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# 300-Level Math Courses

### Math 310 - Chance and Choice

Prerequisites: Sophomore standing and one previous university math course. 3 Credits. The course contents are accessible with mostly precalculus preparation. However, we recommend some calculus background. With its few prerequisites and broad interest, it is also an ideal course for students wanting to explore mathematical thinking at a higher level. The course is a hands-on introduction to various topics in probability. These include basic profitability and combinatorics, conditional probability, expectations, random walks, Poisson distributions and Markov chains. The fundamental, ancient, and deep concepts underlying these  are randomness, fairness, coincidence, and bias. These are all important both in theoretical probability, statistics and real world applications, and the course pursues these ideas from conceptual and applied points of view.

### Math 312 - Applied Modern Algebra

Prerequisites: Math 217 3 Credits. No credit for those who have completed or are enrolled in Math 412. One of the main goals of the course (along with every course in the algebra sequence) is to expose students to rigorous, proof-oriented mathematics. Students are required to have taken Math 217, which should provide a first exposure to this style of mathematics. A distinguishing feature of this course is that the abstract concepts are not studied in isolation. Instead, each topic is studied with the ultimate goal being a real-world application. Sets and functions, relations and graphs, rings, Boolean algebras, semi- groups, groups, and lattices. Applications from areas such as switching, automata, and coding theory, and may include finite and minimal state machines, algebraic decompositions of logic circuits, semigroup machines, binary codes, and series and parallel decomposition of machines.

### Math 316 - Differential Equations

Prerequisites: Math 215 or 285; and 217 3 Credits. Credit is granted for only one course among Math 216, 286, and 316. This is an introduction to differential equations for students who have studied linear algebra (Math 217). It treats techniques of solution (exact and approximate), existence and uniqueness theorems, some qualitative theory, and many applications.  Proofs are given in class; homework problems include both computational and more conceptually oriented problems. First-order equations: solutions, existence and uniqueness, and numerical techniques; linear systems: eigenvector-eigenvalue solutions of constant coefficient systems, fundamental matrix solutions, nonhomogeneous systems; higher-order equations, reduction of order, variation of parameters, series solutions; qualitative behavior of systems, equilibrium points, stability. Applications to physical problems are considered throughout.

### Math 351 - Principles of Analysis

Prerequisites: Math 215 and 217 or permission of instructor 3 Credits.  No credit granted to those who have completed or are enrolled in Math 451. This course has two goals: 1) a rigorous development of the ideas underlying calculus and 2) a future development of the student's ability to handle mathematical abstraction and proofs. The course content is similar to that of Math 451, but Math 351 assumes less background. Math 351 may be used in stead of 451 for the Math of Finance and Risk Management major. Logic and techniques of proof, sequences, continuous functions, uniform continuity, differentiation, integration, and the Fundamental Theorem of Calculus.

### Math 354 - Fourier Analysis and its Applications

Prerequisites: Math 216, 286, or 316 3 credits. No credit granted to those who have completed or are enrolled in Math 450 or 454. This course is an introduction to Fourier analysis with emphasis on applications. The course also can be viewed as a way of deepening one’s understanding of the 100-and 200-level material by applying it in interesting ways. This is an introduction to Fourier Analysis geared towards advanced undergraduate students from both pure and applied areas. It should be particularly suitable for majors in the sciences and engineering. Topics will include properties of complex numbers, the Discrete Fourier Transform, Fourier series, the Dirichlet and Fejer kernals, convolutions, approximations by trigonometric polynomials, uniqueness of Fourier coefficients, Parseval's identity, properties of trigonometric polynomials, absolutely convergent Fourier series, convergence of Fourier series, applications of Fourier series, and the Fourier transform, including the Poisson summation formula and Plancherel's identity. While the main effort will be to establish the foundations of the subject, applications will include the Fast Fourier Transform, the heat equation, the wave equation, sampling, and signal processing.

### Math 371 (Engin 371) - Numerical Methods

Prerequisites: Engin 101; and one of Math 216, 286, or 316; and one of Math 214, 217, 417, or 419. 3 Credits. No credit after Math 471 or Math 472 This is a survey course of the basic numerical methods which are used to solve practical scientific problems. Important concepts such as accuracy, stability, and efficiency are discussed. The course provides an introduction to MATLAB, an interactive program for numerical linear algebra, and may provide practice in FORTRAN programming and the use of software library subroutines.  Convergence theorems are discussed and applied, but the proofs are not emphasized. Floating point arithmetic, Gaussian elimination, polynomial interpolation, spline approximations, numerical integration and differentiation, solutions to non-linear equations, ordinary differential equations, polynomial approximations.  Other topics may include discrete Fourier transforms, two-point boundary-value problems, and Monte-Carlo methods.

### Math 385 - Math for Elementary School Teachers

Prerequisites: One year each of HS algebra and geometry 3 Credits. No credit granted to those who have takend or are enrolled in Math 485. This course, together with its sequel Math 489, 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 are required, a more complete background including pre-calculus or calculus is desirable. Topics covered include problem solving, sets and functions, numeration systems, whole numbers (including some number theory), and integers.  Each number system is examined in terms of its algorithms, its applications, and its mathematical structure.

### Math 389 - Explorations in Mathematics

Prerequisites: Reasonable familiarity with proofs at the level of Math 217 or 295 expected 3 Credits. The course is designed to show you how new mathematics is actually created: how to take a problem, make models and experiment with them, and search for underlying structure. The format involves little formal lecturing, much laboratory work, and student presentations discussing partial results and approaches. Course website: http://www.math.lsa.umich.edu/courses/389/ Problems for projects are drawn from a wide variety of mathematical areas, pure and applied. Problems are chosen to be accessible to undergraduates.

### Math 395 - Honors Analysis I

Prerequisites: Math 296 4 Credits This course is a continuation of the sequence Math 295-296 and has the same theoretical emphasis.  Students are expected to understand and construct proofs. Inverse/implicit function theorems, immersion/submersion theorems.  Quotient and dual spaces, inner product spaces, spectral theory.  Metric spaces, basic point-set topology. Integration in Euclidean space, Fubini's theorem, change of variables formula. Topics in linear algebra: tensor products, exterior and symmetric powers, Jordan and rational canonical forms.

### Math 396 - Honors Analysis II

Prerequisites: Math 395 4 Credits. This course is a continuation of Math 395 and has the same theoretical emphasis. Students are expected to understand and construct proofs. Submanifolds (with or without corners) of Euclidean space, abstract manifolds, tangent and cotangent spaces, immersion/submersion theorems. Partitions of unity, vector fields and differential forms on manifolds, exterior differentiation, integration of differential forms.  Stokes' theorem.  deRham cohomology, Riemannian metrics, Hodge star operator and the standard vector calculus versions of Stokes' theorem.