Applied mathematics exam
The applied mathematics qualifying exam covers topics in dynamical systems, partial differential equations, and applied linear algebra.Material
Linear algebra- Undergraduate linear algebra
- Spaces: Normed spaces, inner product spaces, dual spaces, fundamental subspaces, rank-nullity theorem, direct sums
- Matrices and operators: linear, bounded, self-adjoint, unitary, normal, positive-definite, and compact operators; projections, adjoints, matrix exponential
- Spectra: Eigenvalues, eigenvectors, Rayleigh quotient, eigendecomposition, singular value decomposition, Schur decomposition, generalized eigenvectors
- Orthogonality: orthogonal projections and reflections, orthogonal complement, orthonormality, Gram-Schmidt orthogonalization, orthogonality of range and null space of operators and their adjoints
- Hilbert spaces: completeness, Riesz representation theorem, Parseval’s identity
- ODE theory: Local existence and uniqueness of solutions to initial-value problems, continuous dependence on initial conditions, continuous dependence on parameters, Sturm-Liouville theory for one-dimensional boundary-value problems
- Dynamical systems: continuous and discrete systems, equilibrium states and their stability, flows, orbits, limit sets, invariant sets; linear systems, stable, unstable, center spaces; nonlinear systems, linearization, topological equivalence, center manifold theory, nullclines, Lyapunov function, limit cycles, Poincaré–Bendixson theory, stability of periodic solutions, Poincaré map.
- Bifurcation theory: structural stability, local bifurcations of equilibrium states, genericity, codimension, center manifold theory, global bifurcations (homoclinic, heteroclinic), bifurcations in 1-dimensional maps.
- PDE theory: classification of PDEs, uniqueness of solutions, energy estimates
- Initial-value problems: First-order linear and quasilinear PDEs, method of characteristics, heat equation, wave equation, separation of variables, series solutions, Fourier transform
- Boundary-value problems: Laplace equation, Poisson equation, Sturm–Liouville theory, Green’s functions, fundamental solutions, Dirichlet and Neumann boundary conditions, variational formulations of boundary-value problems.
Textbooks. The following textbooks are recommended:
- Friedberg, Insel, Linear algebra
- Roman, Advanced Linear algebra
- Cheney, Analysis for applied mathematics
- Perko, Differential equations and dynamical systems
- Strogatz, Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering
- Hirsch, Smale, and Devaney, Differential equations, dynamical systems, and an introduction to chaos
- Teschle, Ordinary differential equations and dynamical systems
- Bleecker, Csordas, Basic partial differential equations
- Evans, Partial differential equations
- Haberman, Applied partial differential equations with Fourier series and boundary value problems
- Olver, Introduction to partial differential equations