Graduate Courses in Mathematics Graduate Program Home Page

A typical graduate student entering the pre-PhD program will usually spend one or two years (depending on their previous background) taking classes in preparation for the PhD qualifying examinations.

In the first year (which may be skipped by students with a strong background), students usually take three 500 level sequences, chosen from 544/5/6 algebra, 513/4/5 analysis, 531/2/3 topology/geometry and 564/5/6 probability/statistics. In the second year, students usually take three 600 level sequences, chosen from 616/7/8 real analysis, 634/5/6 algebraic topology, 647/8/9 abstract algebra, 637/8/9 differential geometry (not offered every year, but always offered once every two years) and 671/2/3 probability theory (not offered every year).

The PhD qualifying examinations are taken in two of the 600 level sequences chosen from the following three categories: (1) algebra; (2) analysis; (3) geometry/topology or probability/statistics. Students are expected to have taken three of the 600 level sequences before they take the examinations. After passing the PhD qualifying examinations, students take Advanced Courses and Seminars, as well as reading courses in their chosen area of specialization.

You can follow the links below to get a detailed idea of the graduate courses in mathematics offered at the University of Oregon. For a more detailed description of the requirements in the various graduate programs, see Graduate Programs.


Advanced Graduate Courses 2004-2005 Back to top

Here is the schedule for the graduate courses at the 600 level and above for 2004-2005.

FALL 2004 WINTER 2005 SPRING 2005
607 Non-Commutative Rings 607 Algebraic Number Theory 607 Algebraic Groups
607 Quantum Field Theory 607 Commutative Algebra 607 Algebraic Combinatorics
607 Adv. Topics in Probability
616 Real Analysis 617 Real Analysis 618 Real Analysis
634 Algebraic Topology 635 Algebraic Topology 635 Algebraic Topology
637 Differential Geometry 638 Differential Geometry 639 Differential Geometry
647 Abstract Algebra 648 Abstract Algebra 649 Abstract Algebra
681 Commutative Algebra 682 Algebraic Geometry 683 Adv. Topics in Algebra
684 Fourier Analysis 685 Fourier Analysis 686 Ricci Flow
690 Adv. Topics in Topology 690 Adv. Topics in Topology 692 WETSK

  • 607 Introduction to non-commutative ring theory, Shelton, Fall 2004.

    • We will cover the classical beginning results in non-commutative ring theory including rings of fractions, the Goldie Theorems, Morita equivalence, and perhaps one or more dimension theories (GK-dimension, projective dimension, Krull dimension). Time permitting I will cover the basic theory of filtered rings and almost commutative rings. I will try to keep everything motivated with lots of examples, but I will tend to stick to the Noetherian world.
    Prerequisites: 600 algebra.

  • 607 Quantum field theory and topology, Vaintrob, Fall 2004.
    • Quantum topology of knots and three-manifolds.

    During the last two decades there has been a dramatic change in the relationship between geometry and physics. Ideas coming from quantum field theory have brought spectacular new results and even helped to create new areas of study in geometry and topology. The main focus of this course will be on topological quantum field theories (TQFTs) and their applications in low-dimensional topology. will start by introducing the mathematical formalism of TQFTs and related algebraic structures (tensor categories, Frobenius algebras, quantum groups, e.a.). After constructing first non-trivial examples of TQFTs and a crash course in knot theory, we will study related invariants of links and three-manifolds. These invariants include the famous Jones polynomial and the Witten-Reshetikhin-Turaev invariants of homology spheres. In the third part of the course we will study the perturbation theory approach to the Witten-Chern-Simons TQFT and its relationship with Vassiliev knot invariants and the Kontsevich integral. If time permits, we will include additional topics, such as connection between TQFTs and conformal field theories, moduli spaces and operads.
    Prerequisites: 600 algebra, 600 topology. No prior knowledge of physics or knot theory will be assumed.

  • 607 Algebraic number theory , Polishchuk, Winter 2005.

    • This will be an introduction to modern algebraic number theory. We will concentrate on the study of local fields and their extensions with some applications to number fields. Then we'll review group cohomology and will proceed to the local class field theory.
    Prerequisites: 600 algebra, 600 topology.

  • 681 Introduction to commutative algebra, Vitulli, Fall 2004.

    • Localization of rings and modules. Primary decomposition integral dependence and valuations. ascending and descending chain conditions. Noetherian and Artinian Rings. Discrete rank one valuation rings and Dedekind domains graded rings. Filtrations and completions. Hilbert Functions and dimension theory.
    Text: Introduction to commutative algebra by Atiyah and MacDonald.
    Prerequisites: 600 algebra.

  • 607 Topics in commutative algebra, Vitulli, Winter 2005.

    • Regular sequences and depth Cohen-Macaulay Rings. The canonical module and duality Gorenstein rings. Hilbert functions and multiplicities. Stanley-Reisner rings. Simplicial homology, cellular homology and local cohomology. Semigroup rings and invariant theory. Determinantal rings. Big Cohen-Macaulay Modules. (The precise topics to be covered might depend on the interests of the students in the class.)
    Text: Cohen-Macaulay Rings by Bruns and Herzog supplemented by Notes by Bernd Ulrich.
    Prerequisites: Introduction to commutative algebra.

  • 682 Algebraic geometry I, Kleshchev, Winter 2005.

    • Affine and projective varieties (Zariski topology, irreducible components, product of varieties, flag varieties), dimension, morphisms (fiber of a morphism, constructive sets, open morphisms, birational morphisms), tangent spaces (simple points, local ring of a simple point, separability criterion), complete varieties (completeness of projective varieties), notion of algebraic group, identity component, action of algebraic groups on varieties.
    Text: Linear algebraic groups by J. E. Humphreys.
    Prerequisites: 600 algebra.

  • 683 Algebraic geometry II , Vaintrob, Spring 2005.
    • Varieties, sheaves and schemes.

    Algebraic Geometry is one of the most highly developed and beautiful branches of mathematics. Its ideas and methods play an important role in the development of mathematics as the whole as well as of its various areas, such as number theory, ring theory, representation theory, complex analysis, combinatorics, and more. Recently it found exciting applications in computer science (coding theory) and theoretical physics (string theory, solutions). Many of these developments became possible due to the overhaul of the foundations of Algebraic Geometry in the 1960s which replaced the classical algebraic approach with much more flexible and powerful language of sheaves and schemes. In this course we will study algebraic varieties using the now standard sheaf-theoretic methods. We will start with a general introduction to sheaves and their cohomology and then move to their application in Algebraic Geometry. The topics we will discuss include: divisors, line bundles and rational maps; coherent sheaves on projective varieties and graded modules; Hilbert polynomial; Serre duality; Riemann-Roch theorem for curves and its applications. We will also introduce and discuss schemes (objects generalizing both commutative rings and algebraic varieties), but we will not study them in detail.
    Pre-requisites: Algebraic geometry I

  • 607 Algebraic groups, Kleshchev, Spring 2005.

    • Lie algebra of an algebraic group, derivations, homogeneous spaces, factors, semisimple and unipotent elements, solvable groups, Borel subgroups, centralizers of tori, structure of reductive groups, representations and classification of semisimple groups.
    Text: Linear algebraic groups by J. E. Humphreys.
    Prerequisites: Algebraic geometry I.

  • 607 Advanced topics in probability, Wang, Spring 2005.

    • Probability theory is of major importance to a wide variety of disciplines, including Statistics, Economics, Physics, Chemistry, Biology, Epidemics, Finance, and Insurance. In this course we will give an introduction to the modern probability theory and its applications in different areas through examples. The following topics will be covered. Prerequisites: 500 level real analysis and 400 level probability and statistics.

  • 607 Algebraic combinatorics, Yuzvinsky, Spring 2005.

    • This course is suggested as a part of the second year graduate program in algebra. The general idea of the course consists of two parts: many algebraic (and not only algebraic) problems reduce to combinatorial ones the most interesting results in combinatorics have been obtained by applying theorems from commutative and skew-commutative algebra. The course would tentatively include the following topics: Prerequisites: 600 algebra.

  • 684/5 Introduction to Fourier analysis and wavelets , Bownik, Fall 2004/Winter 2005.

    • Fourier analysis is a subject of mathematics that originated with the study of Fourier series and integrals. Nowadays, Fourier analysis is a vast area of research with applications in various branches of science including signal analysis, tomography, partial differential equations, potential theory, mathematical physics and number theory. A recent noteworthy area of focus in Fourier analysis is orthogonal expansions in wavelet bases. The theory of wavelets is a very active area of research with many real-world applications. This course is an introduction to the theory of Fourier series, Fourier integrals, wavelets, and related topics. More specifically, we are planning to cover the following: There will be a couple of homework assignments. Since there will be no final exam each student will give an oral presentation on a subject of his/her choice related to this course.
    Text:Y. Katznelson, An Introduction to Harmonic Analysis Dover, 1976; P. Wojtaszczyk, A Mathematical Introduction to Wavelets, Cambridge Press, 1997
    Pre-requisites: 600 analysis.

  • 686 Ricci Flow , Lu, Spring 2005.

    • In this course the emphasis will be on the basic geometric pictures and the basic tools used in Ricci flow. The aim is to get greater understanding of sectons 1-10 of Perelman's preprint [2002]. The following topics will be covered.
    • What is Ricci flow? Some special solutions for Ricci flow. Intuitive understanding of what Ricci flow does to Riemannian metrics.
    • Survey of major results before Perelman. Short time existence, Three manifolds with positive Ricci curvature converges to quotient of standard sphere, Shi's derivative estimates.
    • Maximum principle in Ricci flow and its application (Hamilton-Ivey pinching Theorem).
    • Injectivity radius and Bishop-Gromov volume comparison theorem in Riemannian geometry.
    • Ricci flow is a variational flow and non-collapsing theorem (from section 1-4 in Perelman).
    • Dilation limit of singularities in Ricci flow.
    • Remaining sections from Perelman's paper, spending time on any necessary preparation needed to understand this.
    Prerequisites: Elementary Riemannian geometry, some knowledge of PDEs.


    Advanced Graduate Courses 2003-2004 Back to top

    Here is the schedule for the graduate courses at the 600 level and above for 2003-2004.

    FALL 2003 WINTER 2004 SPRING 2004
    616 Real analysis 617 Real analysis 618 Real analysis
    634 Algebraic topology 635 Algebraic topology 636 Algebraic topology
    646 Abstract algebra 647 Abstract algebra 648 Abstract algebra
    671 Probability theory 672 Probability theory 673 Probability theory
    681 Homological algebra 682 Group cohomology 683 Simple rings
    684 Hilbert spaces 685 C* algebras 686 C* algebras
    690 Cohomology/vector bundles 691 K theory 692 WETSK
    607 Cosmology 619 Complex analysis 607 Vertex algebras
    607 Hyperplane arrangements 607 Lie algebras 607 Lie groups

  • 619 Complex Analysis, Gilkey, Winter 2004.
    • Introduction to several complex variables and complex manifolds.

    The theory of several complex variables is quite different from the theory of a single complex variable. We will begin by discussing domains of holomorphy and theorems about removable singularities. We will then discuss topics in the theory of complex manifolds -- the Nirenberg-Newlander theorem (which is a complex analogue of the classical Frobenius theorem), Kaehler manifolds, etc. Specific topics to be covered will depend upon the interests of the class. Perhaps a discussion of the Kodira vanishing theorem or Stein manifolds might be appropriate.
    Pre-requisites: Some familiarity with complex variables and differential geometry.

  • 684/5/6 Advanced Topics in Analysis, Lin, Fall 2003, Winter/Spring 2004.
    • Hilbert spaces and C* algebras.

    In the first quarter, we will present the theory of Hilbert spaces and operators on Hilbert space. Topics includes:

    For the second and the third quarter, we will cover some topics on operator algebras including: Possible text: John B. Conway, A course in Functional Analysis
    Pre-requisites: 600 analysis.

  • 690 Advanced Topics in Geometry/Topology, Gilkey, Fall 2003.
    • Cohomology and the theory of vector bundles.

    The course will begin with a review of the basic properties of cohomology from the 634/5/6 sequence. We will then discuss the Gysin and the Wang sequences. The Gysin sequence will be used to study the cohomology algebras of the real, complex, and quaternionic projective spaces. The Wang sequence will be used to study the real cohomology algebras of some of the classical groups. We will discuss the basic properties of vector bundles (pull back, homotopy axiom, relationship between vector bundles over spheres and homotopy groups of the classical groups, stability results, etc.). We will discuss classifying spaces and give a brief introduction to characteristic classes.
    Pre-requisites: 600 topology.

  • 691 Advanced Topics in Geometry/Topology, Landweber, Winter 2004.
    • K-theory.

    K-theory is a periodic cohomology theory built from vector bundles. This course defines complex K-Theory, proves the Bott periodicity theorem, and moves on to its generalization, the K-theory Thom isomorphism. We then further discuss characteristic classes, using them to construct the Chern character, an isomorphism between rational K-theory and cohomology. The remaining time will be devoted to advanced topics related to K-theory, such as index theory, KO-theory (using real vector bundles), Clifford algebras, equivariant K-theory, and K-homology.
    Pre-requisites: 600 topology and 690.

  • 692 Advanced Topics in Geometry/Topology, Botvinnik, Spring 2004.
    • What Every Topologist Should Know.

    This is a seminar-style course where the students read various classic topology papers and give lectures on them. This is intended to enhance your knowledge of topology and fill gaps in your background. I am expecting that each student will read two papers and give a total of 2-4 lectures. The precise material covered depends on your own interests and the papers you choose. Possible topics include homotopy of spheres, spectral sequences, cobordism theory, generalized cohomologies, topology of Lie groups and homogeneous spaces, power operations in cohomology and K-theory, index theory and many more.

  • 681 Advanced Topics in Algebra, Dugger, Fall 2003.
    • Homological algebra.

    The aim of the course will be to develop a core background knowledge of homological techniques, which the student can then take away and apply to his or her own specialized field---in short, "homological algebra for the practical man". The course will start with the theory of projective and injective resolutions, Yoneda extensions, derived functors, Ext and Tor. It will cover the bar resolution, Koszul complexes, Hochschild (co)homology, and (if time) the derived category of a ring. In addition to the general theory, we'll spend a lot of time learning to work explicit examples. And along the way we'll see some brief applications to the theory of regular local rings, intersection multiplicities, and algebraic deformation theory. Pre-requisites: 600 algebra.

  • 682 Advanced Topics in Algebra, Pevtsova, Winter 2004.
    • Cohomology of finite groups.

    In the first part of this course, we will adopt a purely algebraic point of view. We will introduce group homology and cohomology using projective resolutions and develop the main techniques to work with them. These include bar and minimal projective resolutions, cup product in cohomology, induction and coinduction functors, restriction and corestriction, alternative descriptions of low degree cohomology, and the Lyndon-Hochschild-Serre spectral sequence. We will compute the homology and cohomology rings of finite abelian groups. In the remaining time we will explore a more advanced topic which will be chosen according to the preference and background of the audience. Short presentations by students might constitute part of this exploration into either more algebro-geometric or more topological directions. Plausible topics include (but are not limited to):

    Prerequisites: 600 topology, 607 homological algebra.
    Texts: D. Benson, Group cohomology and Representations. L. Evens. The Cohomology of Groups.

  • 683 Advanced Topics in Algebra, Berenstein, Spring 2004.
    • Simple rings.

    In the modern algebra, ring theory is of great importance. Historically, rings emerged from various branches of mathematics: number theory (rings of integers, orders, Hecke rings), representation theory (group rings, enveloping algebras, rings of matrices), differential Geometry and functional analysis (Algebras of functions and operators, e.g. differential operators), homological algebra (cohomology rings, abelian categories, Grothendieck's K-groups).
    The aim of this course is to give a systematic introduction to non-commutative ring theory with emphasis on semisimple and simple rings (semisimple rings can be thought of as associative analogues of semisimple Lie algebras).
    I am planning to cover the following topics:

    I then plan to emphasize the following additional topics as time permits: Text: Non-commutative Algebra, by Farb and Dennis.
    Pre-requisites: 600 algebra. 607 homological algebra would also be helpful.

  • 607 Cosmology, Isenberg, Fall 2003.
    • The mathematics of black holes, cosmology and gravitational waves.

    Classical (ie, nonquantum) gravitational physics is very accurately modeled by spacetime solutions of Einstein's equations of general relativity. The physics of black holes, the large scale behavior of the universe, and the nature of gravitational radiation can be analyzed by studying various families of spacetime solutions. Assuming that the students are familiar with the basics of Riemannian geometry, this course first introduces the mathematics of Lorentzian geometries and spacetimes, and discusses the Einstein partial differential equations on spacetimes. We then discusses models for black holes, for cosmology, and for gravitational radiation in turn. In each case, we start by considering archetypal explicit solutions (Schwarzschild for black holes, Friedmann for cosmology).
    Pre-requisites: 600 differential geometry.

  • 607 Hyperplane arrangements, Yuzvinsky, Fall 2003.

    • We will discuss hyperplane arrangements arising from reflection groups. Such arrangements have special properties in the world of arbitrary arrangements of hyperplanes. We will discuss the following: Pre-requisites: 600 algebra.

  • 607 Lie algebras, Kleshchev, Winter 2004.

    • This will be a first course on the classification and structure of finite dimensional semisimple Lie Algebras. Note this course is a pre-requisite for the courses on Lie groups and on Vertex algebras in the Spring.
    Text: Introduction to Lie algebras and representation theory by J. E. Humphreys.
    Pre-requisites: 600 algebra.

  • 607 Lie groups, Landweber, Spring 2004.
    • Geometry and topology of Lie groups and homogeneous spaces.

    This course covers the Peter-Weyl and Borel-Weil-Bott theorems, as well as various advanced topics in the geometry and topology of compact Lie groups and homogeneous spaces. The Peter-Weyl theorem decomposes the space of L^2 functions on a compact Lie group G in terms of finite dimensional representations of G, thereby turning analysis into algebra. The Borel-Weil-Bott theorem constructs finite dimensional representations of G as spaces of holomorphic sections (or higher Dolbeault cohomology) on bundles over certain homogeneous spaces G/H. Advanced students can present classic papers by Bott and others during the last two weeks of the quarter.
    Pre-requisites: 607 Lie algebras. 691 K-theory would be helpful but is not required.

  • 607 Vertex algebras, Vaintrob, Spring 2004.

    • Vertex algebras were introduced by physicists in the late 1970s as an algebraic formalism of two-dimensional conformal field theories. Since then they have become an essential tool of contemporary mathematical physics and found important applications in many areas of traditional mathematics such as representation theory, algebraic geometry, combinatorics, number theory, group theory and topology.
    The goal of the course is to give an accessible algebraic introduction to the theory of vertex algebras and discuss some of its applications. Even though I will begin with explaining some physical motivation, no prior knowledge of physics will be necessary. This course might have a follow-up course next year on conformal field theory, where more geometric aspects of the picture will be discussed.
    Pre-requisites: 600 algebra. Familiarity with Lie algebras would be helpful though not essential.


    Advanced Graduate Courses 2002-2003 Back to top

    Here is the schedule for the graduate courses at the 600 level and above for 2002-2003.

    FALL 2002 WINTER 2003 SPRING 2003
    616 Real analysis 617 Real analysis 618 Real analysis
    634 Algebraic topology 635 Algebraic topology 636 Algebraic topology 
    637 Differential geometry 638 Differential geometry 639 Differential geometry
    646 Abstract algebra 647 Abstract algebra 648 Abstract algebra
    681 Rings and modules 682 Topics in representation theory 683 Topics in representation theory
    684 Functional analysis 685 Functional analysis 686 Functional analysis
    690 Homology and cohomology 691 Homotopy theory 692 WETSK
    607 Algebraic geometry 607 Algebraic geometry 619 Complex analysis
    . 607 Commutative algebra 607 Commutative algebra

  • 684/5/6 Functional Analysis, Phillips, Fall-Winter-Spring 2002-2003.

    • In this course I plan to present as much as I can of the analysis behind the Atiyah-Singer index theorem for families of elliptic pseudo-differential operators. Here is a plausible list of topics: I might possibly even get to the proof of the index theorem, though that is probably over optimistic. I would expect to develop in the course parts of the theory of manifolds that I need but which are not familiar. I will state without proof some of the results on K-theory, though the K-theory course is not a prerequisite.

  • 619 Complex Analysis, Phillips, Spring 2003.

    • This course assumes the complex analysis covered in Math 618, about half a quarter, including the Cauchy integral formula, power series and calculus of residues. Depending on the material covered, some other things from Math 616-617 might also be used. We will cover a number of additional topics, probably mostly chosen from either Conway's ``Functions of one complex variable'' or the second half of Rudin's ``Real and complex analysis''. Some fundamental topics would be the Riemann Mapping Theorem, the properties of the gamma function, Runge's Theorem on rational approximation, and the Prime Number Theorem. Additional topics, depending on the interest of the class, could include Picard's Theorem on the range of an entire function, something about Banach spaces of holomorphic functions, or any number of other things.

  • 690 Homology and Cohomology Theory, Sinha, Fall 2002.

    • We continue the 634/5/6 series with topics in homology and cohomology, focusing on manifolds. We start with duality theorems, including those of Poincare, Lefshetz and Alexander, and applications to bordism theory and knot theory. We then develop de Rham cohomology including the de Rham theorem, the generalized Stokes theorem, and applications.

  • 691 Homotopy Theory, Sadofsky, Winter 2002.

    • We'll cover classical (from the 50s, 60s and 70s) topics in homotopy theory. We'll begin with the Freudenthal suspension theorem and the Serre spectral sequence, then move on to Steenrod operations and applications of the Adams spectral sequence. No prior knowledge other than 600 topology and some cohomology (that is 690 would be helpful) will be assumed. Developing facility with spectral sequences will be a primary goal.

  • 692 What Every Topologist Should Know, Landweber, Spring 2003.

    • This is a seminar-style course where the students read various classic topology papers and give lectures on them. This is intended to enhance your knowledge of topology and fill gaps in your background. I am expecting that each student will read two papers and give a total of 2-4 lectures. The precise material covered depends on your own interests and the papers you choose. Possible topics include homotopy of spheres, spectral sequences, cobordism theory, generalized cohomologies, topology of Lie groups and homogeneous spaces, power operations in cohomology and K-theory, index theory and many more. In the spring I will prepare a bibliography of papers that you can choose from, The authors will include Adams, Atiyah, Bott, Chern, Dold, Hirzebruch, P. S. Landweber, Milnor, Ravenel and Serre.

  • 607 Algebraic Geometry, Vaintrob, Fall-Winter 2002-2003.

    • Algebraic geometry is one of the most highly developed and beautiful branches of mathematics with wide spectrum of connections and applications in other disciplines (both within and outside mathematics). Its ideas and methods play an important role in the development of various areas of mathematics, such as number theory, commutative and non-commutative ring theory, representation theory, complex analytic geometry, algebraic combinatorics, and mathematical physics. In essence, it is the study of solutions of polynomial equations in several variables. The algebraic part is in the polynomial nature of the equations, while the geometry lies in the curves, surfaces and higher dimensional objects the equations represent. This course will provide a basic introduction to algebraic geometry. The topics covered will include affine and projective varieties, Hilbert's Nullstellensatz, Zariski topology and regular functions, regular and rational maps of varieties, dimension, degree, blowing up, line bundles and divisors, the Riemann-Roch theorem for curves. We will spend a considerable amount of time looking at specific examples and applications.

  • 607 Commutative Algebra, Yuzvinsky, Winter/Spring 2002-2003.

    • This course is part of the second year graduate program in algebra. It will tentatively include the following topics:
  • 681 Rings and Modules, Anderson, Fall 2002.

    • This will be a second year course on the theory of rings and algebras and their representation theory. The regular 600 algebra sequence will be adequate preparation. Following a brief review of general ring and module theory, including semisimplicity and the radical, we will develop the Morita theory characterizing both equivalence and duality in module categories. Then we will study the Goldie theory and rings of quotients of non-commutative rings; this is the starting point of noncommutative localization theories and algebraic geometry. As an illustration of duality in action, we will include a brief treatment of quasi-Frobenius rings, ring theoretic generalizations of Frobenius algebras and group algebras. Finally, we will carry the Auslander-Reiten representation program as far as time permits.

  • 682/3 Topics in Representation Theory, Brundan, Winter/Spring 2002-2003.

    • I will cover some of the following topics:
    Mathematics Graduate Courses Back to top

    411/511, 412/512 Functions of a Complex Variable I,II (4,4) Complex numbers, linear fractional transformations, Cauchy-Riemann equations, Cauchy's theorem and applications, power series, residue theorem, harmonic functions, contour integration, conformal mapping, infinite products. Sequence. Prereq: MATH 281 or instructor's consent.

    413/513, 414/514, 415/515 Introduction to Analysis I,II,III (4,4,4) Differentiation and integration on the real line and in n-dimensional Euclidean space; normed linear spaces and metric spaces; vector field theory and differential forms. Sequence. Prereq: MATH 282, 315 or instructor's consent.

    420/520 Differential Equations and Fourier Analysis I (4) Ordinary differential equations. General and initial value problems. Explicit, numerical, graphical solutions; phase portraits. Existence, uniqueness, stability. Power series methods. Gradient flow; periodic solutions. Pre- or coreq: MATH 256.

    421/521 Differential Equations and Fourier Analysis II (4) Introduction to PDEs; wave and heat equations. Classical Fourier series on the circle; application of Fourier series. Generalized Fourier series, Bessel and Legendre series. Prereq: MATH 420/520.

    422/522 Differential Equations and Fourier Analysis III (4) General theory of PDEs; the Fourier transform. Laplace and Poisson equations; Green's functions and application. Mean value theorem and max-min principle. Prereq: MATH 421/521.

    431/531, 432/532 Introduction to Topology (4,4) Elementary point-set topology with an introduction to combinatorial topology and homotopy. Sequence. Prereq: upper-division mathematics sequence or instructor's consent.

    433/533 Introduction to Differential Geometry (4) Plane and space curves, Frenet-Serret formula surfaces. Local differential geometry, Gauss-Bonnet formula, introduction to manifolds. Prereq: MATH 281, 341.

    441/541 Linear Algebra (4) Theory of vector spaces over arbitrary fields, theory of a single linear transformation, minimal polynomials, Jordan and rational canonical forms, quadratic forms, quotient spaces. Prereq: MATH 342.

    444/544, 445/545, 446/546 Introduction to Abstract Algebra I,II,III (4,4,4) Theory of groups, rings, and fields. Polynomial rings, unique factorization, and Galois theory. Prereq: MATH 342.

    451/551, 452/552, 453/553 Introduction to Numerical Analysis I,II,III (4,4,4) Methods of numerical analysis with applications. Elementary theory of numerical solutions of differential equations, splines, and fast Fourier transform. Prereq: CIS 210; pre- or coreq: MATH 282.

    455/555 Mathematical Modeling (4) Introduction to discrete and continuous models for various problems arising in the application of mathematics to other disciplines, e.g., biological and social sciences. Prereq: MATH 341. MATH 256 recommended.

    456/556 Networks and Combinatorics (4) Fundamentals of modern combinatorics; graph theory; networks; trees; enumeration, generating functions, recursion, inclusion and exclusion; ordered sets, lattices, Boolean algebras. Prereq: MATH 231 or 346.

    457/557 Discrete Dynamical Systems (4) Linear and nonlinear first-order dynamical systems; equilibrium, cobwebs, Newton's method. Bifurcation and chaos. Introduction to higher-order systems. Applications to economics, genetics, ecology. Prereq: MATH 256 or instructor's consent.

    461/561, 462/562 Introduction to Mathematical Methods of Statistics I,II (4,4) Discrete and continuous probability models; useful distributions; applications of moment-generating functions; sample theory with applications to tests of hypotheses, point and confidence interval estimates. Sequence. Prereq: MATH 252.

    463/563 Mathematical Methods of Regression Analysis and Analysis of Variance (4) Multinomial distribution and chi-square tests of fit, simple and multiple linear regression, analysis of variance and covariance, methods of model selection and evaluation, use of statistical software. Prereq: MATH 462/562.

    464/564, 465/565, 466/566 Mathematical Statistics I,II,III (4,4,4) Random variables; generating functions and characteristic functions; weak law of large numbers and central limit theorem; point and interval estimation; Neyman-Pearson theory and likelihood tests; sufficiency and exponential families; linear regression and analysis of variance. Sequence. Pre- or coreq: MATH 282, 341, 342.

    503 Thesis (1-12R)

    601 Research: [Topic] (1-9R)

    602 Supervised College Teaching (1-16R)

    603 Dissertation (1-16R)

    605 Reading and Conference: [Topic] (1-5R)

    607 Seminar: [Topic] (1-5R) Topics include Classical Groups, Fields, Functional Analysis, Graded Commutative Rings, Lie Groups, Low-Dimensional Topology, Noncommutative Rings, Nonlinear Approximation Theory.

    616, 617, 618 Real Analysis (4-5,4-5,4-5) Measure and integration theory, differentiation, and functional analysis with point-set topology as needed. Sequence.

    619 Complex Analysis (4-5) The theory of Cauchy, power series, contour integration, entire functions, and related topics.

    634, 635, 636 Algebraic Topology (4-5,4-5,4-5) Foundations of homotopy theory, CW-complexes, homology, cohomology theory. Sequence.

    637, 638, 639 Differential Geometry (4-5,4-5, 4-5) Topics include curvature and torsion, Serret-Frenet formulas, theory of surfaces, differentiable manifolds, tensors, forms and integration. Sequence.

    647, 648, 649 Abstract Algebra (4-5,4-5,4-5) Group theory, fields, Galois theory, algebraic numbers, matrices, rings, algebras. Sequence.

    671, 672, 673 Theory of Probability (4-5,4-5, 4-5) Measure and integration, probability spaces, laws of large numbers, central-limit theory, conditioning, martingales, random walks. Sequence.

    681, 682, 683 Advanced Topics in Algebra: [Topic] (4-5,4-5,4-5R) Topics selected from theory of finite groups, representations of finite groups, Lie groups, Lie algebras, algebraic groups, ring theory, algebraic number theory.

    684, 685, 686 Advanced Topics in Analysis: [Topic] (4-5,4-5,4-5R) Topics selected from Banach algebras, operator theory, functional analysis, harmonic analysis on topological groups, theory of distributions.

    690, 691, 692 Advanced Topics in Geometry and Topology: [Topic] (4-5,4-5,4-5R) Topics selected from classical and local differential geometry; symmetric spaces; low-dimensional topology; differential topology; global analysis; homology, cohomology, and homotopy; differential analysis and singularity theory; knot theory.