Courses in LSA Mathematics
Courses are categorized, very roughly, into four tiers:
 introductory courses aimed at advanced undergraduates, M.S. students, and Ph.D. students from other programs;
 courses presenting foundations of the respective fields of mathematics at the beginning graduate level; among them, the core courses forming the basis of the Qualifying Review (QR) exams are shown in boldface;
 advanced courses which require considerable familiarity with the foundations of the corresponding subject; and
 more specialized courses dedicated to topics of current research interest.
The first priority for the incoming firstyear student is to pass the QR, which requires passing approved combinations of written exams and core courses. Consult the table of courses listed here: http://dept.math.lsa.umich.edu/graduate/register.html

Mathematics (MATH)

MATH 404. Intermediate Differential Equations and Dynamics
MATH 216, 256, 286 or 316.
(3).
(BS).
May not be repeated for credit.
Linear systems, qualitative theory of ordinary differential equations for planar and higherdimensional systems, chaos, stability, nonlinear oscillations, periodic orbits, Floquet theory, applications.

MATH 412. Introduction to Modern Algebra
MATH 215, 255 or 285; and 217; only 1 credit after MATH 312.
(3).
(BS).
May not be repeated for credit.
No credit granted to those who have completed or are enrolled in MATH 493. One credit granted to those who have completed MATH 312.
The initial topics include ones common to every branch of mathematics: sets, functions (mappings), relations, and the common number systems (integers, rational, real and complex numbers). These are then applied to the study of particular types of mathematical structures: groups, rings, and fields).

MATH 416. Theory of Algorithms
[MATH 312, 412 or EECS 280] and MATH 465.
(3).
(BS).
May not be repeated for credit.
Many common problems from mathematics and computer science may be solved by applying one or more algorithms, welldefined procedures that accept input data specifying a particular instance of the problem and produce a solution. Students in this course 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. Each term offers varying degrees of emphasis on mathematical proofs and computer implementation of these ideas.

MATH 417. Matrix Algebra I
Three courses beyond MATH 110.
(3).
(BS).
May not be repeated for credit.
No credit granted to those who have completed or are enrolled in MATH 214, 217, 419, or 420.
MATH 417 and 419 not be used as electives in the Statistics concentration.
Matrix operations, vector spaces, Gaussian and GaussJordan algorithms for linear equations, subspaces of vector spaces, linear transformations, determinants, orthogonality, characteristic polynomials, eigenvalue problems and similarity theory. Applications include linear networks, least squares method, discrete Markov processes, linear programming.

MATH 419. Linear Spaces and Matrix Theory
Four courses beyond MATH 110.
(3).
(BS).
May not be repeated for credit.
2 credits granted to those who have completed MATH 214, 217, or 417. No credit for those who have completed or are enrolled in 420.
MATH 417 and 419 not be used as electives in the Statistics concentration. F, W, Su.
Finite dimensional linear spaces and matrix representations of linear transformations. Bases, subspaces, determinants, eigenvectors and canonical forms. Structure of solutions of systems of linear equations. Applications to differential and difference equations. Provides more depth and content than MATH 417. MATH 420 is the proper election for students contemplating research in mathematics.

MATH 420. Advanced Linear Algebra
Linear algebra course (MATH 214, 217, 417, or 419) and one of MATH 296, 412, or 451.
(3).
(BS).
May not be repeated for credit.
This is an introduction to the formal theory of abstract vector spaces and linear transformations. The emphasis is on concepts and proofs with some calculations to illustrate the theory. Students should have some mathematical maturity and in particular should expect to work with and be tested on formal proofs.

MATH 423. Mathematics of Finance
MATH 217 and 425; EECS 183 or equivalent.
(3).
(BS).
May not be repeated for credit.
An introduction to mathematical models used in finance and economics with particular emphasis on models for pricing derivative instruments such as options and futures. Topics include risk and return theory, portfolio theory, capital asset pricing model, random walk model, stochastic processes, BlackScholes Analysis, numerical methods and interest rate models.

MATH 424. Compound Interest and Life Insurance
MATH 215, 255, or 285 or permission of instructor.
(3).
(BS).
May not be repeated for credit.

MATH 425 / STATS 425. Introduction to Probability
MATH 215.
(3).
(BS).
May not be repeated for credit.

MATH 431. Topics in Geometry for Teachers
One of MATH 215, 255, or 285 (completed with a minimum grade of C or better), and MATH 217 (completed with a minimum grade of C or better).
(Prerequisites enforced at registration.)
(3).
(BS).
May not be repeated for credit.
This course is an axiomatic treatment of Euclidean plane geometry and serves as an introduction to the process of doing mathematics rigorously. Intended for prospective geometry teachers, it will place the development of geometry within its historical context with an additional emphasis on expositing proofs.

MATH 433. Introduction to Differential Geometry
MATH 215 (or 255 or 285), and 217.
(3).
(BS).
May not be repeated for credit.

MATH 450. Advanced Mathematics for Engineers I
Permission required after credit earned in MATH 354 or 454.
Consent of department required.
(Prerequisites enforced at registration.)
MATH 215, 255, or 285.
(4).
(BS).
May not be repeated for credit.
No credit granted to those who have completed or are enrolled in MATH 354 or 454.
F, W, Su.
Review of curves and surfaces in implicit, parametric and explicit forms; differentiability and affine approx., implicit and inverse function theorems; chain rule for 3space; multiple integrals, scalar and vector fields; line and surface integrals; computations of planetary motion; work, circulation and flux over surfaces; Gauss' and Stokes' Theorems; heat equation.

MATH 451. Advanced Calculus I
Previous exposure to abstract mathematics, e.g. MATH 217 and 412.
(3).
(BS).
May not be repeated for credit.
No credit granted to those who have completed or are enrolled in MATH 351.

MATH 452. Advanced Calculus II
MATH 217, 419, or 420; and MATH 451.
(3).
(BS).
May not be repeated for credit.
W.
Topics include: (1) partial derivatives and differentiability; (2) gradients, directional derivatives, and the chain rule; (3) implicit function theorem; (4) surfaces, tangent planes; (5) maxmin 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
Permission required after credit earned in MATH 354 or 450.
(Prerequisites enforced at registration.)
MATH 216, 256, 286 or 316.
(3).
(BS).
May not be repeated for credit.
Students with credit for MATH 354 can elect MATH 454 for one credit. No credit granted to those who have completed or are enrolled in MATH 450.
This course is devoted to the use of Fours Series and Transforms in the solution of boundaryvalue problems for 2nd order linear partial differential equations. We study the heat and wave equations in one and higher dimension. We introduce the spherical and cylindrical Bessel functions, Legendre polynomials and analysis of data smoothing and filtering.

MATH 462. Mathematical Models
MATH 216, 256, 286, or 316; and MATH 214, 217, 417, or 419. Students with credit for MATH 463 must have department permission to elect MATH 462.
(3).
(BS).
May not be repeated for credit.
Students with credit for MATH 362 must have department permission to elect MATH 462.
Construction and analysis of mathematical models in physics, engineering, economics, biology, medicine, and social sciences. Content varies considerably 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. Mathematical Modeling in Biology
MATH 214, 217, 417, or 419; and MATH 216, 256, 286, or 316.
(3).
(BS).
May not be repeated for credit.
An introduction to the use of continuous and discrete differential equations in the biological sciences. Modeling in biology, physiology and medicine.

MATH 465. Introduction to Combinatorics
Linear Algebra (one of MATH 214, 217, 256, 286, 296, 417, or 419) or permission of instructor.
(3).
(BS).
May not be repeated for credit.
No credit granted to those who have completed or are enrolled in MATH 565 or 566.
Rackham credit requires additional work.
An introduction to combinatorics, covering basic counting techniques (inclusionexclusion, 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
MATH 217, 417, or 419; MATH 256, 286, or 316; and MATH 450 or 451.
(3).
May not be repeated for credit.
Rackham credit requires additional work.
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 singlespecies populations; and dynamics of interacting populations (perdition, competition, and mutualism), structured populations, and epidemiology. Emphasis is placed on model formulation and techniques of analysis.

MATH 471. Introduction to Numerical Methods
MATH 216, 256, 286, or 316; and 214, 217, 417, or 419; and a working knowledge of one highlevel computer language. No credit granted to those who have completed or are enrolled in MATH 371 or 472.
(3).
(BS).
May not be repeated for credit.
No credit granted to those who have completed or are enrolled in MATH 371 or 472.

MATH 472. Numerical Methods with Financial Applications
Differential Equations (MATH 216, 256, 286, or 316); Linear Algebra (MATH 214, 217, 417, or 419); working knowledge of a highlevel computer language. Recommended: MATH 425.
(3).
(BS).
May not be repeated for credit.
No credit granted to those who have completed or are enrolled in MATH 471 or 371.
Theoretical study and practical implementation of numerical methods for scientific problems, with emphasis on financial applications. Topics: Newton's method for nonlinear equations; Systems of linear equations; Numerical integration; Interpolation and polynomial approximation; Ordinary differential equations; Partial differential equations, in particular the BlackScholes equation; Monte Carlo simulation; Numerical modeling.

MATH 475. Elementary Number Theory
At least three terms of college Mathematics are recommended.
(3).
(BS).
May not be repeated for credit.
W.

MATH 476. Computational Laboratory in Number Theory
Prior or concurrent enrollment in MATH 475 or 575.
(1).
(BS).
May not be repeated for credit.

MATH 481. Introduction to Mathematical Logic
MATH 412 or 451 or equivalent experience with abstract mathematics.
(3).
(BS).
May not be repeated for credit.
F.

MATH 485 / EDUC 485. Mathematics for Elementary School Teachers and Supervisors
One year of high school algebra or permission of the instructor.
(3).
May not be repeated for credit.
No credit granted to those who have completed or are enrolled in MATH 385.
May not be included in a concentration plan in Mathematics. F, Su.
The history, development and logical foundations of the real number system and of numeration systems including scales of notation, cardinal numbers and the cardinal concept, the logical structure of arithmetic (field axioms) and their relations to the algorithms of elementary school instruction.

MATH 486. Concepts Basic to Secondary Mathematics
One of MATH 215, 255, or 285 (completed with a minimum grade of C or better), and MATH 217 (completed with a minimum grade of C or better) OR permission of instructor.
(Prerequisites enforced at registration.)
(3).
(BS).
May not be repeated for credit.
W.
This course examines the principles of analysis and algebra underlying theorems, especially the rationals, reals, and complex numbers. It also considers concerning functions, especially polynomials, exponential functions, and logarithmic functions. The mathematical underpinnings of these ideas serve as an important intellectual resource for students pursuing teacher certification.

MATH 487. Number Theory and Algebra for Secondary Teachers
Math 486 or the prerequisites to 486 (One of MATH 215, 255, or 285 completed with a minimum grade of C or better, AND MATH 217 completed with a minimum grade of C or better OR permission of instructor).
(Prerequisites enforced at registration.)
(3).
May not be repeated for credit.
This course will offer a coherent treatment of topics such as place value, elementary number theory, the geometry of the number line, elementary combinatorics, polynomial algebra, and discrete calculus. The instruction will emphasize rich problem solving, often in small groups, and the discernment of mathematical connections.

MATH 489. Mathematics for Elementary and Middle School Teachers
MATH 385.
(Prerequisites enforced at registration.)
(3).
May not be repeated for credit.
May not be used in any Graduate program in Mathematics.
The course provides an overview of the mathematics underlying the elementary and middle school curriculum. It is required of all students intending to earn an elementary teaching certificate. Concepts are heavily emphasized with some attention given to calculation and proof.

MATH 490. Introduction to Topology
MATH 451 or equivalent experience with abstract mathematics.
(3).
(BS).
May not be repeated for credit.
Knots, orientable and nonorientable surfaces, Euler characteristic, open sets, connectedness, compactness, metric spaces. The topics covered are fairly constant but the presentation and emphasis will vary significantly with the instructor.

MATH 493. Honors Algebra I
MATH 296.
(3).
(BS).
May not be repeated for credit.
Description and indepth study of the basic algebraic structures: groups, rings, fields including: set theory, relations, quotient groups, permutation groups, Sylow's Theorem, quotient rings, field of fractions, extension fields, roots of polynomials, straightedge and compass solutions, and other topics.

MATH 494. Honors Algebra II
MATH 493.
(Prerequisites enforced at registration.)
(3).
(BS).
May not be repeated for credit.
Vector spaces, linear transformations and matrices, equivalence or matrices and forms, canonical forms, and applications to linear differential equations.

MATH 497. Topics in Elementary Mathematics
MATH 489 or permission of instructor.
(3).
(BS).
May be repeated for a maximum of 6 credits.
F.
Selected topics in geometry, algebra, computer programming, logic and combinatorics for prospective and inservice elementary, middle, or juniorhigh school teachers. Content will vary from term to term.

MATH 498. Topics in Modern Mathematics
Senior Mathematics concentrators and Master Degree students in Mathematical disciplines.
(3).
(BS).
May not be repeated for credit.

MATH 499. Independent Reading
Consent of instructor required.
Graduate standing in a field other than Mathematics and permission of instructor.
(1  4).
(INDEPENDENT).
May not be repeated for credit.

MATH 501. Applied & Interdisciplinary Mathematics Student Seminar
At least two 300 or above level math courses, and Graduate standing; Qualified undergraduates with permission of instructor only.
(1).
May be repeated for a maximum of 6 credits.
Offered mandatory credit/no credit.
MATH 501 is an introductory and overview seminar course in the methods and applications of modern mathematics. The seminar has two key components: (1) participation in the Applied and Interdisciplinary Math Research Seminar; and (2) preparatory and postseminar discussions based on these presentations. Topics vary by term.

MATH 506 / IOE 506. Stochastic Analysis for Finance
Graduate students or permission of instructor.
(3).
(BS).
May not be repeated for credit.
The aim of this course is to teach the probabilistic techniques and concepts from the theory of stochastic processes required to understand the widely used financial models. In particular concepts such as martingales, stochastic integration/calculus, which are essential in computing the prices of derivative contracts, will be discussed. Pricing in complete/incomplete markets (in discrete/ continuous time) will be the focus of this course as well as some exposition of the mathematical tools that will be used such as Brownian motion, Levy processes and Markov processes.

MATH 507. Mathematical Methods for Algorithmic Trading
(3).
May not be repeated for credit.
This is a graduate level course focused on the Mathematical methods for designing automated trading strategies. The aim is to teach the relevant techniques from Probability, Statistics, PDEs, and Optimization, as well as to introduce students to the wide range of specific problems and existing models related to algorithmic trading.

MATH 520. Life Contingencies I
MATH 424 and 425 with minimum grade of C, plus declared Actuarial/Financial Mathematics Concentration.
(Prerequisites enforced at registration.)
(3).
(BS).
May not be repeated for credit.
Quantifying the financial impact of uncertain events is the central challenge of actuarial mathematics. The goal of this course is to teach the basic actuarial theory of mathematical models for financial uncertainties, mainly the time of death. The main topics are the development of (1) probability distributions for the future lifetime random variable; (2) probabilistic methods for financial payments on death or survival; and (3) mathematical models of actuarial reserving.

MATH 521. Life Contingencies II
MATH 520 with a grade of C or higher.
(Prerequisites enforced at registration.)
(3).
(BS).
May not be repeated for credit.
This course extends the single decrement and single life ideas of MATH 520 to multidecrement and multiplelife applications directly related to life insurance. The sequence 520521 covers the Part 4A examination of the Casualty Actuarial Society and covers the syllabus of the Course 150 examination of the Society of Actuaries. Concepts and Calculation are emphasized over proof.

MATH 523. Loss Models I
MATH/STATS 425.
(Prerequisites enforced at registration.)
(3).
(BS).
May not be repeated for credit.
Risk management and modeling of financial losses. Review of random variables (emphasizing parametric distributions), review of basic distributional quantities, continuous models for insurance claim severity, discrete models for insurance claim frequency, the effect of coverage modification on severity and frequency distributions, aggregate loss models, and simulation.

MATH 524. Loss Models II
STATS 426 and MATH 523.
(Prerequisites enforced at registration.)
(3).
(BS).
May not be repeated for credit.
Risk management and modeling of financial losses. Frequentist and Bayesian estimation of probability distributions, model selection, credibility, and other topics in casualty insurance.

MATH 525 / STATS 525. Probability Theory
MATH 451 (strongly recommended). MATH 425/STATS 425 would be helpful.
(3).
(BS).
May not be repeated for credit.

MATH 526 / STATS 526. Discrete State Stochastic Processes
MATH 525 or STATS 525 or EECS 501.
(3).
(BS).
May not be repeated for credit.

MATH 531. Transformation Groups in Geometry
MATH 451 or equivalent experience with abstract mathematics.
(3).
(BS).
May not be repeated for credit.

MATH 537. Introduction to Differentiable Manifolds
MATH 420, and 590 or 591.
(3).
(BS).
May not be repeated for credit.
This course is intended for students with a strong background in topology, linear algebra, and multivariable advanced calculus equivalent to the courses 420 and 590. Its goal is to introduce the basic concepts and results of differential topology and differential geometry. Content covered includes Manifolds, vector fields and flows, differential forms, Stokes' theorem, Lie group basics, Riemannian metrics, LeviCivita connection, geodesics.

MATH 540 / BIOINF 540. Mathematics of Biological Networks
(BS).
May not be repeated for credit.
This course addresses methods and principles involved in constructing and studying the structure and function of biological networks using examples from real datasets (emphasis on genomewide chromosome conformation capture (HiC), RNAsew, and imaging). Basic mathematical principles required for understanding biological networks included. Topics include: review of basic probability theory, spectral graph theory, higherorder singular value decomposition, graphs and their Laplacians. Fiedler number and Fiedler vector, clustering network controllability.

MATH 550 / CMPLXSYS 510. Introduction to Adaptive Systems
MATH 215, 255, or 285; MATH 217; and MATH 425.
(3).
(BS).
May not be repeated for credit.

MATH 555. Introduction to Functions of a Complex Variable with Applications
MATH 451 or equivalent experience with abstract mathematics.
(3).
(BS).
May not be repeated for credit.
Intended primarily for students of engineering and of other cognate subjects. Doctoral students in mathematics elect Mathematics 596. Complex numbers, continuity, derivative, conformal representation, integration, Cauchy theorems, power series, singularities, and applications to engineering and mathematical physics.

MATH 556. Applied Functional Analysis
MATH 217, 419, or 420; MATH 451; and MATH 555.
(3).
(BS).
May not be repeated for credit.
This is an introduction to methods of applied functional analysis. Students are expected to master both the proofs and applications of major results. The prerequisites include linear algebra, undergraduate analysis, advanced calculus and complex variables. This course is a core course for the Applied and Interdisciplinary Mathematics (AIM) graduate program.

MATH 557. Applied Asymptotic Analysis
MATH 217, 419, or 420; MATH 451; and MATH 555.
(3).
(BS).
May not be repeated for credit.
Topics include: asymptotic sequences and (divergent) series; asymptotic expansions of integrals and Laplace's method; methods of steepest descents and stationary phase; asymptotic evaluation of inverse Fourier and Laplace transforms; asymptotic solutions for linear (nonconstant coefficient) differential equations; WBK expansions; singular perturbation theory; and boundary, initial, and internal layers.

MATH 558. Applied Nonlinear Dynamics
MATH 451.
(3).
(BS).
May not be repeated for credit.
This course is an introduction to dynamical systems (differential equations and iterated maps). The aim is to survey a broad range of topics in the theory of dynamical systems with emphasis on techniques and results that are useful in applications, including chaotic dynamics. This is a core course for the Applied and Interdisciplinary Mathematics (AIM) graduate program.

MATH 559. Selected Topics in Applied Mathematics
MATH 451; and 217, 419, or 420.
(3).
(BS).
May be repeated for a maximum of 6 credits.
This course will focus on particular topics in emerging areas of applied mathematics for which the application field has been strongly influenced by mathematical ideas. It is intended for students with interests in mathematical, computational, and/or modeling aspects of interdisciplinary science, and the course will develop the intuitions of the field of application as well as the mathematical concepts.

MATH 561 / IOE 510 / TO 518. Linear Programming I
MATH 217, 417, or 419.
(3).
(BS).
May not be repeated for credit.
F, W, Sp.
Formulation of problems from the private and public sectors using the mathematical model of linear programming. Development of the simplex algorithm; duality theory and economic interpretations. Postoptimality (sensitivity) analysis application and interpretations. Introduction to transportation and assignment problems; special purpose algorithms and advanced computational techniques. Students have opportunities to formulate and solve models developed from more complex case studies and to use various computer programs.

MATH 562 / IOE 511. Continuous Optimization Methods
MATH 217, 417, or 419.
(3).
(BS).
May not be repeated for credit.
Survey of continuous optimization problems. Unconstrained optimization problems: unidirectional search techniques; gradient, conjugate direction, quasiNewton methods. Introduction to constrained optimization using techniques of unconstrained optimization through penalty transformations, augmented Langrangians, and others. Discussion of computer programs for various algorithms.

MATH 563 / BIOINF 563. Advanced Mathematical Methods for the Biological Sciences
Graduate standing.
(3).
(BS).
May not be repeated for credit.
This course focuses on discovering the way in which spatial variation influences the motion, dispersion, and persistence of species. Specific topics may include i) Models of Cell Motion: Diffusion, Convection, and Chemotaxis; ii) Transport Processes in Biology; iii) Biological Pattern Formation; and iv) Delaydifferential Equations and Agestructured Models of Infectious Diseases.

MATH 564. Topics in Mathematical Biology
(MATH 217 or 417 or 419) and (450 or 454) and (216 or 316).
(3).
May be repeated for a maximum of 9 credits.
This is an advanced course on topics in mathematical biology. Topic will vary according to the instructor. Possible topics include modeling infectious diseases, cancer modeling, mathematical neurosciences, and biological oscillators. A sample description is available for a course in biological oscillators.

MATH 565. Combinatorics and Graph Theory
MATH 465.
(3).
(BS).
May not be repeated for credit.
Topics in the graph theory part of the course include (if time permits) trees, kconnectivity, Eulerian and Hamiltonian graphs, tournaments, graph coloring, planar graphs, Euler's formula, the 5Color theorem, Kuratowski's theorem, and the matrixtree theorem. The second part of the course will deal with topics in the theory of finite partially ordered sets. This will include material about Mobius functions, lattices, simplicial complexes, and matroids.

MATH 566. Combinatorial Theory
MATH 465.
(3).
(BS).
May not be repeated for credit.
Permutations, combinations, generating functions, and recurrence relations. The existence and enumeration of finite discrete configurations. Systems of representatives, Ramsey's Theorem, and extremal problems. Construction of combinatorial designs.

MATH 567. Introduction to Coding Theory
One of MATH 217, 419, 420.
(3).
(BS).
May not be repeated for credit.
Introduction to coding theory focusing on the mathematical background for errorcorrecting codes. Topic include: Shannon's Theorem and channel capacity; review of tools from linear algebra and an introduction to abstract algebra and finite fields; basic examples of codes such and Hamming, BCH, cyclic, Melas, ReedMuller, and ReedSolomon; introduction to decoding starting with syndrome decoding and covering weight enumerator polynomials and the MacWilliams Sloane identity

MATH 568 / BIOINF 568. Mathematical and Computational Neuroscience
MATH 463 or 462 (for undergraduate students) or Graduate standing.
(Prerequisites enforced at registration.)
(3).
(BS).
May not be repeated for credit.
Computational neuroscience provides a set of quantitative approaches to investigate the biophysical mechanisms and computational principles underlying the function of the nervous system. This course introduces students to mathematical modeling and quantitative techniques used to investigate neural systems at many different scales, from single neuron activity to the dynamics of large neuronal networks.

MATH 571. Numerical Linear Algebra
MATH 214, 217, 417, 419, or 420; and one of MATH 450, 451, or 454.
(3).
(BS).
May not be repeated for credit.
Direct and iterative methods for solving systems of linear equations (Gaussian elimination, Cholesky decomposition, Jacobi and GaussSeidel iteration, SOR, introduction to multigrid methods, steepest descent, conjugate gradients), introduction to discretization methods for elliptic partial differential equations, methods for computing eigenvalues and eigenvectors.

MATH 572. Numerical Methods for Differential Equations
MATH 214, 217, 417, 419, or 420; and one of MATH 450, 451, or 454.
(3).
(BS).
May not be repeated for credit.
Finite difference methods for ordinary differential equations and hyperbolic partial differential equations. Topics include onestep and multistep methods, Astability, root condition, LaxRichtmyer Equivalence Theorem, CFLcondition, von Neumann stability condition, discrete energy methods, Kreiss matrix theorem.

MATH 573. Financial Mathematics I
(3).
(BS).
May not be repeated for credit.
This is a core course for the quantitative finance and risk management masters program and introduces students to the main concepts of Financial Mathematics. This course emphasizes the application of mathematical methods to the relevant problems of financial industry and focuses mainly on developing skills of model building.

MATH 574. Financial Mathematics II
MATH 526 and MATH 573.
(Prerequisites enforced at registration.)
Although MATH 506 is not a prerequisite for MATH 574, it is strongly recommended that either these courses are taken in parallel, or MATH 506 precedes MATH 574.
(3).
(BS).
May not be repeated for credit.
This is a core course for the quantitative finance and risk management masters program and is a sequel to Math 573. This course emphasizes the application of mathematical methods to the relevant problems of financial industry and focuses mainly on developing skills of model building.

MATH 575. Introduction to Theory of Numbers I
MATH 451 and 420 or permission of instructor.
(1  3).
(BS).
May not be repeated for credit.
Students with credit for MATH 475 can elect MATH 575 for 1 credit.
Topics covered include divisibility and prime numbers, congruences, quadratic reciprocity, quadratic forms, arithmetic functions, and Diophantine equations. Other topics may be covered as time permits or by request.

MATH 582. Introduction to Set Theory
MATH 412 or 451 or equivalent experience with abstract mathematics.
(3).
(BS).
May not be repeated for credit.
The main topics covered are set algebra (union, intersection), relations and functions, orderings (partial, linear, well), the natural numbers, finite and denumerable sets, the Axiom of Choice, and ordinal and cardinal numbers.

MATH 583. Probabilistic and Interactive Proofs
MATH 412, 451 or permission of instructor.
May not be repeated for credit.
Probabilisticallycheckable proofs, zeroknowledge proofs, and other interactive proofs are studied and their computational and other advantages discussed. Appropriate background in mathematics and computer science is presented.

MATH 590. Introduction to Topology
MATH 451.
(3).
(BS).
May not be repeated for credit.
Rackham credit requires additional work.
Topics include metric spaces, topological spaces, continuous functions and homeomorphisms, separation axioms, quotient and product topology, compactness, and connectedness. We will also cover a bit of algebraic topology (e.g., fundamental groups) as time permits.

MATH 591. Differentiable Manifolds
MATH 451, 452 and 590.
(3).
(BS).
May not be repeated for credit.
This is one of the basic courses for students beginning the PhD program in mathematics. The approach is rigorous and emphasizes abstract concepts and proofs. The first 23 weeks of the course will be devoted to general topology, and the remainder of the course will be devoted to differential topology.

MATH 592. Introduction to Algebraic Topology
MATH 591.
(3).
(BS).
May not be repeated for credit.
Fundamental group, covering spaces, simplicial complexes, graphs and trees, applications to group theory, singular and simplicial homology, EilenbergSteenrod axioms, Brouwer's and Lefschetz' fixedpoint theorems, and other topics.

MATH 593. Algebra I
MATH 412, 420, and 451 or MATH 494.
(3).
(BS).
May not be repeated for credit.
Topics include basics about rings and modules, including Euclidean rings, PIDs, UFDs. The structure theory of modules over a PID will be an important topic, with applications to the classification of finite abelian groups and to Jordan and rational canonical forms of matrices. The course will also cover tensor, symmetric, and exterior algebras, and the classification of bilinear forms with some emphasis on the field case.

MATH 594. Algebra II
MATH 593.
(3).
(BS).
May not be repeated for credit.
Topics include group theory, permutation representations, simplicity of alternating groups for n>4, Sylow theorems, series in groups, solvable and nilpotent groups, JordanHolder Theorem for groups with operators, free groups and presentations, fields and field extensions, norm and trace, algebraic closure, Galois theory, and transcendence degree.

MATH 596. Analysis I
MATH 451.
(3).
(BS).
May not be repeated for credit.
Students with credit for MATH 555 may elect MATH 596 for two credits only.

MATH 597. Analysis II
MATH 451 and 420.
(3).
(BS).
May not be repeated for credit.
Topics include: Lebesgue measure on the real line; measurable functions and integration on R; differentiation theory, fundamental theorem of calculus; function spaces, Lp (R), C(K), Holder and Minkowski inequalities, duality; general measure spaces, product measures, Fubini's Theorem; RadonNikodym Theorem, conditional expectation, signed measures, introduction to Fourier transforms.

MATH 602. Real Analysis II
MATH 590 and 597.
(3).
(BS).
May not be repeated for credit.

MATH 605. Several Complex Variables
MATH 596 and 597. Graduate standing.
(3).
(BS).
May not be repeated for credit.

MATH 606. Advanced Stochastic Analysis for Finance
Consent of instructor required.
Strong background in Probability Theory (Math 625), Measure Theory, Real Analysis & Functional Analysis.
(3).
May not be repeated for credit.
This is a PhD level course in Stochastic Analysis and applications to Quantitative Finance. The aim of this course is to teach the advanced probabilistic techniques and concepts from the theory of stochastic processes, in order to provide a sufficient knowledge base for the students to conduct research in Financial Mathematics. It is a researchlevel extension of Math 506.

MATH 612. Algebra II
MATH 593 and 594; and Graduate standing.
(3).
(BS).
May not be repeated for credit.

MATH 613. Homological Algebra
MATH 590 and 594 or permission of instructor.
(3).
(BS).
May not be repeated for credit.

MATH 614. Commutative Algebra
MATH 593 and Graduate standing.
(3).
(BS).
May not be repeated for credit.

MATH 615. Commutative Algebra II
MATH 614 or permission of instructor. Graduate standing.
(3).
(BS).
May not be repeated for credit.

MATH 623 / IOE 623. Computational Finance
MATH 316 and MATH 425 or 525.
(3).
(BS).
May not be repeated for credit.
This is a course in computational methods in finance and financial modeling. Particular emphasis will be put on interest rate models and interest rate derivatives. The specific topics include; BlackScholes theory, no arbitrage and complete markets theory, term structure models: Hull and White models and Heath Jarrow Morton models, the stochastic differential equations and martingale approach: multinomial tree and Monte Carlo methods, the partial differential equations approach: finite difference methods.

MATH 625 / STATS 625. Probability and Random Processes I
MATH 597 and Graduate standing.
(3).
(BS).
May not be repeated for credit.

MATH 626 / STATS 626. Probability and Random Processes II
MATH 625/STATS 625 and Graduate standing.
(3).
(BS).
May not be repeated for credit.

MATH 627 / BIOSTAT 680. Applications of Stochastic Processes I
Graduate standing; BIOSTAT 601, 650, 602 and MATH 450.
(3).
(BS).
May not be repeated for credit.

MATH 631. Introduction to Algebraic Geometry
MATH 594 or permission of instructor. Graduate standing.
(3).
(BS).
May not be repeated for credit.

MATH 632. Algebraic Geometry II
MATH 631 and Graduate standing.
(3).
(BS).
May not be repeated for credit.

MATH 635. Differential Geometry
MATH 537 or permission of instructor. Graduate standing.
(3).
(BS).
May not be repeated for credit.

MATH 636. Topics in Differential Geometry
MATH 635 and Graduate standing.
(3).
(BS).
May not be repeated for credit.

MATH 637. Lie Groups
MATH 635 and Graduate standing.
(3).
(BS).
May not be repeated for credit.

MATH 650. Fourier Analysis
MATH 596, 602, and Graduate standing.
(3).
(BS).
May not be repeated for credit.

MATH 651. Topics in Applied Mathematics I
MATH 451, 555 and one other 500level course in analysis or differential equations. Graduate standing.
(3).
(BS).
May be elected twice for credit.
Topics such a celestial mechanics, continuum mechanics, control theory, general relativity, nonlinear waves, optimization, statistical mechanics.

MATH 654. Introduction to Fluid Dynamics
MATH 450; MATH 555 or 596; and MATH 454 or 556. Graduate standing.
(3).
(BS).
May not be repeated for credit.
This is an introduction to fluid dynamics. The syllabus includes a derivation of the governing equations, as well as basic background information on potential flow, boundary layers, vortex dynamics, hydrodynamic stability, and turbulence. This course uses a variety of mathematical techniques including asymptotic methods and numerical simulations.

MATH 656. Introduction to Partial and Differential Equations
MATH 558, 596 and 597 or permission of instructor. Graduate standing.
(3).
(BS).
May not be repeated for credit.

MATH 657. Nonlinear Partial Differential Equations
MATH 656.
(3).
(BS).
May not be repeated for credit.

MATH 658. Ordinary Differential Equations
A course in differential equations (e.g., MATH 404 or 558). Graduate standing.
(3).
(BS).
May not be repeated for credit.

MATH 660 / IOE 610. Linear Programming II
MATH 561 and Graduate standing.
(3).
(BS).
May not be repeated for credit.

MATH 663 / IOE 611. Nonlinear Programming
MATH,MATH 561.
(3).
(BS).
May not be repeated for credit.

MATH 665. Combinatorial Theory II
MATH 664 or equivalent. Graduate standing.
(3).
(BS).
May not be repeated for credit.

MATH 669. Topics in Combinatorial Theory
MATH 565, 566, or 664; and Graduate standing.
(3).
(BS).
May not be repeated for credit.

MATH 671. Analysis of Numerical Methods I
MATH. 571, 572, or permission of instructor. Graduate standing.
(3).
(BS).
May not be repeated for credit.

MATH 675. Analytic Theory of Numbers
MATH 575, 596, and Graduate standing.
(3).
(BS).
May not be repeated for credit.

MATH 676. Theory of Algebraic Numbers
MATH 575, 594, and Graduate standing.
(3).
(BS).
May not be repeated for credit.

MATH 677. Diophantine Problems
MATH 575 and Graduate standing.
(3).
(BS).
May be repeated for credit.

MATH 678. Modular Forms
MATH 575, 596, and Graduate standing.
(3).
(BS).
May be repeated for credit.

MATH 679. Arithmetic of Elliptic Curves
MATH 594 and Graduate standing.
(3).
(BS).
May be repeated for credit.

MATH 681. Mathematical Logic
Mathematical maturity appropriate for a 600level MATH course. Graduate standing.
(3).
(BS).
May not be repeated for credit.

MATH 682. Set Theory
MATH 681 or equivalent.
(3).
(BS).
May not be repeated for credit.

MATH 684. Recursion Theory
MATH 681 or equivalent.
(3).
(BS).
May not be repeated for credit.

MATH 695. Algebraic Topology I
MATH 591 or permission of instructor. Graduate standing.
(3).
(BS).
May not be repeated for credit.

MATH 696. Algebraic Topology II
MATH 695 or permission of instructor.
(3).
(BS).
May not be repeated for credit.

MATH 697. Topics in Topology
Graduate standing.
(2  3).
(BS).
May not be repeated for credit.

MATH 700. Directed Reading and Research
Consent of instructor required.
Graduate standing and permission of instructor.
(1  3).
(INDEPENDENT).
May be elected three times for credit.

MATH 703. Topics in Complex Function Theory I
MATH 604 and Graduate standing.
(3).
(BS).
May not be repeated for credit.

MATH 709. Topics in Modern Analysis I
MATH 597 and Graduate standing.
(3).
(BS).
May not be repeated for credit.

MATH 710. Topics in Modern Analysis II
MATH 597 and Graduate standing.
(3).
(BS).
May not be repeated for credit.

MATH 711. Advanced Algebra
MATH 594 or 612 or permission of instructor. Graduate standing.
(3).
(BS).
May not be repeated for credit.

MATH 715. Advanced Topics in Algebra
MATH 594, 612, and Graduate standing.
(3).
(BS).
May not be repeated for credit.

MATH 731. Topics in Algebraic Geometry
Graduate standing.
(3).
(BS).
May not be repeated for credit.

MATH 732. Topics in Algebraic Geometry II
MATH 631 or 731.
(3).
(BS).
May not be repeated for credit.

MATH 756. Advanced Topics in Partial Differential Equations
MATH 597 and Graduate standing.
(3).
(BS).
May not be repeated for credit.

MATH 775. Topics in Analytic Number Theory
MATH 675.
(3).
(BS).
May be repeated for credit.

MATH 776. Topics in Algebraic Number Theory
MATH 676 and Graduate standing.
(3).
(BS).
May be repeated for credit.

MATH 821. Actuarial Math
(1).
May be repeated for a maximum of 2 credits.
This course has a grading basis of "S" or "U".

MATH 929. Internship
Consent of department required.
(INDEPENDENT).
May be repeated for a maximum of 3 credits.
This course has a grading basis of "S" or "U".
Credit is granted for a fulltime 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 under consultation with a faculty advisor.

MATH 990. Dissertation/Precandidate
Election for dissertation work by doctoral student not yet admitted as a Candidate. Graduate standing.
(1  8; 1  4 in the halfterm).
(INDEPENDENT).
May be repeated for credit.
This course has a grading basis of "S" or "U".

MATH 993. Graduate Student Instructor Training Program
Graduate standing and appointment as GSI in Mathematics Department.
(1).
May not be repeated for credit.
This course has a grading basis of "S" or "U".

MATH 995. Dissertation/Candidate
Graduate School authorization for admission as a doctoral Candidate.
(Prerequisites enforced at registration.)
(8; 4 in the halfterm).
(INDEPENDENT).
May be repeated for credit.
This course has a grading basis of "S" or "U".
