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400-Level Math Courses

Math 403 - Introduction to Discrete Mathematics

Prerequisites:	Math 214, 217, 286, 296, 417, or 419
Credit:	3 Credits
Background and Goals:	This course is intended for students in the Data Science Masters program, or undergraduate students who are not mathematics majors. This course will not count towards the Mathematics Major.
Content:	Propositional logic; quantifiers; basic logical deduction rules; fundamental properties of natural numbers, especially methematical induction and well-ordering; sets; set operations and their algebraic properties; functions, including properties like injectivity and surjectivity; relations; patial and total orders; equivalence relations and partitions; elementary combinatorics, including permutations, binomial coefficients, and inclusion-exclusion; graphs; and discrete probability, including conditional probability, Bayes's theorem, independence, expectation, variance, and standard deviation.

Math 404 - Intermediate Differential Equations

Prerequisites:	Math 216, 286, or 316
Credit:	3 Credits
Background and Goals:	This course is an introduction to the modern qualitative theory of ordinary differential equations with emphasis on geometric techniques and visualization. Much of the motivation for this approach comes from applications. Examples of applications of differential equations to science and engineering are a significant part of the course. There are relatively few proofs.
Content:	Geometric representation of solutions, autonomous systems, flows and evolution, linear systems and phase portraits, nonlinear systems, local and global behavior, linearization, stability, conservation laws, periodic orbits. Applications: free and forced oscillations, resonance, relaxation oscillations, competing species, Zeeman’s models of heartbeat and nerve impulse, chaotic orbits, strange attractors.

Math 412 - Introduction to Modern Algebra (IBL)

Prerequisites:	Math 205 or 215 or 285; and Math 217
Credit:	3 credits. 1 credit after Math 312
Background and Goals:	This course is designed to serve as an introduction to the methods and concepts of abstract mathematics. A typical student entering this course has substantial experience in using complex mathematical (calculus) calculations to solve physical or geometrical problems, but is inexperienced at analyzing carefully the content of definitions and the logical flow of ideas which underlie and justify these calculations. Although the topics discussed here are quite distinct from those of calculus, an important goal of the course is to introduce the student to this type of analysis. Much of the reading, homework exercises, and exams consist of theorems (propositions, lemmas, etc.) and their proofs. Math 217, or equivalent, required as background.
Content:	The initial topics include ones common to every branch of mathematics: sets, functions (mappings), relations, and the common number systems (integers, rational numbers, real numbers, complex numbers). These are then applied to the study of particular types of mathematical structures: groups, rings, and fields. These structures are presented as abstractions from many examples such as the common number systems together with the operations of addition or multiplication, permutations of finite and infinite sets with function composition, sets of motions of geometric figures, and polynomials. Notions such as generator, subgroup, direct product, isomorphism, and homomorphism are defined and studied.

Math 416 - Theory of Algorithms

Prerequisites:	Math 312, 412; or EECS 280 and Math 465; or permission of instructor
Credit:	3 credits.
Background and Goals:	Many common problems from mathematics and computer science may be solved by applying one or more algorithms — well-defined procedures that accept input data specifying a particular instance of the problem and produce a solution. Students entering Math 416 typically have encountered some of these problems and their algorithmic solutions in a programming course. The goal here is to develop the mathematical tools necessary to analyze such algorithms with respect to their efficiency (running time) and correctness. Different instructors will put varying degrees of emphasis on mathematical proofs and computer implementation of these ideas.
Content:	Typical problems considered are: sorting, searching, matrix multiplication, graph problems (flows, traveling salesman), and primality and pseudo-primality testing (in connection with coding questions). Algorithm types such as divide-and-conquer, backtracking, greedy, and dynamic programming are analyzed using mathematical tools such as generating functions, recurrence relations, induction and recursion, graphs and trees, and permutations. The course often includes a section on abstract complexity theory including NP completeness.

Math 417 - Matrix Algebra I

Prerequisites:	Three mathematics courses beyond Math 110
Credit:	3 credits. No credit granted to those who have completed or are enrolled in Math 214, 217, 419, or 420. Math 417 and Math 419 may not be used as electives in the Statistics major.
Background and Goals:	Many problems in science, engineering, and mathematics are best formulated in terms of matrices - rectangular arrays of numbers. This course is an introduction to the properties of and operations on matrices with a wide variety of applications. The main emphasis is on concepts and problem-solving, but students are responsible for some of the underlying theory. Diversity rather than depth of applications is stressed. This course is not intended for mathematics majors, who should elect Math 217, and/or Math 493-494 if pursuing the honors major.
Content:	Topics include matrix operations, echelon form, general solutions of systems of linear equations, vector spaces and subspaces, linear independence and bases, linear transformations, determinants, orthogonality, characteristic polynomials, eigenvalues and eigenvectors, and similarity theory. Applications include linear networks, least squares method (regression), discrete Markov processes, linear programming, and differential equations.

Math 419 - Linear Spaces and Matrix Theory

Prerequisites:	4 courses beyond Math 110
Credit:	3 credits. Credit is granted for only one course among Math 214, 217, 417, and 419. No credit granted to those who have completed or are enrolled in Math 420. Math 417 and Math 419 may not be used as electives in the Statistics major.
Background and Goals:	Math 419 covers much of the same ground as Math 417 but presents the material in a somewhat more abstract way in terms of vector spaces and linear transformations instead of matrices. There is a mix of proofs, calculations, and applications with the emphasis depending somewhat on the instructor. A previous proof-oriented course is helpful but by no means necessary.
Content:	Basic notions of vector spaces and linear transformations: spanning, linear independence, bases, dimension, matrix representation of linear transformations; determinants; eigenvalues, eigenvectors, Jordan canonical form, inner-product spaces; unitary, self-adjoint, and orthogonal operators and matrices, applications to differential and difference equations.

Math 420 - Advanced Linear Algebra

Prerequisites:	Math 214, 217, 417, or 419 and one of 296, 412, or 451
Credit:	3 credits
Background and Goals:	This is an introduction to the formal theory of abstract vector spaces and linear transformations. It is expected that students have completed at least one prior linear algebra course. The emphasis is on concepts and proofs with some calculations to illustrate the theory. Students should have significant mathematical maturity, at the level of Math 412 or 451. In particular, students should expect to work with and be tested on formal proofs.
Content:	Topics are selected from: vector spaces over arbitrary fields (including finite fields); linear transformations, bases, and matrices; inner product spaces, duals and spaces of linear transformations, theory of determinants, eigenvalues and eigenvectors; applications to linear differential equations; bilinear and quadratic forms; spectral theorem; Jordan Canonical Form, least squares, singular value theory.

Math 422 (BE 440) - Risk Management and Insurance

Prerequisites:	Math 115, Junior standing, and permission of instructor
Credit:	3 credits. Satisfies the Upper Level Writing Requirement (ULWR)
Background and Goals:	This course is designed to allow students to explore the insurance mechanism as a means of replacing uncertainty with certainty. A main goal of the course is to explain, using mathematical models from the theory of interest, risk theory, credibility theory, and ruin theory, how mathematics underlies many important individual and societal problems.
Content:	We will explore how much insurance affects the lives of students (automobile insurance, social security, health insurance, theft insurance) as well as the lives of other family members (retirements, life insurance, group insurance). While the mathematical models are important, an ability to articulate why the insurance options exist and how they satisfy the consumer’s needs are equally important. In addition, there are different options available (e.g., in social insurance programs) that offer the opportunity of discussing alternative approaches. This course may be used to satisfy the LS&A upper-level writing requirement.

Math 423 - Mathematics of Finance

Prerequisites:	Math 217 and Math 425 or 525, and EECS 183 or equivalent
Credit:	3 credits.
Background and Goals:	This course is an introduction to the mathematical models used in finance and economics with particular emphasis on models for pricing derivative instruments such as options and futures. The goal is to understand how the models derive from basic principles of economics and to provide the necessary mathematical tools for their analysis. A solid background in basic probability theory is necessary.
Content:	Topics include risk and return theory, portfolio theory, the capital asset pricing model, the random walk model, stochastic processes, Black-Scholes Analysis, numerical methods, and interest rate models.

Math 424 - Compound Interest and Life Insurance

Prerequisites:	Math 205 or 215 or 285
Credit:	3 credits.
Background and Goals:	This course explores the concepts underlying the theory of interest and then applies them to concrete problems. The course also includes applications of spreadsheet software. The course is a prerequisite to advanced actuarial courses. It also helps students prepare for some of the professional actuarial exams.
Content:	The course covers compound interest (growth) theory and its application to valuation of monetary deposits, annuities, and bonds. Problems are approached both analytically (using algebra) and geometrically (using pictorial representations). Techniques are applied to real-life situations: bank ac- counts, bond prices, etc. The text is used as a guide because it is prescribed for the professional examinations; the material covered will depend some- what on the instructor.

Math 425 (Stats 425) - Introduction to Probability

Prerequisites:	Math 205 or 215 or 285
Credit:	3 credits
Background and Goals:	This course introduces students to both useful and interesting ideas from the mathematical theory of probability and to a number of applications of probability to a variety of fields including genetics, economics, geology, business, and engineering. The theory developed together with other mathematical tools such as combinatorics and calculus are applied to everyday problems. Concepts, calculations, and derivations are emphasized. The course will make essential use of the material of Math 116 and 215.
Content:	Topics include the basic results and methods of both discrete and continuous probability theory: conditional probability, independent events, random variables, joint distributions, expectations, variances, and covariances. The culminating results are the Law of Large Numbers and the Central Limit Theorem. Beyond this, different instructors may add additional topics of interest.

Math 427 - Retirement Plans and Other Employee Benefits

Prerequisites:	Math 115, Junior standing or permission of instructor
Credit:	3 credits. Satisfies the Upper Level Writing Requirement (URLW).
Background and Goals:	An overview of the range of employee benefit plans, the considerations (actuarial and others) which influence plan design and implementation practices, and the role of actuaries and other benefit plan professionals and their relation to decision makers in management and unions. This course is certified for satisfaction of the upper-level writing requirement.
Content:	Particular attention will be given to government programs which provide the framework, and establish requirements, for privately operated benefit plans. Relevant mathematical techniques will be reviewed, but are not the exclusive focus of the course.

Math 429 - Internship

Prerequisites:	Major in Mathematics
Credit:	1 Credit (Credit/No Credit grading scheme)
Background and Goals:	Credit is granted for a full-time internship of at least eight weeks that is used to enrich a student's academic experience and/or allows the student to explore careers related to his/her academic studies. Internship credit is not retroactive and must be prearranged. May not apply toward a Mathematics concentration. May be used to satisfy the Curriculum Practical Training (CPT) required of foreign students. Y grade may be posted at the end of the term to indicate work in progress.
Content:	Course content is determined by student's internship. After completing the internship, the student submits a report (2 to 3 pages, double-spaced) describing the work done in the internship, how the internship was valuable to the math major, and how it provided experiences that couldn't be achieved in a classroom. The grade will be changed to "CR" after the report is received.

Math 431 - Explorations in Euclidean Geometry (IBL)

Prerequisites:	Math 205 or 215 or 285; Math 217
Credit:	3 credits
Background and Goals:	This course deepens students' understanding of the structure and practice of mathematics through an axiomatic development of Euclidean geometry. Throughout the course, students will write proofs, make conjectures, communicate mathematics verbally and in writing, and critique mathematical arguments. At instructor discretion, examples of non-Euclidean geometries may be studied for comparison.
Content:	Topics selected depend heavily on the instructor but may include classification of isometries of the Euclidean plane; similarities; rosette, frieze, and wallpaper symmetry groups; tessellations; triangle groups; finite, hyperbolic, and taxicab non-Euclidean geometries.

Math 433 - Introduction to Differential Geometry

Prerequisites:	Math 205 or 215 or 285; and Math 217
Credit:	3 credits.
Background and Goals:	This course is about the analysis of curves and surfaces in 2- and 3-space using the tools of calculus and linear algebra. There will be many examples discussed, including some which arise in engineering and physics applications. Emphasis will be placed on developing intuition and learning to use calculations to verify and prove theorems. Students need a good background in multivariable calculus (Math 215) and linear algebra (preferably Math 217). Some exposure to differential equations (Math 216 or Math 316) is helpful but not absolutely necessary.
Content:	Curves and surfaces in three-space using calculus. Curvature and torsion of curves. Curvature, covariant differentiation, parallelism, isometry, geodesics, and area on surfaces. Gauss-Bonnet Theorem. Minimal surfaces.

Math 440 - LoG(M) | Laboratory of Geometry

Prerequisites:	Math 205 or 215 or 285 and Math 217
Credit:	3 Credits.
Background and Goals:	The LoG-M course aims to provide undergraduates with mathematics research-type projects which focus on methods in computation and visualization. Projects differe each semester and are designed and led by experienced research faculty in the department. Undergrads are assigned to teams which work through one research problem through the semester. Lab of Geometry at Michigan LoG(M)
Content:	Varies, centered on geometry & visualization.

Math 450 - Advanced Mathematics for Engineers I

Prerequisites:	Math 205 or 215 or 285; and Math 216, 286, or 316
Credit:	4 Credits. No credit granted for those who have completed or are enrolled in Math 354 or Math 454.
Background and Goals:	This course is an introduction to some of the main mathematical techniques in engineering and physics. It is intended to provide some background for courses in those disciplines with a mathematical requirement that goes beyond calculus. Model problems in mathematical physics are studied in detail. Applications are emphasized throughout.
Content:	Topics covered include: Fourier series and integrals; the classical partial differential equations (the heat, wave and Laplace’s equations) solved by separation of variables; an introduction to complex variables and conformal mapping with applications to potential theory. A review of series and series solutions of ODEs will be included as needed. A variety of basic diffusion, oscillation, and fluid flow problems will be discussed.

Math 451 - Advanced Calculus I

Prerequisites:	Previous exposure to abstract mathematics, e.g. MATH 217 and 412
Credit:	3 credits. No credit granted to those who have completed or are enrolled in MATH 351.
Background and Goals:	This course has two complementary goals: (1) a rigorous development of the fundamental ideas of calculus, and (2) a further development of the student’s ability to deal with abstract mathematics and mathematical proofs. The key words here are “rigor” and “proof;” almost all of the material of the course is geared toward understanding and constructing definitions, theorems (propositions, lemmas, etc.), and proofs. This is considered one of the more difficult among the undergraduate mathematics courses, and students should be prepared to make a strong commitment to the course. In particular, it is strongly recommended that some course which requires proofs (such as Math 412) be taken before Math 451.
Content:	Topics covered include: logic and techniques of proofs; sets, functions, and relations; cardinality; the real number system and its topology; infinite sequences, limits, and continuity; differentiation; integration, the Fundamental Theorem of Calculus, infinite series; sequences and series of functions.

Math 452 - Advanced Calculus II

Prerequisites:	Math 217, 417, 419 or 420 (may be concurrent) and Math 451
Credit:	3 credits.
Background and Goals:	This course gives a rigorous development of multivariable calculus and elementary function theory with some view towards generalizations. Concepts and proofs are stressed. This is a relatively difficult course, but the stated prerequisites provide adequate preparation.
Content:	Topics include: (1) partial derivatives and differentiability; (2) gradients, directional derivatives, and the chain rule; (3) implicit function theorem; (4) surfaces, tangent planes; (5) max-min theory; (6) multiple integration, change of variable, etc.; (7) Greens' and Stokes’ theorems, differential forms, exterior derivatives.

Math 454 - Boundary Value Problems for Partial Differential Equations

Prerequisites:	Math 215 or 285; and Math 216, 286, or 316
Credit:	3 Credits. 1 credit after Math 354. No credit granted to those who have completed or are enrolled in Math 450.
Background and Goals:	This course is devoted to the use of Fourier series and other orthogonal expansions in the solution of initial-value and boundary-value problems for second-order linear partial differential equations. Emphasis is on concepts and calculation. The official prerequisite is ample mathematical preparation.
Content:	Classical representation and convergence theorems for Fourier series; meth- od of separation of variables for the solution of the one-dimensional heat and wave equation; the heat and wave equations in higher dimensions; eigen-function expansions; spherical and cylindrical Bessel functions; Legendre polynomials; methods for evaluating asymptotic integrals (Laplace’s method, steepest descent); Laplace’s equation and harmonic functions, including the maximum principle. As time permits, additional topics will be selected from: Fourier and Laplace transforms; applications to linear input-output systems, analysis of data smoothing and filtering, signal processing, time-series analysis, and spectral analysis; dispersive wave equations; the method of stationary phase; the method of characteristics.

Math 462 - Mathematical Models

Prerequisites:	Math 216, 286, or 316; and Math 217, 417, or 419
Credit:	3 Credits. 1-3 credits after 463 depending on overlap
Background and Goals:	The focus of this course is the application of a variety of mathematical techniques to solve real-world problems. Students will learn how to model a problem in mathematical terms and use mathematics to gain insight and eventually solve the problem. Concepts and calculations, using applied analysis and numerical simulations, are emphasized.
Content:	Construction and analysis of mathematical models in physics, engineering, economics, biology, medicine, and social sciences. Content varies consider- ably with instructor. Recent versions: Use and theory of dynamical systems (chaotic dynamics, ecological and biological models, classical mechanics), and mathematical models in physiology and population biology.

Math 463 (Bioinf 463/Biophys 463) - Math Modeling in Biology

Prerequisites:	Math 214, 217, 417, or 419; and 216, 286, or 316
Credit:	3 Credits.
Background and Goals:	The complexities of the biological sciences make interdisciplinary involvement essential and the increasing use of mathematics in biology is inevitable as biology becomes more quantitative. Mathematical biology is a fast growing and exciting modern application of mathematics that has gained world- wide recognition. In this course, mathematical models that suggest possible mechanisms that may underlie specific biological processes are developed and analyzed. Another major emphasis of the course is illustrating how these models can be used to predict what may follow under currently untested conditions. The course moves from classical to contemporary models at the population, organ, cellular, and molecular levels. The goals of this course are: (i) Critical understanding of the use of differential equation methods in mathematical biology and (ii) Exposure to specialized mathematical and computational techniques which are required to study ordinary differential equations that arise in mathematical biology. By the end of this course students will be able to derive, interpret, solve, understand, discuss, and critique discrete and differential equation models of biological systems.
Content:	This course provides an introduction to the use of continuous and discrete differential equations in the biological sciences. Biological topics may include single species and interacting population dynamics, modeling infectious and dynamic diseases, regulation of cell function, molecular interactions and receptor-ligand binding, biological oscillators, and an introduction to biological pattern formation. There will also be discussions of current topics of interest such as Tumor Growth and Angiogenesis, HIV and AIDS, and Control of the Mitotic Clock. Mathematical tools such as phase portraits, bifurcation diagrams, perturbation theory, and parameter estimation techniques that are necessary to analyze and interpret biological models will also be covered. Approximately one class period each week will be held in the mathematics computer laboratory where numerical techniques for finding and visualizing solutions of differential and discrete systems will be discussed.

Math 464 - Inverse Problems

Prerequisites:	Math 217, 417, or 419 and Math 216, 286, or 316
Credit:	3 credits.
Background and Goals:	Solution of an inverse problem is a central component of fields such as medical tomography, geophysics, non-destructive testing, and control theory. The solution of any practical inverse problem is an interdisciplinary task. Each such problem requires a blending of mathematical constructs and physical realities. Thus, each problem has its own unique components; on the other hand, there is a common mathematical framework for these problems and their solutions. This framework is the primary content of the course. This course will allow students interested in the above-named fields to have an opportunity to study mathematical tools related to the mathematical foundations of said fields.
Content:	The course content is motivated by a particular inverse problem from a field such as medical tomography (transmission, emission), geophysics (remote sensing, inverse scattering, tomography), or non-destructive testing. Mathematical topics include ill-posedness (existence, uniqueness, stability), regularization (e.g., Tikhonov, least squares, modified least squares, variation, mollification), pseudoinverses, transforms (e.g., k-plane, Radon, X-ray, Hilbert), special functions, and singular-value decomposition. Physical aspects of particular inverse problems will be introduced as needed, but the emphasis of the course is investigation of the mathematical concepts related to analysis and solution of inverse problems.

Math 465 - Introduction to Combinatorics

Prerequisites:	Linear Algebra (one of Math 214, 217, 296, 417, or 419) or permission of instructor
Credit:	3 Credits. No credit granted to those who have completed or are enrolled in Math 565 or 566.
Background and Goals:	Combinatorics is the study of finite mathematical objects, including their enumeration, structural properties, design, and optimization. Combinatorics plays an increasingly important role in various branches of mathematics and in numerous applications, including computer science, statistics and statistical physics, operations research, bioinformatics, and electrical engineering. This course provides an elementary introduction to the fundamental notions, techniques, and theorems of enumerative combinatorics and graph theory.
Content:	An introduction to combinatorics, covering basic counting techniques (inclusion-exclusion, permutations and combinations, generating functions) and fundamentals of graph theory (paths and cycles, trees, graph coloring). Additional topics may include partially ordered sets, recurrence relations, partitions, matching theory, and combinatorial algorithms.

Math 466 (EEB 466) - Mathematical Ecology

Prerequisites:	Math 217, 417, or 419; Math 286, or 316; and Math 450 or 451
Credit:	3 credits.
Background and Goals:	This course gives an overview of mathematical approaches to questions in the science of ecology. Topics include: formulation of deterministic and stochastic population models; dynamics of single-species populations; and dynamics of interacting populations (predation, competition, and mutualism), structured populations, and epidemiology. Emphasis is placed on model formulation and techniques of analysis.
Content:	Why do some diseases become pandemic? Why does the introduction of certain species result in widespread invasions? Why do some populations grow while others decline and still others cycle rhythmically? How are all of these phenomena influenced by climate change? These and many other fundamental questions in the science of ecology are intrinsically quantitative and concern highly complex systems. To answer them, ecologists formulate and study mathematical models. This course is intended to provide an overview of the principal ecological models and the mathematical techniques available for their analysis. Emphasis is placed on model formulation and techniques of analysis. Although the focus is on ecological dynamics, the methods we discuss are readily applicable across the sciences. The course presumes mathematical maturity at the level of advanced calculus with prior exposure to ordinary differential equations, linear algebra, and probability.

Math 471 - Introduction to Numerical Methods

Prerequisites:	Math 216, 286, or 316; Math 214, 217, 417, or 419; and a working knowledge of one high-level computer language
Credit:	3 Credits. No credit granted to those who have completed or are enrolled in Math 371 or 472.
Background and Goals:	This is a survey of the basic numerical methods which are used to solve scientific problems. The emphasis is evenly divided between the analysis of the methods and their practical applications. Some convergence theorems and error bounds are proved. The course also provides an introduction to MATLAB, an interactive program for numerical linear algebra, as well as practice in computer programming. One goal of the course is to show how calculus and linear algebra are used in numerical analysis.
Content:	Topics may include computer arithmetic, Newton’s method for non-linear equations, polynomial interpolation, numerical integration, systems of linear equations, initial value problems for ordinary differential equations, quadrature, partial pivoting, spline approximations, partial differential equations, Monte Carlo methods, 2-point boundary value problems, Dirichlet problem for the Laplace equation.

Math 472 - Numerical Methods with Financial Applications

Prerequisites:	Math 216, 256, 286, or 316; Math 214, 217, 417, or 419; and a working knowledge of one high-level computer language. Math 425 or 525 is recommended.
Credit:	3 Credits. No credit granted to those who have completed or are enrolled in Math 371 or 471.
Background and Goals:	This is a survey of the basic numerical methods which are used to solve scientific problems. The goals of the course are similar to those of Math 471, but the applications are chosen to be of interest to students in the Actuarial Mathematics and Financial Mathematics programs.
Content:	Topics may include: Newton’s method for non-linear equations, systems of linear equations, numerical integration, interpolation and polynomial approximation, ordinary differential equations, partial differential equations - in particular the Black-Scholes equation, Monte Carlo simulation, and numerical modeling.

Math 474 - Introduction to Stochastic Analysis for Finance

Prerequisites:	Math 525 and 423
Credit:	3 credits
Background and Goals:	This is an undergraduate level course in Stochastic Analysis and applications to Quantitative Finance. The aim of this course is to teach the probabilistic techniques and concepts from the theory of continuous-time stochastic processes and their applications to modern methematical finance. It is a continuation of Math 423.
Content:	The course starts with the basic theory of diffusion processes. Specifically, it covers the topics: stochastic integrals, continuous-time martingales, stochastic calculus, and stochastic differential equations. It introduces the students to Ito's formula and geometric Brownian motion, which are fundamental concepts in the theory of mathematical finance. Afterwards, the course focuses on mathematical finance models in continuous-time. First, basic definitions and models are being introduced: portfolio dynamics, arbitrage theory (including the celebrated Black-Scholes' equation and formula), and hedging. Then, the course covers more advanced models, using the martingale approach to arbitrage theory. This includes martingale pricing, stochastic discounting, Girsanov's theorem, revisiting the Black-Scholes model. Finally, the course introduces multidimensional models and the concepts of complete and incomplete markets.

Math 475 - Elementary Number Theory

Prerequisites:	None
Credit:	3 credits.
Background and Goals:	This is an elementary introduction to number theory, especially congruence arithmetic. Number Theory is one of the few areas of mathematics in which problems easily describable to a layman (is every even number the sum of two primes?) have remained unsolved for centuries. Recently some of these fascinating but seemingly useless questions have come to be of central importance in the design of codes and ciphers. In addition to strictly number- theoretic questions, concrete examples of structures such as rings and fields from abstract algebra are discussed. Concepts and proofs are emphasized, but there is some discussion of algorithms which permit efficient calculation. Students are expected to do simple proofs and may be asked to perform computer experiments. Although there are no special prerequisites and the course is essentially self-contained, most students have some experience in abstract mathematics and problem solving and are interested in learning proofs. At least three semesters of college mathematics are recommended. A Computational Laboratory (Math 476, 1 credit) will usually be offered as an optional supplement to this course.
Content:	Topics usually include the Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Hensel’s Lemma, Diophantine equations, arithmetic in polynomial rings, primitive roots, quadratic reciprocity, and quadratic fields. This material corresponds to Chapters 1-3 and selected parts of Chapter 5 of Niven, Zuckerman, and Montgomery.

Math 476 - Computational Laboratory in Number Theory

Prerequisites:	Prior or concurrent enrollment in Math 475 or 575
Credit:	1 credit
Background and Goals:	Intended as a companion course to Math 475 (Elem. Number Theory) or 575 (Intro to Theory of Numbers) Participation should boost the student’s performance in either of those classes. Students in the Lab will see mathematics as an exploratory science (as mathematicians do).
Content:	Students will be provided with software with which to conduct numerical explorations. No programming necessary, but students interested in programming will have the opportunity to embark on their own projects. Students will gain a knowledge of algorithms which have been developed for number theoretic purposes, e.g., for factoring.

Math 481 - Introduction to Mathematical Logic

Prerequisites:	Math 412 or 451 or equivalent experience with abstract mathematics
Credit:	3 credits.
Background and Goals:	All of modern mathematics involves logical relationships among mathematical concepts. In this course we focus on these relationships themselves rather than the ideas they relate. Inevitably this leads to a study of the (formal) languages suitable for expressing mathematical ideas. The explicit goal of the course is the study of propositional and first-order logic; the implicit goal is an improved understanding of the logical structure of mathematics. Students should have some previous experience with abstract mathematics and proofs, both because the course is largely concerned with theorems and proofs and because the formal logical concepts will be much more meaningful to a student who has already encountered these concepts informally. No previous course in logic is prerequisite.
Content:	In the first third of the course the notion of a formal language is introduced and propositional connectives (‘and,’ ‘or,’ ‘not,’ ‘implies’), tautologies, and tautological consequence are studied. The heart of the course is the study of first-order predicate languages and their models. The new elements here are quantifiers (‘there exists’ and ‘for all’). The study of the notions of truth, logical consequence, and probability leads to the completeness and compactness theorems The final topics include some applications of these theorems, usually including non-standard analysis. This material corresponds to Chapter 1 and sections 2.0-2.5 and 2.8 of Enderton.

Math 486 - Concepts Basic is Secondary School Math (IBL)

Prerequisites:	Math 205 or 215 or 285; and 217; or permission of the Instructor. Prerequisites must be completed with a minimum grade of C- or better.
Credit:	3 credits.
Background and Goals:	This course is designed for students who intend to teach middle or high school mathematics. The course focuses on the reasoning and justification behind middle and high school topics, taking the content beyond just calculation and application. The syllabus consists of high school mathematics from an advanced perspective. The course is conducted in a discussion (Inquiry Based Learning) format. Class participation is essential and constitutes a significant part of the course grade.
Content:	Topics covered have included number systems and their axiomatics; number theory, particularly a study of divisibility, primes, and prime factorizations; the abstract theory of sets, operators, and functions; and the epsilon-delta underpinnings of limits and derivatives.

Math 487 - Number Theory and Algebra for Secondary Teachers (IBL)

Prerequisites:	Math 205 or 215 or 285; and 217 or 486
Credit:	3 credits.
Background and Goals:	This course is a companion course of Math 486. It has two mutually supportive aims: To cultivate what can be called "connected mathematical thinking", largely through ambitious problem solving activities, and to provide a rigorous and coherent treatment of some of the foundational domains of the school mathematics curriculum (see the content below). The ethos of the course is the making of mathematical connections between topics or concepts that are often not made explicit, by working on problems whose solution draws upon resources from different domains of mathematics, and by identifying and making use of common mathematical structure underlying different mathematical situations.
Content:	Place value (in-depth); modular arithmetic, basic n-itions of commutative rings; discrete additive subgroups of real numbers; commensurability, Euclidian algorithm, gcd and Icm; primes and prime factorization; elementary combinatorics; polynomials; Lagrange interpolation, binomial theorem, inclusion-exclusion formula; discrete calculus.

Math 489 - Math for Elementary and Middle School Teachers (IBL)

Prerequisites:	Math 385 or permission of instructor
Credit:	3 credits.
Background and Goals:	This course, together with its predecessor, Math 385, provides a coherent overview of the mathematics underlying the elementary and middle school curriculum. It is required of all students intending to earn an elementary teaching certificate and is taken almost exclusively by such students. Concepts are heavily emphasized with some attention given to calculation and proof. The course is conducted using a discussion format. Class participation is expected and constitutes a significant part of the course grade. Enrollment is limited to 30 students per section. Although only two years of high school mathematics is required, a more complete background including pre-calculus or calculus is desirable.
Content:	Topics covered include fractions and rational numbers, decimals and real numbers, probability and statistics, geometric figures, and measurement. Algebraic techniques and problem-solving strategies are used throughout the course.

Math 490 - Introduction to Topology (IBL)

Prerequisites:	Math 351, 451; or previous exposure to real analysis; or permission of instructor
Credit:	3 credits
Background and Goals:	Topology is a fundamental area of mathematics that provides a foundation for analysis and geometry. Once a set has a topology (so it becomes a "topological space"), we can start to build on it. For example, the notion of a continuous function makes sense on a topological space, and in fact, this is the most general setting where the idea of a continuous function makes sense. The goal of this course is to introduce you to the world of topology, with emphasis placed on careful reasoning, and understanding and constructing proofs. Math 490 is an Inquiry Based Learning course. This means that the students work in groups, guided by worksheets, to explore and develop new material with minimal guidance from the instructor. Depending on the instructor, students may also be asked to regularly present material in class.
Content:	We will generalize important concepts like continuity and compactness from the setting of real analysis into the more general setting of topological spaces. We draw motivation from two major theorems from real analysis: 1) the Extreme Value Theorem, and 2) the Intermediate Value Theorem. Another thing we will do in this course is rigorously explore what it means for two topological spaces to be "the same". To this end, we will develop tools that help to distinguish topological spaces from each other. Topics include, but are not limited to: metric spaces, compactness, connectedness, productspaces, quotient spaces.

Math 493 - Honors Algebra I

Prerequisites:	Math 296 (enforced) or permission of instructor
Credit:	3 credits.
Background and Goals:	Math 493-494 is one of the more abstract and difficult sequences in the undergraduate program. Its goal is to introduce students to the basic structures of modern abstract algebra (groups, rings, fields, and modules) in a rigorous way. Emphasis is on concepts and proofs; calculations are used to illustrate the general theory. Exercises tend to be quite challenging. Students must have some previous exposure to rigorous proof-oriented mathematics and be prepared to work hard.
Content:	The course covers basic definitions and properties of groups, fields, and vector spaces including homomorphisms, isomorphisms, subgroups, and bilinear forms. Further topics are selected from: Sylow theorems; structure theorem for finitely-generated abelian groups; permutation representation; the symmetric and alternating groups; vector spaces over arbitrary fields; spectral theorem; and linear groups.

Math 494 - Honors Algebra II

Prerequisites:	Math 493 (enforced)
Credit:	3 credits.
Background and Goals:	Math 493-494 is one of the more abstract and difficult sequences in the undergraduate program. Its goal is to introduce students to the basic structures of modern abstract algebra (groups, rings, fields, and modules) in a rigorous way. Emphasis is on concepts and proofs; calculations are used to illustrate the general theory. Exercises tend to be quite challenging. Students must have some previous exposure to rigorous proof-oriented mathematics and be prepared to work hard.
Content:	This course is a continuation of Math 493 (Honors Algebra I). It covers basic definitions and properties of rings and modules including quotients, ideals, factorization, and field extensions. Further topics are selected from: representation theory; structure theory of modules over a PID; Jordan canonical form; Galois theory, Nullstellensatz; finite fields; Euclidean, Principal ideals, and unique factorization domains; polynomial rings in one and several variables; and algebraic varieties.

Math 497 - Topics in Elementary Mathematics (IBL)

Prerequisites:	Math 489 or permission of instructor
Credit:	3 credits.
Background and Goals:	This is a required course for elementary teaching certificate candidates that extends and deepens the coverage of mathematics begun in the required two-course pair Math 385&489. Topics are chosen from geometry, algebra, computer programming, logic, and combinatorics. Applications and problem-solving are emphasized. The class meets three times per week in recitation sections. Grades are based on class participation, two one-hour exams, and a final exam.
Content:	Selected topics in geometry, algebra, computer programming, logic, and combinatorics for prospective and in-service elementary, middle, or junior- high school teachers. Content will vary from term to term.

Math 498 - Topics in Modern Mathematics

Prerequisites:	Junior or Senior standing
Credit:	3 credits.
Background and Goals:	As a topics course, this course will vary greatly from term to term. In one re- cent offering, the aim of the course was to introduce, at an elementary level, the basic concepts of the theory of dynamical systems.
Content:	Varies.