem. Joint probability mass functions and densities. Independent random variables and the distribution of their sums. Covariance. Conditional expectations and distributions (discrete case). Moment generating functions. Law of large numbers. Central limit theorem. Additional topics (time permitting): the Poisson process, finite state space Markov chains, entropy. Prereq: MATH 122 or MATH 126.
MATH 381. Introduction to Mathematical Methods in Finance (3)
Mathematical finance in discrete time. Single period market models. Arbitrage. Risk-neutral valuation of contingent claims. Complete markets. Summary of results from probability theory and stochastic processes in discrete time. Conditional expectation. Discrete parameter martingales. Multiperiod market modes. Equivalent martingale measures. Risk-neutral valuation. Hedging strategies. Complete markets. The Cox-Ross-Rubinstein model. European options. American options. The Black-Scholes model. Binomial approximation. The pricing formula for European call options. Prereq: MATH 380.
MATH 399. Special Topics (3)
Special Topics in Mathematics
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