Graduate Math Courses Offered Every Year

• Primarily for first-year students:
  • Fall semester
    • Math 216: Applied Stochastic Processes
    • Math 224: Scientific Computing I
    • Math 231: Ordinary Differential Equations
    • Math 241: Real Analysis I
    • Math 251: Groups, Rings and Fields
  • Spring semester
    • Math 219: Introduction to Stochastic Calculus or Math 287: Probability Theory
    • Math 225: Scientific Computing II
    • Math 232: Partial Differential Equations I
    • Math 245: Complex Analysis
    • Math 252: Commutative Algebra
    • Math 261: Algebraic Topology I
    • Math 267: Differential Geometry
• Primarily for second-year students:
  • Fall semester
    • Math 226: Numerical Partial Differential Equations I
    • Math 233: Asymptotic and Perturbation Methods
    • Math 242: Real Analysis II (Functional Analysis)
    • Math 253: Representation Theory
    • Math 262: Algebraic Topology II
    • Math 272: Riemann Surfaces
    • Math 282: Elliptic Partial Differential Equations
    • Math 288: Topics in Probability
  • Spring semester
    • Math 228: Mathematical Fluid Dynamics
    • Math 229: Mathematical Modeling
    • Math 236: General Relativity
    • Math 268: Topics in Differential Geometry
    • Math 273: Algebraic Geometry or Math 274: Number Theory
    • Math 281: Hyperbolic Partial Differential Equations

Course Descriptions

• MATH 216 - APPLIED STOCHASTIC PROCESSES
  An introduction to stochastic processes without measure theory. Topics selected from: Markov chains in discrete and continuous time, queuing theory, branching processes, martingales, Brownian motion, stochastic calculus.
• MATH 219 - INTRODUCTION TO STOCHASTIC CALCULUS
  An introduction to the theory of stochastic differential equations with an eye towards those topics useful in applications. Brownian motion, stochastic integrals, and diffusions as solutions of stochastic differential equations. Functionals of diffusions and their connection with partial differential equations. Ito's formula, Girsanov's theorem, Feynman-Kac formula, Martingale representation theorem. Additional topics have included one dimensional boundary behavior, stochastic averaging, stochastic numerical methods.
• MATH 224 - SCIENTIFIC COMPUTING I
  Structured scientific programming in C/C++ and FORTRAN. Floating point arithmetic and interactive graphics for data visualization. Numerical linear algebra, direct and iterative methods for solving linear systems, matrix factorizations, least squares problems and eigenvalue problems. Iterative methods for nonlinear equations and nonlinear systems, Newton's method.
• MATH 225 - SCIENTIFIC COMPUTING II
  Approximation theory: Fourier series, orthogonal polynomials, interpolating polynomials and splines. Numerical differentiation and integration. Numerical methods for ordinary differential equations: finite difference methods for initial and boundary value problems, and stability analysis. Introduction to finite element methods.
• MATH 226 - NUMERICAL PARTIAL DIFF EQUATIONS I
  Numerical solution of hyperbolic conservation laws. Conservative difference schemes, modified equation analysis and Fourier analysis, Lax-Wendroff process. Gas dynamics and Riemann problems. Upwind schemes for hyperbolic systems. Nonlinear stability, monotonicity and entropy; TVD, MUSCL, and ENO schemes for scalar laws. Approximate Riemann solvers and schemes for hyperbolic systems. Multidimensional schemes. Adaptive mesh refinement.
• MATH 227 - NUMERICAL PARTIAL DIFFERENTIAL EQUATIONS II
  Numerical solution of parabolic and elliptic equations. Diffusion equations and stiffness, finite difference methods and operator splitting (ADI). Convection-diffusion equations. Finite element methods for elliptic equations. Conforming elements, nodal basis functions, finite element matrix assembly and numerical quadrature. Iterative linear algebra; conjugate gradients, Gauss-Seidel, incomplete factorizations and multigrid. Mixed and hybrid methods. Mortar elements. Reaction-diffusion problems, localized phenomena, and adaptive mesh refinement.
• MATH 228 - MATHEMATICAL FLUID DYNAMICS
  Properties and solutions of the Euler and Navier-Stokes equations, including particle trajectories, vorticity, conserved quantities, shear, deformation and rotation in two and three dimensions, the Biot-Savart law, and singular integrals. Additional topics determined by the instructor.
• MATH 229 - MATHEMATICAL MODELING
  Formulation and analysis of mathematical models in science and engineering. Emphasis on case studies; may include individual or team research projects.
• MATH 231 - ORDINARY DIFFERENTIAL EQUATIONS
  Existence and uniqueness theorems for nonlinear systems, well-posedness, two-point boundary value problems, phase plane diagrams, stability, dynamical systems, and strange attractors.
• MATH 232 - INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS
  Fundamental solutions of linear partial differential equations, hyperbolic equations, characteristics, Cauchy-Kowalevski theorem, propagation of singularities.
• MATH 233 - ASYMPTOTICS AND PERTURBATION METHODS
  Asymptotic solution of linear and nonlinear ordinary and partial differential equations. Asymptotic evaluation of integrals. Singular perturbation. Boundary layer theory. Multiple scale analysis.
• MATH 241 - REAL ANALYSIS
  Measures; Lebesgue integral; L^p spaces; Daniell integral, differentiation theory, product measures.
• MATH 242 - FUNCTIONAL ANALYSIS
  Metric spaces, fixed point theorems, Baire category theorem, Banach spaces, fundamental theorems of functional analysis, Fourier transform.
• MATH 245 - COMPLEX ANALYSIS
  Complex calculus, conformal mapping, Riemann mapping theorem, Riemann surfaces.
• MATH 251 - GROUPS RINGS & FIELDS
  Groups including nilpotent and solvable groups, p-groups and Sylow theorems; rings and modules including classification of modules over a PID and applications to linear algebra; fields including extensions and Galois theory.
• MATH 252 - COMMUTATIVE ALGEBRA
  Extension and contraction of ideals, modules of fractions, primary decomposition, integral dependence, chain conditions, affine algebraic varieties, Dedekind domains, completions.
• MATH 253 - REPRESENTATION THEORY
  Representation theory of finite groups, Lie algebras and Lie groups, roots, weights, Dynkin diagrams, classification of semisimple Lie algebras and their representations, exceptional groups, examples and applications to geometry and mathematical physics.
• MATH 261 - ALGEBRAIC TOPOLOGY I
  Fundamental group and covering spaces, singular and cellular homology, Eilenberg-Steenrod axioms of homology, Euler characteristic, classification of surfaces, singular and cellular cohomology.
• MATH 262 - ALGEBRAIC TOPOLOGY II
  Universal coefficient theorems, Kunneth theorem, cup and cap products, Poincare duality, plus topics selected from: higher homotopy groups, obstruction theory, Hurewicz and Whitehead theorems, and characteristic classes.
• MATH 267 - DIFFERENTIAL GEOMETRY
  Differentiable manifolds, fiber bundles, connections, curvature, characteristic classes, Riemannian geometry including submanifolds and variations of length integral, complex manifolds, homogeneous spaces.
• MATH 272 - RIEMANN SURFACES
  Compact Riemann Surfaces, maps to projective space, Riemann-Roch Theorem, Serre duality, Hurwitz formula, Hodge theory in dimension one, Jacobians, the Abel-Jacobi map, sheaves, Cech cohomology.
• MATH 273 - ALGEBRAIC GEOMETRY
  Affine varieties, projective varieties, Riemann surfaces, algebraic curves, algebraic groups, sheaf cohomology, singularities, Hodge theory, or computational algebraic geometry.
• MATH 281 - HYPERBOLIC PDE
  Linear wave motion, dispersion, stationary phase, foundations of continuum mechanics, characteristics, linear hyperbolic systems, and nonlinear conservation laws.
• MATH 282 - ELLIPTIC PDE
  Fourier transforms, distributions, elliptic equations, singular integrals, layer potentials, Sobolev spaces, regularity of elliptic boundary value problems.
• MATH 283 - TOPICS IN PARTIAL DIFFERENTIAL EQUATIONS
  Hyperbolic conservation laws, pseudo-differential operators, variational inequalities, theoretical continuum mechanics.
• MATH 287 - PROBABILITY THEORY
  Measure theoretic probability. Triangular arrays, weak laws of large numbers, variants of the Central Limit Theorem, rates of convergence of limit theorems, local limit theorems, stable laws, infinitely divisible distributions, general state space Markov chains, Ergodic theorems, Large deviations, Martingales, Brownian Motion and Donsker's Theorem.
• MATH 288 - TOPICS IN PROBABILITY
  Probability tools and theory, geared towards topics of current research interest. See synopsis for current description.

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