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- Graduates
- Admissions
- Awards & Fellowships
- Courses
- Programs
- Recent Ph.D. Recipients
- Student Handbook - AIM & Math
- Students On the Job Market - AIM & Math
- Student Spotlight - AIM & Math
- Thesis Defense Schedule

### Actuarial/Financial

**Math 423 Mathematics of Finance (3).**- (Prerequisite: Math 217, 425, and EECS 183 or equivalents.)
- Topics include risk and return theory, portfolio theory, capital asset pricing model, random walk model, stochastic processes, Black-Scholes Analysis, numerical methods and interest rate models.

**Math 424 Theory of Interest (3)**- (Prerequisite: Mathematics 215, 255, or 285; or permission of instructor.)
- 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 accounts, bond prices, etc.

**Math 520 Life Contingencies I (3).**- (Prerequisite: Math 424 and Math 425, or permission of instructor.)
- The main topics are the development of (1) probability distributions for the future lifetime random variable, (2) probabilistic methods for financial payments depending on death or survival, and (3) mathematical models of actuarial reserving.

**Math 521 Life Contingencies II (3).**- (Prerequisite: Math 520 with a grade of C- or higher.)
- Topics include multiple life models--joint life, last survivor, contingent insurance; multiple decrement models---disability, withdrawal, retirement, etc.; and reserving models for life insurance.

**Math 523 Loss Models I (3).**- (Prerequisite: Math 425 or equivalent.)
- 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 credibility.

**Math 623 Computational Finance (3).**- (Prerequisite: Math 316 and 425 or 525.)
- 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. Specific topics include Black-Scholes theory, no-arbitrage and complete markets theory, term structure models, Hull and White models, 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.

### Algebra/Group Theory

**Math 412 Introduction to Modern Algebra (3).**- Prerequisite: Math 215 or 285; and Math 217.
- Only 1 credit after Math 312.
- Initial topics include ones common to every branch of mathematics: sets, functions (mappings), relations, & 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 gometric figures, and polynomials. Notions such as generator, subgroup, direct product, isomorphism and homomorphism are defined and studied.

**Math 419 Linear Spaces and Matrix Theory (3)**- Advisory Prerequisite: Four courses beyond MATH 110.

- No credit for those who have completed or are enrolled in 214, 217, 419, or 420.

- 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.

- Advisory Prerequisite: Four courses beyond MATH 110.
**Math 420 Advanced Linear Algebra (3).**- Advisory Prerequisites: Linear algebra course (MATH 214, 217, 417, or 419) and one of MATH 296, 412, or 451.

- 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.

- Advisory Prerequisites: Linear algebra course (MATH 214, 217, 417, or 419) and one of MATH 296, 412, or 451.

**Math 593 Algebra I (3).**- Prerequisite: Math 494; or Math 412, 420, and 451.
- 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 (3).**- Prerequisite: Math 593
- Topics include group theory, permutation representations, simplicity of alternating groups for n > 4. Sylow theorems, series in groups, solvable and nilpotent groups. Jordan-Holder Theorem for groups with operators, free groups and presentations, fields and field extensions, norm and trace, algebraic closure, Galois theory, and transcendence degree.

**Math 613 Homological Algebra (3).**- Prerequisite: Math 590 and 594 or permission of instructor.
- Categories and functors; abelian categories and the imbedding theorem; homology and cohomology theories.

**Math 619 Topics in Algebra (3).**- Prerequisite: Math 593.
- Selected topics.

**Math 711 Advanced Algebra (3).**- Prerequisite: Math 594 and 612 or permission of instructor.
- Topics of current research interest, such as groups, rings, lattices, etc., including a thorough study of one such topic.

**Math 715 Advanced Topics in Algebra (3).**- May be taken more than once for credit.

- May be taken more than once for credit.
**Math 790 Transformation Groups (3).**- Prerequisite: Math 591.
- Selected topics from the theory of topological or differential transformation groups.

### Analysis/Functional Analysis

**Math 450 Advanced Mathematics for Engineers I (4).**- Prerequisite: Math 215 or 285 and Math 216, 286, or 316.
- 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 iwth applications to potiential 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 (3).**- Prerequisite: A thorough understanding of calculus and one of Math 217, 312, 412, or permission of instructor
- 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; differentitation; integration, the Fundamental Theorem of Calculus, infinite series; sequences and series of functions.

**Math 452 Advanced Calculus II (3).**- Prerequisite: Math 217, 417, 419, or 420 (may be taken concurrently) and Math 451.
- 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) Green's & Stokes' theorems, differential forms, exterior derivatives.

**Math 597 Analysis II (Real)(3).**- Prerequisite: Math 451 and 420; or Math 395.
- Topics include: Lebesgue measure on the real line.; measurable functions and integration on
**R**, differentiation theory, fundamental theorem of calculus; function spaces, L^{p}(**R**), C(K), Holder and Minkowski inequalities, duality; general measure spaces, product measures, Fubini's Theorem. Radon-Nikodym Theorem, conditional expectation, signed measures, introduction to Fourier transforms.

**Math 602 Real Analysis II (3).**- Prerequisite: Math 590 and 597.
- Introduction to functional analysis; metric spaces, completion, Banach spaces, Hilbert spaces, $L^p$ spaces; linear functionals, dual spaces, Riesz representation theorems; principle of uniform boundedness, closed graph theorem, Hahn-Banach theorem, B aire category theorem, applications to classical analysis.

**Math 609 Topics in Analysis (3).**- Prerequisite: Math 451.
- Selected topics in analysis. Content and prerequisites will vary from year to year. May be taken for credit more than once.

**Math 650 Fourier Analysis (3).**- Prerequisite: Math 602 and 596.
- General properties of orthogonal systems. Convergence criteria for Fourier series. The Fourier integral, Fourier transform and Plancherel theorem. Wiener's Tauberian theorem. Elements of harmonic analysis. Applications.

**Math 701 Functional Analysis I (3).**- Prerequisite: Math 602.
- Geometry of Hilbert space; basic properties of linear operators; self-adjoint, unitary, and normal operators; spectral theorem; compact operators; unbounded operators; Banach spaces, Banach algebras, topological vector spaces.

**Math 702 Functional Analysis II (3).**- Prerequisite: Math 602 and sometimes Math 701.
- Further topics in Functional Analysis.

**Math 707 Calculus of Variations (3).**- Prerequisite: Math 597.
- Modern theory of calculus of variations. Topics will be taken from: critical point theory, Morse theory, bifurcation theory, geometric measure theory, \relax $\mathsurround =\z@ \mathinner {\ldotp \ldotp \ldotp }\mskip \thinmuskip $, etc.

**Math 710 Topics in Modern Analysis, II (3).**- Prerequisite: Math 597.
- Selected advanced topics in analysis.

### Applied Mathematics

**Math 464 Inverse Problems (3)**- Prerequisite: Math 217, 417, or 419 and Math 216, 286, or 316
- 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 sqares, 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 i investigation of the mathematical concepts related to analysis and solution of inverse problems.

**Math 550 (CMPLXSYS 510) Intro to Adaptive Systems (3)**- Prerequisite: Math 215, 255, or 285; Math 217; and Math 425
- The course will start with classical differential equation and game theory approaches. It will then focus on the theory and application of particular models of adaptive systems such as models of neural systems, genetic algorithms, classifier systems, and cellular automata. Time permitting, we will discuss more recent developments such as sugarscape and echo.

**Math 623 Computational Finance (3)**- Prerequisite: Math 316 and Math 425 or 525
- 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. Specific topics include Black-Scholes theory, no-arbitrage and complete markets theory, term structure models, Hull and White models, 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 651 Topics in Applied Mathematics I (3).**- Prerequisite: Math 451, 555 and one other 500-level course in analysis or differential equations.
- Topics such as celestial mechanics, continuum mechanics, control theory, general relativity, nonlinear waves, optimization, statistical mechanics.

**Math 652 Topics in Applied Mathematics II (3).**- Prerequisite: Math 451, 555 and one other 500-level course in analysis or differential equations.
- Topics such as celestial mechanics, continuum mechanics, control theory, general relativity, nonlinear waves, optimization, statistical mechanics.

### Combinatorics

**Math 565 Combinatorics and Graph Theory (3).**

Prerequisite: Math 465 or equivalent experience with abstract mathematics.

Math 565 and 566 can be taken in any order. These courses introduce the basic notions and techniques of combinatorics and graph theory at the beginning graduate level. Math 565 emphasizes the aspects connected with computer science, geometry, and topology. Graph theory: connectivity, planarity, graph coloring, extremal graph theory, and Ramsey theory. Other topics: combinatorial designs, partially ordered sets, lattices, matroids, simplicial complexes, and convex polytopes.

**Math 566 Combinatorial Theory (3).**

Prerequisite: Math 465 or equivalent experience with abstract mathematics.

Math 565 and 566 can be taken in any order. These courses introduce the basic notions and techniques of combinatorics and graph theory at the beginning graduate level. Math 566 emphasizes the enumerative and algebraic aspects of the subject. Algebraic graph theory: graph eigenvalues, enumeration of walks and tilings, electric networks, the matrix-tree theorem. Partitions and Young tableaux: generating functions, q-analogues, hooklength formula, the Schensted correspondence.

**Math 567 Introduction to Coding Theory (3).**

Prerequisite: Math 412 or equivalent experience with abstract mathematics.

Mathematical foundations of modern coding and information theory. When data is being transmitted over a noisy channel, the goal is to find efficient ways to encode/decode it to deal with data loss and errors. Topics include: entropy, Huffman codes, channel capacity, Shannon’s theorem, error correcting block codes, various constructions of linear codes over finite fields (Hamming codes, Golay codes, Reed-Muller codes, cyclic codes, etc.), bounds for codes, and more.

**Math 664 Combinatorial Theory I (3).**

Prerequisite: Math 465 or equivalent experience with abstract combinatorial mathematics.

Introduction to the techniques of enumeration. Basic material is found in Stanley's "Enumerative Combinatorics,'' chapters 1-2 and 4-5. Sieve methods. Ordinary and exponential generating functions. Partitions and $q$-series. Polya theory. The transfer-matrix method. Other optional topics as time permits.

**Math 665 Combinatorial Theory II (3).**

Prerequisite: Math 565 or 566 or 664 or permission of instructor.

This is a topics course whose content varies significantly from year to year. Topics in recent years have included combinatorial representation theory, symmetric functions, Schubert calculus, Coxeter groups and root systems, cluster algebras, and total positivity.

**Math 668 Advanced Combinatorics (3)**

Prerequisite: Math 565 or 566 or equivalent

This introductory course in combinatorial matrix theory focuses on connections between linear algebra and algebraic combinatorics. Topics may include: Combinatorial techniques in linear algebra. Basic algebraic graph theory. Determinantal identities. Matroids and projective geometry. Grassmannians and Schubert cells. Canonical forms and factorizations. Polynomials with real roots. Totally positive matrices.

**Math 669 Topics in Combinatorial Theory (3)**.

Prerequisite: Math 565 or 566 or 664 or permission of instructor.

This is a topics course whose content varies significantly from year to year. Topics in recent years have included combinatorial matrix theory, convex polytopes, lattice points in polyhedra, combinatorics and complexity of partition functions, and combinatorial representation theory.

### Commutative Algebra/Algebraic Geometry

**Math 614 Commutative Algebra I (3).**- Prerequisite: Math 593.
- Review of commutative rings and modules. Local rings and localization. Noetherian and Artinian rings. Integral independence. Valuation rings, Dedekind domains, completions, graded rings. Dimension theory.

**Math 615 Commutative Algebra II (3).**- Prerequisite: Math 614 or permissions of instructor.
- This is a continuation of Math 614: structure of complete local rings, regular, Cohen-Macaulay, and Gorenstein rings, excellent rings, Henselian rings, etale maps, equations over local rings.

**Math 631 Algebraic Geometry I. (3).**- Prerequisite: Math 594 or 614 or permission of instructor).
- Theory of algebraic varieties: affine and projective varieties, dimension of varieties and subvarieties, singular points, divisors, differentials, intersections. Schemes, cohomology, curves and surfaces, varieties over the complex numbers.

**Math 632 Algebraic Geometry II. (3).**- Prerequisite: Math 631).
- Continuation of Math 631.

**Math 731 Topics in Algebraic Geometry I (3).**- Selected topics in algebraic geometry.

**Math 732 Topics in Algebraic Geometry II (3).**- Prerequisite: Math 631 or 731.
- Selected topics in algebraic geometry.

### Complex Analysis

**Math 555 Introduction to Complex Variables (3)**- Prerequisite: Math 450 or 451. Students who had 450 (or equivalent) but not 451 are encouraged to take 451 simultaneously wiht 555.
- Differentiation and integration of complex-valued functions of a complex variable, series, mappings, residues, applications. Evaluation of improper real integrals, applications in ideal fluid dynamics. This corresponds to Chapters 1-9 of Churchill & Brown

**Math 596 Analysis I (3).**- Prerequisite: Math 451. 2 hours credit for those with credit for 555.
- Review of analysis in
**R**^{2}including metric spaces, differentiable maps, Jacobians; analytic functions. Cauchy-Riemann equations, conformal mappings, linear fractional tranformations; Cauchy's theorem, Cauchy integral formula; power series and Laurent expansions, residue theorem and applications, maximum modulus theorem, argument principle; harmonic functions; global properties of analytic functions; analytic continuation; normal families, Riemann mapping theorem.

**Math 604 Complex Analysis II (3).**- Prerequisite: Math 596.
- Selected topics such as potential theory, geometric function theory, analytic continuation, Riemann surfaces, uniformization and analytic varieties.

**Math 605 Several Complex Variables (3).**- Prerequisite: Math 604 or consent of instructor.
- Power series in several complex variables, domains of holomorphy, pseudo convexity, plurisubharmonic functions, the Levi problem. Domains with smooth boundary, tangential Cauchy-Riemann equations, the Lewy and Bochner extension theorems. The $\overlin e {\partial }$-operator and Hartog's Theorem, Dol beault-Grothendieck lemma, theorems of Runge, Mittag-Leffler and Weierstrass. Analytic continuation, monodromy theorem, uniformization and Koebe's theorem, discontinuous groups.

**Math 606 Riemann Surfaces (3).**- Prerequisite: Math 590, 604, and some knowledge of group theory.
- Introduction to the theory of Riemann surfaces. The Riemann surface of an analytic function. Covering surfaces, monodromy theorem, groups of cover transformations, uniformization theorem. Differentials and integrals, Riemann-Roch theorem.

**Math 703 Topics in Complex Function Theory I (3).**- Prerequisite: Math 604.
- Selected advanced topics in function theory. May be taken for credit more than once, as the content will vary from year to year.

**Math 704 Topics in Complex Function Theory II (3).**- Prerequisite: Math 604.
- Selected advanced topics in function theory. May be taken for credit more than once, as the content will vary from year to year.

### Differential Equations

**Math 454 Boundary Value Problems for Partial Differential Equations (3)**- 1 Credit after Math 354, No credit to those who have complete Math 450.
- Prerequisite: Math 215 or 285 and Math 216, 286, or 316.
- Classical representation & convergence theorems for Fourier series; method of separation of variables for the solution of the one-dimensional heat & wave equation; heat & wave equationss in higher dimensions; eigen function expansions; spherical & cylindrical Bessel functions; Legendre polynomials; methods for evaluating asympttic 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 transofrms; 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 558 Ordinary Differential Equations (3).**- Prerequisite: Math 451
- The basic results on qualitative behavior, centered on themes of stability and phase plane analysis will be presented in a context that includes applications to a variety of classic examples. The proofs of the fundamental facts will be presented, along with discussions of examples.

**Math 656 Introduction to Partial Differential Equations (3).**- Prerequisite: Math 558, 596 and 597 or permission of instructor.
- Characteristics, heat, wave and Laplace's equation, energy methods, maximum principles, distribution theory.

**Math 657 Nonlinear Partial Differential Equations (3).**- Prerequisite: Math 656 or permission of instructor.
- A survey of ideas and methods arising in the study of nonlinear partial differential equations, nonlinear variational problems, bifurcation theory, nonlinear semigroups, shock waves, dispersive equations.

**Math 658 Topics in Ordinary Differential Equations (3).**- Prerequisite: Senior undergraduate or graduate course
- This is a topics course whose content varies significantly from year to year. Topics in recent years have included: the qualitative theory of ODE’s on manifolds, symplectic and Poisson geometry, nonlinear stability theory, Lagrangian and Hamiltonian mechanics, integrable systems, reduction and symmetries, mechanical systems with constraints and controls, and optimal control.

**Math 756 Advanced Topics in Partial Differential Equations (3).**- May be taken more than once for credit.

### Differential Geometry

**Math 433 Introduction to Differential Geometry (3).**- Prerequisite: Math 215 or 285 and Math 217.
- 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 537 Introduction to Differentiable Manifolds (3).**- Prerequisite: Math 590 and 420.
- The following topics will be discussed: smooth manifolds and maps, tangent spaces, submanifolds, vector fields and flows, basic Lie group theory, group actions on manifolds, differential forms, de Rham cohomology, orientation and manifolds with boundary, integration of differential forms, Stokes' theorem

**Math 635 Differential Geometry (3).**- Prerequisite: Math 537 or permission of instructor.
- Second fundamental form, Hadamard manifolds, spaces of constant curvature, first and second variational formulas, Rauch comparision theorem, and other topics chosen by the instructor

**Math 636 Topics in Differential Geometry (3).**- Prerequisite: Math 635.

### Lie Theory/Representation Theory

**Math 612 Lie Algebras and Their Representations.**

- Prerequisite: Math 593 and 594 or consent of instructor.
- Representation Theory of semisimple Lie algebras over the complex numbers. Weyl's Theorem, root systems, Harish Chandra's Theorem, Weyl's formulae and Kostant's Multiplicity Theorem. Lie groups, their Lie algebras and further examples of representatio ns.

### Logic and Foundations

**Math 481 Introduction to Mathematical Logic (3).**- Prerequisite: Math 412 or 451 or equivalent experience with abstract mathematics.
- 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 1st order predicate languages and their models . New elements here are quantifiers ('there exists' and 'for all'). The study of notions of truth, logical consequences, & provability leads to completeness & 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 fo Enderton.

**Math 582 Introduction to Set Theory (3).**- Prerequisite: Math 412 or 451 or equivalent experience with abstract mathematics.
- The main topics 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 681 Mathematical Logic (3).**- Prerequisite: Mathematical maturity appropriate to a 600-level course. (No previous knowledge of mathematical logic is needed.)
- Syntax and semantics of first-order logic. Formal deductive systems. Soundness and completeness theorems. Compactness principle and applications. Decision problems for formal theories. Additional topics may include non-standard models and logical syst ems other than classical first-order logic.

**Math 682 Set Theory (3).**- Prerequisite: Math 681 or Equivalent.
- Axiomatic development of set theory including cardinal and ordinal numbers. Constructible sets and the consistency of the axiom of choice and the generalized continuum hypothesis. Forcing and the independence of choice and the continuum hypothesis. Ad ditional topics may include combinatorial set theory, descriptive set theory, or further independence results.

**Math 683 Model Theory (3).**- Prerequisite: Math 681 or equivalent.
- Connections between classes of mathematical structures and the sentences (in first-order logic) describing them. Definable sets within structures; definable classes of structures. Methods for producing structures with prescribed properties. Categorica l and complete theories. Methods for analyzing the first-order properties of structures. Introduction to some concepts of classification theory.

**Math 684 Recursion Theory (3).**- Prerequisite: Math 681 or equivalent.
- Elementary theory of recursive functions, sets, and relations and recursively enumerable sets and relations. Definability and incompleteness in arithmetic. Godel's incompleteness theorems. Properties of r.e. sets. Relative recursiveness, degrees of un solvability and the jump operator. Oracle constructions. The Friedberg-Muchnik Theorem and the priority method.

**Math 781 Topics in Mathematical Logic (3).**- Prerequisite: Varies according to content.
- Advanced topics in mathematical logic. Content will vary from year to year. May be repeated for credit.

### Mathematical Physics

**Math 556 Applied Functional Analysis (3)**- Prerequisites: Math 217, 419, or 420; Math 451 and Math 555.
- Topics in functional analysis that are used in the analysis of ordinary and partial differential equations. Metric and normed linear spaces., Banach spaces and the contraction mapping theorem, HIlbert spaces and spectral theory of compact operators, distributions and Fourier transforms, Sobolev spaces and applications to elliptic PDEs.

**Math 557 Applied Asymptotic Analysis (3).**- Prerequisites: Math 217, 419, or 420; Math 451; and Math 555.
- Topics include stationary phase, steepest descent, characterization of singularities in terms of the Fourier transform, regular and singular perturbation problems, boundary layers, multiple scales, WKB method. Additional topics depend on the instructor bu may include non-linear stability theory, bifurcations, applications in fluid dynamics (Rayleigh-Benard convection), combustion (flame speed).

### Number Theory

**Math 475 Elementary Number Theory (3).**- Prerequisites: None
- 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 (3).**- Prerequisite: Math 475 or 575; may be concurrently.
- 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 538 Lie Algebras (3).**- Prerequisite: Algebra: (Math 493, 494) or (Math 593, 594)
- Lie algebras arise naturally in many areas of mathematics and physics. This course covers the basic structuretheory of Lie algebras and finishes with a discussion of their representation theory. Intended for advanced mathundergraduates, math master's, and math PhDs. (Primarily graduate students)

**Math 575 Introduction to Theory of Numbers (3; 1 hour credit for students with credit for Math 475.)**- Prerequisite: Math 451 and one of Math 420 or 494, or permission of instructor.
- This is a first course in number theory. Topics covered include divisibility and prime numbers, congruences, quadratic reciprocity, quadratic forms, arthmetic functions, and Diophantine equations. Other topics may be covered as time permits or by request.

**Math 576 Algebraic Number Theory (3)**.- Prerequisites: Abstract algebra (groups, rings, field, Galois theory), linear algebra
- This course is an introduction to algebraic number theory, focused on number fields (i.e., finite-degreeextensions of Q), their subrings, and their completions. Intended for advanced math undergraduates, mathmaster's, and math PhDs. (Primarily graduate students)

**Math 637 Lie Groups (3).**- Prerequisites: Topology (at least one undergraduate class), Algebra (Math 493/494 or 593/594), Analysis (at least one undergraduate class)
- Lie groups arise naturally in many areas of mathematics (representation theory, geometry, dynamics, harmonicanalysis, number theory) and physics. This course covers the structure and representation theory of compact Liegroups. Intended for Math PhDs.

**Math 638 Algebraic Groups (3).**- Prerequisites: Algebraic geometry (Math 631)
- Introduction to the basic theory of affine algebraic groups over an algebraically closed field. Discuss the structure theory of reductive groups. Discuss non-algebraically closed fields if time permits.

**Math 675 Analytic Theory of Numbers (3).**- Prerequisite: Math 575, 596.
- Theory of the Riemann zeta-function and the L-functions, distribution of primes, Dirichlet's theorem on primes in a progression, quadratic forms, transcendental numbers.

**Math 676 Class Field Theory (3).**- Prerequisite: Field theory and Galois theory. Algebraic number theory: factorization of prime ideals, class groups, ramification, p-adic numbers.
- This course is an introduction to class field theory, which is the study of abelian extensions of local and global fields. It is a basic prerequisite to understanding the Langlands program which deals with the more difficult case of nonabelian extensions. The focus of the course will typically be on explaining the structure of the theory using examples and applicatoins, with intricate proofs often omitted. Intended for Math PhDs.

**Math 677 Arithmetic Geometry (3).**- Prerequisite: Algebra, algebraic number theory, and algebraic geometry.
- This course is a topics course in artihmetic geometry. Topics will change depending on the year. Possible topics include etale cohomology, introduction to the Langlands program, and rational points on varieties (syllabi available). Intended for Math PhDs.

**Math 678 Modular Forms (3).**- Prerequisite: Math 596 and 575.
- A basic introduction to modular functions, modular forms, modular groups. Hecke operators, Selberg trace formula. Applications to theory of partitions, quadratic forms, class field theory and elliptic curves.

**Math 679 Arithmetic of Elliptic Curves (3).**- Prerequisite: Familiarity with algebra at the level of Math 493/494 or Math 593/594
- This course is an introduction to elliptic curves, focusing on the arithmetic side of the theory. It develops the theory of elliptic curves over various types of fields and in particular over algebraic number fields.

**Math 738 Topics in Representation Theory (3).**- Prerequisites: Lie algebra, Lie groups, algebraic groups
- This is an advanced topics course in representation theory. Topics will change each semester. Possible topicsinclude Bruhat-Tits theory, p-adic representation theory, etc. Intended for Math PhDs.

**Math 775 Topics in Analytic Number Theory (3).**- Prerequisite: Math 675.
- Selected topics in analytic number theory.

**Math 776 Topics in Algebraic Number Theory (3).**- Prerequisites: Algebra, Algebraic number theory.
- This course is an advanced topics course in algebraic number theory. Topics will change depending on the year. Possible topics include lwasawa theory, modular curves, etc. Intended for Math PhDs.

**Math 777 Topics in Arithmetic Geometry (3).**- Prerequisites: Algebra, algebraic number theory, algebraic geometry
- This is an advanced topics course in arithmetic geometry. Topics will change each semester, but could include p-adic Hodge theory, perfectoid spaces, etc. Intended for Math PhDs.

### Numerical Analysis

**Math 471 Introduction to Numerical Methods (3).**- Prerequisite: Math216, 286, or 316; Math 214, 217, 417, or 419; and a working knowledge of one high-level ocmputer language.
- Topics may include computer arithmetic, Newton's method for nonlinear 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 571 Numerical Linear Algebra (3).**- Prerequisite: Math 214, 217, 417, 419, or 420 and one of Math 450, 451, or 454, or permission from the instructor.
- This course is an introduction to numerical linear algebra, a core subject in scientific computing. Three types of problems are considered: (1) linear systems, (2) eigenvalues and eigenvectors, and (3) least squares problems. Topics include: Gram-Schmidt orthogonalization, QR factorization, singular value decomposition (SVD), normal equations, vector and matrix norms, condition number, backward error analysis, LU factorization, Cholesky factorization, reduction to Hessenberg and tridiagional form, power method, inverse iteration, rayleigh quotient iteration, QR algorithm, Krylov subspace methods, Arnoldi iteration, GMRES, steepest descent, conjugate gradient method, preconditioning, applications to image compression, finite-difference schemes for two-point boundary value problems, Dirichlet problem for the Laplace equation, least squares data fitting.

**Math 572 Numerical Methods for Differential Equations (3).**- Prerequisite: Math 214, 217, 417, 419, or 420 and one of Math 450, 451, or 454 or permission of the instructor.
- content varies somewhat with the instructor. Numerical methods for ordinary differential equations; Lax's equivalence theorem; finite difference and spectral methods for linear time dependent PDEs: diffusion equations, scalar first order hyperbolic equations, symmetric hyperbolic systems.

**Math 671 Analysis of Numerical Methods I (3).**- Prerequisite: Math 571, 572, or permission of instructor
- This is a course on special topics in numerical analysis and scientific computing. Subjects of current research interest will be included. Recent topics have been: Finite difference methods for hyperbolic problems, Multigrid methods for elliptic bound ary value problems. Students can take this class for credit repeatedly.

### Probability Theory

**Math 425 (Stat. 425) Introduction to Probability (3).**- Prerequisite: Math 215 or 285.
- Topics include the basic results and methods of both discrete and continous 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 525 (Stat. 525). Probability Theory (3).**- Prerequisite: Math 297, 351, or 451 (strongly recommended). Math 425/Stats 425 would be helfpul.
- The following topics will be covered: sample space and events, random variables, concept and definition of probability and expectation, conditional probability and expectation, independence, momen generating functions, Law of large numbers, Central limit theorem, Markov chains, Poisson process and exponential distribution.

**Math 526 (Stat. 526). Discrete State Stochastic Processes (3).**- Prerequisite: Math 525 or EECS 501.
- The material is divided between discrete and continuous time processes. In both, a general theory is developed and detailed study is made of some special classes of processes and their applications. Some specific topics include generating functions; recurrent events and the renewal theorem; random walks; Markov chains; branching processes; limit theorems; Markov chains in continuous time with emphasis on birth and death processes and queuieing theory; an introduction to Brownian motion; stationary processes and martingales..

**Math 625 (Math. Stat. 625) Probability and Random Processes I (3).**- Prerequisite: Math 597.
- Axiomatics; measures and integration in abstract spaces. Fourier analysis, characteristic functions. Conditional expectation, Kolmogoroff extension theorem. Stochastic processes; Wiener-Levy, infinitely divisible, stable. Limit theorems, law of the it erated logarithm.

**Math 626 (Math. Stat 626) Probability and Random Processes II (3).**- Prerequisite: Math 625.
- Selected topics from among: diffusion theory and partial differential equations; spectral analysis; stationary processes, and ergodic theory; information theory; martingales and gambling systems; theory of partial sums.

### Topology

**Math 590 An Introduction to Topology (3).**- Prerequisite: Math 451.
- 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 General and Differential Topology (3).**- Prerequisite: Math 451, 452, 590.
- Topics include: Product and quotient topology, CW-complexes, group actions, topological manifolds, smooth manifolds, manifolds with boundary, smooth maps, partitions of unity, tangent vectors and differentials, the tangent bundle, submersions, immersions and embeddings, smooth submanifolds, Sard's Theorem, the Whitney embedding theorem, transversality, Lie groups, vector fields, Lie brackets, Lie algebra, multilinear algebra, vector bundles, differential forms, exterior derivatives, orientation, De Rham cohomology groups, homotopy invariance, degree theory.

**Math 592 An Introduction to Algebraic Topology (3).**- Prerequisite: Math 591.
- Fundamental group, covering spaces, simplicial complexes, graphs and trees, applications to group theory, singular and simplicial homology, Eilenberg-Steenrod axioms, Brouwer's and Lefschetz' fixed-point theorems and other topics.

**Math 690 Topological Groups (3).**- Prerequisite: Math 590.
- Group theory, general topology, integration. Elementary properties, Haar measure, representation (Peter-Weyl), positive definite functions, Fourier transforms.

**Math 691 Combinatorial and Geometric Topology I (3).**- Prerequisite: Math 591.
- Selected topics in the theory of piecewise linear and topological manifolds.

**Math 692 Combinatorial and Geometric Topology II (3).**- Prerequisite: Math 691.
- Selected topics in the theory of piecewise linear and topological manifolds.

**Math 694 Differential Topology (3).**- Prerequisite: Math 537 and 591 or permission of instructor.
- Transversality, embedding theorems, vector bundles and selected topics from the theories of cobordism, surgery, and characteristic classes.

**Math 695 Algebraic Topology I (3).**- Prerequisite: Math 591 or permission of instructor.
- Cohomology Theory, the Universal Coefficient Theorems, Kunneth Theorems (product spaces and their homology and cohomology), fiber bundles, higher homotopy groups, Hurewicz' Theorem, Poincar{\accent 19 e} and Alexander duality.

**Math 696 Algebraic Topology II (3).**- Prerequisite: Math 695 or permission of instructor.
- Further topics in algebraic topology typically taken from: obstruction theory, cohomology operations, homotopy theory, spectral sequences and computations, cohomology of groups, characteristic classes.

**Math 697 Topics in Topology (3).**- An intermediate level topics course.

**Math 791 Advanced Topics in the Topology of Manifolds I (3).**- Prerequisite: Permission of instructor.