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Math

Courses and Seminars

Courses and Seminars

SEMINARS

Part of becoming an independent mathematician is becoming exposed to a broad range of mathematical research as well as studying some specific areas in greater depth. It is expected that students attend colloquia and a seminar of their choice. Students receiving tuition awards are expected to attend the graduate seminar or, with the approval of their advisor, a research seminar at least 11 weeks per semester. Students are expected to attend at least five department colloquia a semester.

The graduate seminar is a weekly seminar organized by the Graduate Student Advisory Committee. Its purpose is to acquaint students with the faculty and their research, prepare students for the department colloquia, and to permit students an opportunity to share their research with other students. As students advance in the program, it is generally expected that they will attend a regular research seminar instead of or in addition to the graduate seminars.

The research seminars usually include the following:

    • Algebra
    • Algebraic Geometry
    • Dynamical Systems
    • Mathematical Biology
    • Mathematical Physics
    • Number theory
    • Partial Differential Equations
    • Stochastic Differential Equations
    • Topology

Other seminars also run from time to time. Please check on the math department homepage for more information.

COURSES

We have 20 core courses that are regularly scheduled and that are mainly for first year students. The remaining courses (about 10) are determined according to the needs of the current graduate students.

CORE COURSES

Twenty courses will be taught every year:

Math 510, 511 Numerical Analysis

Math 532 Complex Analysis

Math 540, 541 Real Analysis

Math 543,544 Probability

Math 553, 554 Topology

Math 570 Matrix Analysis

Math 571, 572 Algebra

Non-core courses

Math 586, 587 Number Theory

Math 634, 635 ODEs/Dynamical Systems

Math 640, 641 Analysis

Math 655, 656 Algebraic/Differential Topology

Non-core courses

Other courses are offered less regularly. Consult the course schedule to check availability.

All students should submit their program of study by the end of November of their first year. Students must also submit an additional list to the secretary, giving a schedule of when they plan to take the courses in their programs of study. Students should consult with their advisors to select the most beneficial courses, and to decide when they should take them. The graduate committee uses these lists to determine which classes will be taught the next year, and to ensure that students’ coursework needs are met.

DESCRIPTIONS OF APPROVED COURSES

510. Numerical Methods for Linear Algebra. (3)Prerequisite(s): Math 410 or equivalent.Numerical matrix algebra, orthogonalization and least squares methods, unsymmetric and symmetric eigenvalue problems, iterative methods, advanced solvers for partial differential equations.

511. Numerical Methods for Partial Differential Equations. (3)Prerequisite(s): Math 303 or 447; 410; or equivalents.Finite difference and finite volume methods for partial differential equations. Stability, consistency, and convergence theory.

513R. Advanced Topics in Applied Mathematics. (3)Prerequisite(s): Instructor’s consent.

521. Methods of Applied Mathematics 1. (3)Prerequisite(s): Math 334 or equivalent.Possible topics include variational, integral, and partial differential equations; spectral and transform methods; nonlinear waves; Green’s functions; scaling and asymptotic analysis; perturbation theory; continuum mechanics.

522. Methods of Applied Mathematics 2. (3)Prerequisite(s): Math 521 or equivalent.Possible topics include variational, integral, and partial differential equations; spectral and transform methods; nonlinear waves; Green’s functions; scaling and asymptotic analysis; perturbation theory; continuum mechanics.

532. Complex Analysis. (3)Prerequisite(s): Math 352 or instructor’s consent.Introduction to theory of complex analysis at beginning graduate level. Topics: Cauchy integral equations, Riemann surfaces, Picard’s theorem, etc.

534. Introduction to Dynamical Systems 1. (3)Prerequisite(s): Math 334, 341; or equivalents.Discrete dynamical systems; iterations of maps on the line and the plane; bifurcation theory; chaos, Julia sets, and fractals. Computational experimentation.

540. Linear Analysis. (3)Normed vector spaces and linear maps between them.

541. Real Analysis. (3)Prerequisite(s): Math 341; 314 or 342; or equivalents.Rigorous treatment of differentiation and integration theory; Lebesque measure; Banach spaces.

543. Advanced Probability 1. (3)Prerequisite(s): Math 314 or equivalent.Foundations of the modern theory of probability with applications.Probability spaces, random variables, independence, conditioning, expectation, generating functions, and Markov chains.

544. Advanced Probability 2. (3)Prerequisite(s): Math 543.Advanced concepts in modern probability. Convergence theorems and laws of large numbers. Stationary processes and ergodic theorems. Martingales. Diffusion processes and stochastic integration.

547. Partial Differential Equations 1. (3)Prerequisite(s): Math 334, 342; or equivalents.Methods of analysis for hyperbolic, elliptic, and parabolic equations, including characteristic manifolds, distributions, Green’s functions, maximum principles and Fourier analysis.

548. Partial Differential Equations 2. (3)Prerequisite(s): Math 547.Tools for PDEs and special topics: spherical means, method of descent, subharmonic functions, Hamilton-Jacobi equations, Riemann invariants, conservation laws for linear and nonlinear waves.

553. Foundations of Topology 1. (3)Prerequisite(s): Math 451 or instructor’s consent.Naive set theory, topological spaces, product spaces, subspaces, continuous functions, connectedness, compactness, countability, separation axioms, metrization, complete metric spaces, function spaces, and Baire spaces.

554. Foundations of Topology 2. (3)Prerequisite(s): Math 553 or instructor’s consent.Fundamental group, retractions and fixed points, homotopy types, separation theorems, classification of surfaces, Seifert-van Kampen Theorem, classification of covering spaces, and applications to group theory.

561. Introduction to Algebraic Geometry 1. (3)Prerequisite(s): Math 571 or concurrent enrollment.Basic definitions and theorems on affine, projective, and quasi-projective varieties.

562. Introduction to Algebraic Geometry 2. (3)Prerequisite(s): Math 561.Local properties of quasi-projective varieties. Divisors and differential forms.

565. Differential Geometry. (3)Prerequisite(s): Math 342 or equivalent.A rigorous treatment of the theory of differential geometry.

570. Matrix Analysis. (3)Prerequisite(s): Math 302 or 313 or equivalent.Special classes of matrices, canonical forms, matrix and vector norms, localization of eigenvalues, matrix functions, applications.

571. Algebra 1. (3)Prerequisite(s): Math 372 or equivalent.

572. Algebra 2. (3)Prerequisite(s): Math 571.

586. Introduction to Algebraic Number Theory. (3)Prerequisite(s): Math 372 or equivalent.Algebraic integers; different and discriminant; decomposition of primes; class group; Dirichlet unit theorem; Dedekind zeta function; cyclotomic fields; valuations; completions.

587. Introduction to Analytic Number Theory. (3)Prerequisite(s): Math 352 or equivalent.Arithmetical functions; distribution of primes; Dirichlet characters; Dirichlet’s theorem; Gauss sums; primitive roots; Dirichlet L-functions; Riemann zeta-function; prime number theorem; partitions.

621. Matrix Theory 1. (3)Prerequisite(s): Math 570.Symmetric matrices, spectral graph theory, interlacing, the Laplacian matrix of a graph.

622. Matrix Theory 2. (3)Prerequisite(s): Math 621.Research topics in combinatorial matrix theory.

634. Theory of Ordinary Differential Equations. (3)Prerequisite(s): Math 334, 341; or equivalents.

635. Dynamical Systems. (3)Prerequisite(s): Math 634.

640. Nonlinear Analysis. (3)Differential calculus in normed spaces, fixed point theory, and abstract critical point theory.

641. Functions of a Real Variable. (3)Prerequisite(s): Math 541 or instructor’s consent.Abstract measure and integration theory; L(p) spaces; measures on topological and Euclidean spaces.

643R. Special Topics in Analysis. (3)Prerequisite(s): Math 641 or instructor’s consent.Advanced topics in analysis drawn from pure and applied mathematics.

644. Harmonic Analysis. (3)Prerequisite(s): Math 532, 541.Harmonic analysis on the torus and in Euclidean space; pointwise and norm convergence of Fourier series and functional-analytic aspects of Fourier transforms emphasized.

647. Theory of Partial Differential Equations 1. (3)Prerequisite(s): Math 541, 547.

648. Theory of Partial Differential Equations 2. (3)Prerequisite(s): Math 647.

651. Topology 1. (3)Prerequisite(s): Math 553, 554.Advanced topics in topology. Topics may include, but are not limited to, piecewise linear topology, 3-manifold theory, homotopy theory, differential topology, Riemannian geometry, and geometric group theory.

652. Topology 2. (3)Prerequisite(s): Math 651.Advanced topics in topology. Topics may include, but are not limited to, piecewise linear topology, 3-manifold theory, homotopy theory, differential topology, Riemannian geometry, and geometric group theory.

655. Differential Topology. (3)Prerequisite(s): Math 342 or equivalent; Math 554 or equivalent.Topological and smooth manifolds, tangent vectors, vector bundles, cotangent bundles, submersions, immersion, and embeddings of submanifolds, transversality, embedding and approximation theorems, differential forms, wedge products, exterior derivative, orientation, Stokes’ theorem and integration on manifolds.

656. Algebraic Topology. (3)Prerequisite(s): Math 655.Fundamental group and homotopy, Von Kampen theorem, covering spaces, group actions, higher homotopy; simplicial, singular, and cellular homology, homology with coefficients, exact sequences, excision, Mayer-Vietoria; cohomology, universal coefficients, cup product, Poincare duality.

663. Algebraic Geometry 1. (3)Prerequisite(s): Math 676 or concurrent enrollment.Basic definitions and theorems on varieties, sheaves, and schemes.

664. Algebraic Geometry 2. (3)Prerequisite(s): Math 663.Cohomology of schemes. Classification problems. Applications.

673. Algebra 3. (3)Prerequisite(s): Math 572 or equivalent.

675R. Special Topics in Algebra. (3)Prerequisite(s): Math 572.

676. Commutative Algebra. (3)Prerequisite(s): Math 572.Commutative rings, modules, tensor products, localization, primary decomposition, Noetherian and Artinian rings, application to algebraic geometry and algebraic number theory.

677. Homological Algebra. (3)Prerequisite(s): Math 572.Chain complexes, derived functors, cohomology of groups, ext and tor, spectral sequences, etc. Application to algebraic geometry and algebraic number theory.

686R. Topics in Algebraic Number Theory. (3)Prerequisite(s): Math 372, 487 and instructor’s consent.Current topics of research interest.

687R. Topics in Analytic Number Theory. (3)Prerequisite(s): Math 352, 487; or equivalents.Current topics of research interest.

691R. Graduate Mathematics Colloquium. (1)Prerequisite(s): Math 371 or equivalent.A diverse set of talks at the graduate level. Students will broaden their knowledge of recent and current research in mathematics. Speakers will be faculty, visitors, and students reporting on thesis work.

695R. Readings in Mathematics. (1-2)

698R. Master’s Project. (2)

699R. Master’s Thesis. (1-9)

751R. Advanced Special Topics in Topology. (3)Prerequisite(s): Math 652.Current topics in topology of research interest

799R. Doctoral Dissertation. (1-9)

GRADUATE FACULTY AND AREAS OF INTEREST

Algebra and Algebraic Geometry

  • Stephen Humphries: Group Theory, Low-dimensional Topology
  • Tyler Jarvis: Algebraic Curves, Moduli Spaces
  • Pace Nielsen: Ring Theory, Module Theory, and Number Theory
  • Nathan Priddis: Algebraic Geometry

    Combinatorics

    • Rodney W. Forcade: Combinatorics, Number Theory, and Graph Theory

    Differential Equations and Dynamical Systems

    • Mark Allen: Partial Differential Equations
    • Lennard Bakker: Dynamical Systems and Celestial Mechanics
    • Todd Fisher: Dynamical Systems
    • Christopher P. Grant: Nonlinear Partial Differential Equations, Dynamical Systems
    • Kenneth Kuttler: Abstract Methods for Nonlinear Partial Differential Equations and Inclusion
    • Kening Lu: Nonlinear Partial Differential Equations, Dynamical Systems, and Applied Mathematics
    • Tiancheng Ouyang: Nonlinear Partial Differential Equations and Celestial Mechanics
    • Benjamin Webb: Dynamical Systems and Applied Mathematics

      Minimal Surfaces

      • Michael Dorff: Minimal Surfaces, Geometric Function Theory
      • Gary Lawlor: Minimal Surfaces

        Number Theory

        • Roger Baker: Number Theory
        • David A. Cardon: Analytic Number Theory, Algebraic Number Theory, Automorphic Forms
        • Jasbir S. Chahal: Number Theory
        • Darrin Doud: Number Theory
        • Paul Jenkins: Number Theory, Modular Forms, Partitions
        • Xian-Jin Li: Analytic Number Theory

          Numerical Analysis and Applied Mathematics

          • Blake Barker: Applied Mathematics, Partial Differential Equations
          • Shue-Sum Chow: Numerical Analysis
          • John Dallon: Mathematical Biology
          • Emily Evans: Applied Mathematics
          • Scott Glasgow: Optics both Classical and Quantum
          • Jeff Humpherys: Applied Mathematics, Nonlinear Partial Differential Equations, and Dynamical Systems
          • Robin Roundy: Operations Research
          • Vianey Villamizar: Numerical Solution of Partial Differential Equations, Wave Scattering, and Asymptotic Methods
          • Jared Whitehead: Applied Mathematics, Fluid Dynamics

          Topology and Geometric Group Theory

          • Gregory Conner: Geometric Group Theory, Topology, and Combinatorial Group Theory
          • Denise Halverson: Geometric Topology
          • Curtis Kent: Geometric Group Theory
          • Eric Swenson: Geometric Group Theory, Topology